45
N/kg ~n ~ p n = d ~ F dA - ~ p n = -p~n, P = -p 1 0 0 0 1 0 0 0 1 ≡-pE ~ p n P ~ F = - Z A p~n dA ~ F ~ P

Fluidi handout MF

  • Upload
    nikola

  • View
    243

  • Download
    0

Embed Size (px)

DESCRIPTION

Book

Citation preview

� � ��������� � � �����

����� ���������� �"! � #%$'&)(+*,(+-.$0/2143'(+5 � 6'187:9)$'�';<(>=��?9)1�6'#4@A1:;B1�9)$C@A$D5)(E7:9)$'�';F/2#+GH1I/2#>! #C;B1�9)� J+�K5)()L• M%N?OQP2R,S:TQUQVXWYRZV+T\[2R^] ;B149)$C@A$_5)(F=�*%(X6 �`7:9)$'�';B5 �a;B149)�)b`$'5 $'Jc(X�a! #%=I/d(+J+�B(+5 1feI(+!)� 1I/2� 5)10gX1IeD5 1I6,#%h;B� � 1I6 J+5 #%hF7�� e��)& 6'#Xhi6'#X5 Jj(X6 Jj(lkm=4�?9<(_e�1I/Qn�� 5 1oJ+14GH1Xpq� 5 1I� r �j@s(+9)5)(a=4� 9B( pq149B1�6 J+�)#%/d(+hI5 1�JX5)(a=4�?9<(%tvuw *X1�=4�?9B1x=418#Xg%� & 5 #H� eI�)(jG>(+*%(4@A$E! #:@A1v;B�)5 � r �F/d(%=41Ey

N/kg z u• OQ{B|<P\}_TQUQVXWYR�VXT~[dR:] ;B149)$C@A$85)(�#C;BhI#X*%(X�)(v@A$'b 1�! #X*%� ���x�H#%=�J+*%(X� $C@A$8=41�;B� � 1I6 J+5 �)/^6'#X5 Jj(X6 J+#%/�k�;B1s@s=4J+*%#5 1�6'1:& *X�B=4J+1�! #%*X� ���05)(H7:9)$'�';\p?�?9B�D;B1s@s=4J+*%#E7:9)$'�';<(�5)(�g%�?9)#H6'#4@A$EeI(%/2� �xn�1�5)$D! #X*%� �Ht4u���(�#%*X1�=4�?9B1=41E/2#X�B(f;B1I7�� 5)�)=�(XJ+�^�_#C;BhI#X*,(+�)(4@A$,bB(D! #%*X� ��5)(K6'#4@A$i;B149)$C@A$�u0��� ��! � #%$'&)(+*,(+-.$F! #X*%� ���)5)=46 � 3^=4�?9<(;B1I7��)5 � ��1>=41H! #�@s(X/ |�R,W\� {BPQN �?9)� � R,U\M,{)P2N 5B(+! #%5)( ] ! #%*X� ��� 5)=46?(f=4�?9B(f$i5 1I6,#�@�Jj(+& 6)��=�1H=�*X#C;B�F5B(#C;Bh�#%*%(X�)(v@A$'b $>� 5)7��)5 � J+1Ie��)/d(+9)5 $0!)#X*X�)��6'#�@s(�@A18#C;<� 1��'1�5B(�=�*X#4@A�B/�*%1�6 J+#%� #%/�5 #X�B/d(j9)1

~nu

~pn =d~F

dA− *X1�6)J+#X�"5)(+!)#X5)( kA�%u��ct

�8(oJj(4@H5)(X& � 5�=41%p�$`5)1�6'#%/�#X! ��J+1I/�=v9)$'&)(4@A$_;B#%gX�j@s(_7�� eI� & 6?(�*X149B� & � 5)(�eC(o6'#4@A$F@A1i! #XJ+�)1�gX5)#o;<1�*X1IJ=46?(j9<(+� 5 � 3D*X149B� & � 5)(�;<(Hg%�fg%�?9B(H$D! #%J+! $'5 #%=�J+�K#C;B� 1I�,1I5)(�$D5 1�6'#�@.Jj(+&)6 �f$f7:9)$'�';B$�u���#%/2#%b $EJ+1�5)e�#X�B(5)(+!)#X5)(�=�1�5)(�*%�?9)#�1I7�1I6 Jj(+585)(X& � 5x/2#XG�1�#%! �)=�(XJ+�8! � 1I5 #X��1�-:12;B1s@s=4J+*,(�! #X*%� ���)5)=46 � 3:=4� 9B(�6 � #%e�7:9)$'�';\u

�l#X6 *%� � $x=�Jc(XJ+� 6'1q7:9)$'�';<(�! � #X$'&)(X*%(�=41q(+!)=�#+9)$'J+5 #����)149B(XJ+� *X5)#:/2� �)#X*%(X-�1q7:9)$'�';<()u\�8(X6,#�b 1q��5)(4@s/d(+-:�Jj(+5)h�1�5)r �j@s(j9B5 ��5)(+! #%5�;B#X*%1I=4JX��;B#Z6)� 1�Jj(+-�(Z7:9)$'�';<()p�! � ��/2� � #%*%(+-.$�7:9B$,�?;<(lJc(X5 h�1I5 r �j@�9<(+5 ��5)(X! #X5)��$7:9)$'�';<$�/2#X�B(v@A$�g%� J+��@A14;B5)(+6)��5 $C9)�Ypx��5)(X! #X5B=46'#Z=4Jj(+-:1a$�-:1I/2$�=41`=�*X#C;B��=�(%/2#Z5)(�5 #%�)/d(j9)5)1a5)(+! #%5 1Xu�8(X6,#H7:9)$'�';f5 1�/2#+GH1�;�(��+JX� ! �)�X5 #X�)/d(+9)5 1:5)(+!)#X5 1��)=4J+1IeI(X-:("k�g%� b 1:5)(+� $%��1I5)(8! � 1IJ+! #,=4Jj(+*%6'(�#�7:9)$'�';<$E6'(X#6'#X5)J+� 5 $'$?/2$ tvpQ! � �?9)�)6,#,/�/2�)� #X*,(+-�("7:9B$,�?;<("! #%=�Jc(4@s(+b 1E=I(X/2#K5 #X�B/d(j9)5 �_5)(X! #X5)��!)� � J+�)=�6?( ] Jv@4uD5)(X! #X5B=46'#=�Jc(X-�18$f7:9)$'�';B$f=�18/2#+G�1�#%! �)=�(XJ+�H@A1v;B5)#%=4Jj(X*X5 #,/`� 1�9B(+r �+@A#%/qL

~pn = −p~n, Jv@4uP = −p

1 0 0

0 1 0

0 0 1

≡ −pE ks�Xu��Xt

h�;B1q@A1�=�(~pn

#XeI5)(+&)1�5K*X1I6 J+#X�K5)(X! #X5B( p =�(PJ+1I5 e�#%�i5)(+!)#X5)()u2 �1�Jj(jn�5)(E(X5)(j9)�)eI(H5)(+!)#X5)=�6,#%h8=4Jj(+-�(>$f7:9)$'�';<$<p

6?(+#K�^;B1C7�� 5 � r �+@A1D*X1I6 J+#X�B(f�_J+1I5 e�#%�)(i=4$F;<(+JX��5B(K! � 1v;�(+*%(X-��B/d( uH �(+6 9)1Xp\$o=4Jj(+JX� r �`7:9)$'�';<(K=��?9)$o6'#�@A#%/7:9)$'�';K;B1�9)$C@A185)(H5 1�6)$D! #X*%� �0p?�B/d(v@A$'b �K$0*%�';B$oks�Xu¡�Ct.�_kA�Xu��Xt4p #C;B� 1I�'$I@A1C/2#>! #%/2#%b $D� eI�)(+eC( L

~F = −∫

A

p~ndAkA�Xu ¢%t

£ �?9B$ ~Fb)1I/2#E& 1C=4J+#E#%e�5)(X&)(+*,(+J+�f�"=I( ~P

�K5)(+eI� *%(XJ+� V+T\[2{<S¤OQP\TQ�?TQVXWYN u

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼���¾½ ¿�À�Á�ÂiÃÄÂiÀ8ÅiÃiÂiÃÄÆFÇKÈ.�^ÉqÊ�À8�8Ë�À8ÌÍÀÎÆFÇKÃKÊ8ÃiÉ�ÏDÀÐÒÑ�ÓFÐ0Ãf�oÐ

�8(X#���J+#:@A18� 1�& 1I5 #E$D$'*X#C;<$<p'! �)�K/2�)� #X*,(+-.$f7:9)$'�';<(H5)(X! #X5B=46'#H=�Jj(+-:1:$"7:9)$'�';B$0=�18=4*X#C;<�f5)(H! �)� J+�)=I(+6Yuw =�5 #X*%5)(.@A1v;B�)5 � r)(�$

SI=4�)=�J+1I/2$>eI(x! � � J+�B=�(+6x@A1 O2N V+WYN'[_] [Pa]

u ¥ 9)JX1�� 5)(XJ+� *%5)(�@A14;B� 5 � rB(xeI(x! � � J+�B=�(+6~p,6'#�@s(=�1E& 1I=�J+#K6,#%� �)=�J+�_$^/2143'(+5 �)r ��7:9B$,�?;<(�@A1

bar ] 1bar = 100000 Pa ≡ 105 PauH �1I7��)5 � ���B/2#K=I(c;<("! #�@s/2#X*%1

*%1�eI(X5 1�eI(H! � �)J+�)=�(X6Yp)(H6'#4@A18b 1C/2#>& 1I=�J+#D=4� 1IJj(+J+�Yu�Ô~#D=4$�L• « � S:{<VXÕiR,P2V+WQT�OQP\TQ�?T2VXN?W_] ! � 14;<=4Jj(X*+n8(8! � � JX�)=�(X6f#%6,#X9)5 #%h�*%(Xev;B$,3,()pX�D#%5x@A1�7�$'5 6 r �j@s(�*X� 1C/21�5)=�6 �?3$%=49)#X*,(���5B(c;</2#%�)=46'1.*X�)=�� 5 1Xu w e�5)(X&)(+*,(�=�1.=I(

pa�?9)�

pbu~��5)(X���)/o! � #%�)(+& $'5 �B/d(�5)(4@A& 1���b 1q=41.$,eI�)/d(+JX�

;<(>@A1pa = const

p2��;<(oJc(F*X� 14;B5 #%=�Ji� e�5)#%=4�a!)� � g+9B�?GE5 #pa ≈ 1 bar

p2Jj(+& 5)�j@A1pa = 101325 Pa

u w *%(*X� 14;B5 #,=4JH! � 1v;�=4Jj(+*Xn�(HJ+eI*)u�=4Jj(X5';<(+�%;B5)$0(+JX/2#,=�7�1I� $�u

• « O2VX{)[Q¸��?UQT_O\PQTQ�?TQV%N?WF] (X!)=4#X9)$'J+5 �H! � � J+�B=�(+6E=�1q/21I� �H$x#C;B5 #,=4$85)(�(+!B=4#+9B$,JX5 $x5 $C9)$x!)� � J+�)=�6?(Ek�*X�?;B�=v9)�)6 $o�%u �%tvp?�"#X5K$07:9)$'�';B$f/2#+G�18g%� J+�i/d(+-:�Ypj@A14;B5)(X6"� 9)�K*X1�b)�f#C;F(+JX/2#,=�7�1I�)=46'#Xh%u w e�5)(X&)(+*,(�=418=I(

pu

• ± N?�?OQP\TQ�?TQV%N?W�] (+6'#"@A1o$Ö5 1�6'#�@>Jj(X& 6 ��7:9)$'�';B$Ö! � � JX�)=�(X6 |<R'×\T #C;�(+JX/2#,=�7�1I�)=46'#Xh%p�#%5';<(a=�1F$�J+#�@Jj(+& 6 ��/2#+GH18;B1C7�� 5 �)=I(+J+�^5)(+J+!)� � J+�)=I(+6�6'#4@A�F! �)1v;<=�J+*+n8(0�B(+e49B� 6 $F(+!B=4#+9B$,JX5 #Xh�! � � J+�B=46?(0$KJ+#�@�Jj(+&)6 ��i(+JX/2#,=�7�1��B=46'#Xh:! � � JX�)=46?( u w eI5)(+&B(+*%(�=41x=I(

pmu

• ¯ {)�?OQPQT\�?T2VXN?W�] (+6'#i@A1�$�5 1�6'#�@DJc(X& 6 ��$�7:9B$,�?;B$�! � � J+�B=�(+6 S�N?Ø0T #C;Ù(XJX/2#%=I7�1��)=�6,#%hXp�#%5';<(�=�1o$J+#4@"Jj(+& 6 �Ù/2#+GH1o;<1I7�� 5 �B=�(+JX��! #%J+! � � JX�)=�(X6�6'#4@A�Ù! � 14;<=4Jj(X*+n8(��)(Xe49)�)6 $�(+JX/2#,=�7�1I�)=46'#Xh^! � � JX�)=46?(l�(+!)=�#+9)$'J+5)#Xh:! � � J+�B=46?(>$0J+#�@.Jj(X& 6 �Yu w eI5)(+&)(X*%(�=41x=�(

pvu

p

pA

(pm)A

(pv)B

pB

pa

Ú

Û

Ü�Ý)Þ4ßXà)á'â+ã)äHã áCàBä>Ý å æ)â+æ)Þ4ç?äièêé%ä+ç)á,á?ë�ì

ÜqâXë2ß%ÞIí�î�å)Þ�ç æKÝ å æ âXæ)Þ�äXç

£ 9)�)6'("�%u��%L2ïd(Xe�5 �"5)(X& � 5 �K! � 14;<=4Jj(X*+n8(+-�(�!)� � J+�)=�6?(

pA = pa + (pm)A − (+!B=4#+9B$,JX5 �K! � � J+�B=�(+6K$fJc(X& 6 � ¥pB = pa − (pv)B − (+!)=�#+9)$'J+5 �K! �)� J+�)=I(+6K$0Jj(+&)6 �"ð

(pm)A − 5)(XJ+! � � JX�)=�(X6"$0Jc(X& 6 � ¥(pv)B − ! #%J+! � �)J+�)=�(X6K$0Jj(+& 6)�Kð

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ����ò óôÉ�Âi����ÂiÀ�õ~È.�^ÂiÀ8ÅiÃiÂiÀ¤ö�ÃFÇK����À8ÌÍÀ¤Ñ�ÓFÐ0Ãf�^À�÷ ó�õYÓoÈ�ÇK����ÀZõ~È.�^ÂiÀ8ÅiÃiÂiÀHø

��#,=�/d(+JX�)(v@s/2#^! � #X�)e�*X#Xn:5)$aeI(+!)� 1I/2� 5)$V

7:9)$'�';�(o6'#4@s(0@A1"#Xh��B(+5 � &)1�5)(^eI(XJ+*X#%� 1�5 #,/�! #X*%� ���A

$a6'#4@A#�@=�105)(+9B(+eI��7:9)$'�';ÖhI$%=4JX� 5 1

ρkê=49)� 6?(Ö�Xu��Xtvu���#C;�6'#4@A�)/�$,=v9)#%*X#%/Ùb 1"#X*,(KeI(+!)� 1I/2� 5B(o7:9)$'�';<(ogX� JX�`$`=4Jj(+-.$

/2� �)#X*%(X-:(,ù w ;BhI#X*%#X��@A1:;B#Xg%� #>! #XeI5)(+J>�>@A1v;<5 #%=�Jc(X*%(X5 ] *%1�6 JX#X�)=�6 �Ke�gX�)�K=�*X�?3"=4�?9B(>6'#�@A185)(>-�$0;B1�9)$C@A$f/2#X�B(g%� J+�H@A1v;B5B(+6K5 $C9)�Yu

A

V

dA

dV

~n

−p~ndA

~f

£ 9)�)6'("�%u � L2���)#X� eI*X#+n�5)(�eI(X! � 1I/2�)5)(>7:9)$'�';<(H$f=�Jc(X-�$f/2� � #X*,(+-�(

��6 $'! 5)(>/d(X=�1�5)(>=4� 9B(>6'#4@s(�;B149B$I@A1�5)(>eC(+! �)1I/2� 5 $V@A1XL

��6 $'! 5)(>eC(+! �)1I/2� 5)=�6?(H=��?9B(=

V

ρ ~f dV

;B#%6H@A18$,6)$,!)5)(H! #X*%� ��� 5B=46?(�=��?9B(Kk�;<149)$C@A1x5)(H! #%*X� �At.#C;B� 1��'1I5)(��)e��)(Xe�#%/qL

��6 $'! 5)(>! #%*X� ��� 5)=�6'(H=4� 9B(= −

A

p~n dA

��5B(+6�/2� 5)$%=�$�! #,=v9)14;B-�#�@,@A14;B5)(X& � 5 ��! � #X�)e��?9B(Xe��x�)e�& � -:1�5)� r 1d;�(d@A1�=��?9B(�! � � J+�B=46?(�$'*X1I6x$%=I/21�� 1I5)(�=�$,!)� #XJ+5)#5 #%�)/d(j9B��! #%*X� ����5)(Ö6'#4@A$Ö;B149B$I@A1%uú �(+6 9)1Xp:(+6'#Ö=I/2#`;B1C7�� 5 �)=I(j9)�Ù=4*%1�=��?9)1�6,#�@A1F;<149)$C@A$Z5)(Ö! #%=I/d(+J+�B(+5 $eC(+! � 1C/2� 5 $�p)=I(c;<(>/2#+GH1I/2#>*X� 9)#H9B(+6'#�;<(H�)=46?(+G�1I/2#:@A1Ie�� 6'#%/l/d(+JX1I/d(+JX� 6'186,#�@A�H@A18J+#E$%=49)#X*H5 1�#%!?3%#C;�(+5E;�(�g%�7:9)$'�';Fg%� #�$"=4Jj(+-.$f/2�)� #X*,(+-�( udÔY#�@A1q@A14;B5)(+&)� 5)(>�)(X*X5 #%J+14GH1:=��?9B( L

V

ρ ~f dV +

−∮

A

p~n dA

= 0kA�Xu û,t

��#%� �)=4JX1�b �iJ+1I#X� 1C/2$"ü?(c$,= ] w =4J+�)#Xh��B(c;<=�6,#%hXp'6'#4@s(>5)(%/lh�#X*%#X� �"#EJ+#,/21�6?(+6'#>! � #,/21�5 1x5)1�6'1�7�� eI� & 6'18*X1�9)� & � 5)15)(>eC(+J+*%#X� 1I5 #4@�! #X*%� ���

A6'#4@s(H#XhI�)(+5 �)&)(+*,(�eC(+! � 1C/2� 5 $

V$,JX� & $05B(�!)� #%/21�5)$0J+1x7�� eI� & 6'18*X1�9)� & �)5 1�$,5)$,Jj(X�

eC(+! � 1C/2� 5 1Vp';B� $'h��"&?9B(X5K$8@A1v;B5B(+& � 5)�a�Xu ûH=�1x/2#+G�1�5)(X! �)=I(+J+�K$0=49)1v;B1Ib 1I/Ö#XgX9)� 6 $�L∮

A

p~n dA =

V

∇ p dV, ∇ =∂

∂x~i+

∂y~j +

∂z~k − ý (X/2� 9)J+#X5)#X*>#X! 1I�)(+JX#X�

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R þ

Jj(+6'#�;<(>=41q@A14;B5)(X& � 5)("�Xu ûE/2#+GH1:5B(+! �)=I(+J+�K$0=49)1v;<1�b 1C/�#%g+9)� 6)$<L∫

V

(

ρ ~f −∇ · p)

dV = 0kA�Xu�ÿXt

�8(X6,#H@A1DeC(+! �)1I/2� 5)(V

! � #X�)e�*X#Xn:5)#i� eI(XgX�)(X5)( p~9)1�*,("=�J+�)(X5)(>@A1v;<5)(+& �)5 1okA�%u û't:b 1Dg%� J+�K@A1v;<5)(+6?(K5 $C9)�`$^5)(v@ ]#%! ��J+�+@A1I/�=v9)$'&)(4@A$D6?(c;<(�@A18! #C;<� 5 J+1Ih��)(+9)5)(>7�$'5 6 r �j@s(�@A14;B5)(+6?(H5 $C9)�Yu

ρ ~f −∇ · p = 0 =⇒ ρ ~f = grad pks�Xu �%t

�8(�#%=45)#X*X$:@A14;B5)(+&)� 5 1>kA�Xu �%t�/2#+GH1�=�1�eC(+6?nq$'& � J+�0;<(�@A1�! #Xn:1:! � �)J+�)=46?(H$D7:9)$'�';B$>6'#4@A�"/2� � $C@A1�$E! #XJX! $'5 #%=�J+�#C;B�)1��'1�5 #>! #Xn:1C/�eC(+! �)1I/2� 5)=�6 �?3f=��?9B(>6'#4@A1�5)(H-:1�h4(�;B1�9)$C@A$<u

�l� 1��H(+*,(+-.$8! �)(+6)J+� & 5 � 3�! � #%g+9)1C/d(�& 1C=4J+#�=41.6'#X� �B=4J+�H�x;<� $'h���#%g+9)�)6�@A1v;<5)(+& �)5 18kA�Xu �%t4pC;B#:6,#%h4(�=�1�;<#+9B(Xe��5)(E=49)1v;<1�b �"5)(+&)� 5Yu��81�*,(H�D;B1C=45)(E=�J+�)(X5)(�@A1v;B5)(X& � 5 1x=�1�5)(+! �)��$0$0�)(+eI*X�j@A1I5 #%/�#Xg+9B� 6 $�p?Jv@4u

ρ(

fx~i+ fy~j + fz ~k)

=∂p

∂x~i+

∂p

∂y~j +

∂p

∂z~k

h�;B1�=4$fxpfy

�fz

!)� #4@A1I6 r �j@A1:eI(X! � 1I/2�)5)=46'1�=4�?9)1:5)(�6'#X#X�%;B�)5)(+J+5)1�#%=�1� �1I6'(X� J+#%*X#Xh�6,#%#X�%;B� 5B(+5 J+5)#Xh�=��)=4JX1I/d( u£ (j;<(K=41E�_9)1I*%("�^;B1I=�5)(i=4J+�B(+5 1�@A14;B5)(+&)� 5 1E! #,/25 #+GE�a=�(K1�9)1I/21I5 Jj(+� 5 �B/�*%1�6 JX#X� #,/

d~r = dx~i+ dy~j + dz ~kp

!)(E=�1�;B#Xg%�j@s(>=v9)14;B1�b)(�@A14;B5)(+&)� 5)( L

ρ (fx dx+ fy dy + fz dz ) =∂p

∂xdx+

∂p

∂ydy +

∂p

∂zdz

kA�%u��Xt �1C=45)(�=�J+�)(+5)1Y@A1v;<5)(+& �)5 1�kA�%u��Xt�! � 14;<=4Jj(X*+n8(qJ+#XJj(j9B5 ��;B�)7�1I� 1�5)r �j@s(j9�7�$'5 6)r �j@A1

p = p(x, y, z)pCJc(X6'#.;<(�6'#X5)(X& 5 #

;B#%gX�j@s(%/2#)Ldp = ρ (fx dx+ fy dy + fz dz )

kA�Xu �%t �(X6?9)1Xp\;B#%gX�j@A1I5)(H@A1x@A14;B5)(i;B�)7�1��)1�5 r �+@s(j9)5)(E@A1v;B5)(X& � 5)()p2� eD6'#4@A10=�10� 5)J+1�hI�)(jn�1�-:1I/��a9B1�*X10��;B1I=�5 1f=�J+�)(+5)1@A14;B5)(+&)� 5 1DkA�%u �,t./2#+G�1q@A14;B5 #%e�5)(X& 5 #>#C;B� 14;B� J+�K! #Xn:1�! � �)J+�)=46?()p Jv@4u

p = p(x, y, z) =

ρ (fx dx+ fy dy + fz dz ) + CkA�Xu �%t

h�;B1�@A1C

6'#X5B=4Jj(+5 Jj("6'#4@s(K=41E#C;<� 1��'$C@A1E� eEh��B(+5 � &)5 �?3^$,=v9)#%*%()pYJv@4u0�)eE! #XeI5)(+J+1E*%� 1v;<5 #%=�J+�_! � � J+�B=46?(K$o5 1I6'#4@Jj(+&)6 �

(x, y, z)$E7:9)$'�';B$�ud��� �)/21I5)(�@A14;B5)(X& � 5 1HkA�Xu �%t�5)(v@Ag%#+n�1�b 1:=41��?9)$%=�J+� #%*%(+JX�E5)(x! �)�)/21�� �B/d(�6'#4@A�f=v9)14;B1Xu

�%1v;B5B(+& � 5)1�kA�%u �,tY�"kA�%u �,tY*,(jG�1���/2� �)#X*%(X-�1q=4JX� �xn�� *X#%h:kρ 6= const

t~��eC(�/2� �)#X*%(X-�1.5 1C=4J+� ��n:�)*X#Xhq7:9)$'�';<(kρ = const

t4p%@A1I�^5)�j@A1�$'*%1v;B1I5)(D5 �)6'(X6 *%(D! � 1I=�J+! #,=4Jj(+*%6'(0*X1IeI(+5B(EeI(0h�$%=�J+� 5 $F7:9B$,�?;<(D6'#4@A��=41�5)(j9B(Xe��F$'5 $'Jj(+�eC(+! � 1C/2� 5 1x=I(H=49)� 6'1E�Xu���! � �?9)�)6,#,/��)e�*X#j�'1I-:( w @�9)1��)#X*%(�@A14;B5)(+&)� 5 1Xu

��e�@A1v;B5)(X& � 5 1_kA�%u �,tx/2#XG�1I/2#^� e�*%$'b �Ö=49)1v;B1Ib 10eC(+6?nq$'& 6'1�kê*X�';<��! � 1Ie�1I5 Jj(+r �j@A$`�%ï21�! 1IJ+� J+#%� �j@A$?/�J+1I#X� �j@A1! #Xn�(%�%t4L

• *X1I6 J+#X� ~f ! #%6'(Xe�$C@A1d! �)(X*%(Xr:�x=I/21��85)(4@A*X1Ib 1�! � #,/21�5 1�! � � J+�B=46?(�$�7:9)$'�';B$�6'#�@A�x/2� � $C@A1Xp�@A1���@A1�6'#+9)� 5)1I(+�B(+5=�(H*X1I6 J+#%� #%/gradp

• $H7:9B$,�?;B$�6'#4@A�D/2�)� $C@A1�*X1I6 J+#%� ~f@A1�5)#X�)/d(+9B(+5>5)(8!)#X*X�)���>6'#X5)=�Jc(X5 J+5 #%h�! � � JX�)=46?(0kê� e�#%g%(X�)=46'1q! #X*%� ���<t

¯ PQTQS�R,P�� ���� � � #%*%(X-�1�5 1C=4J+� ��n:�)*X#Xhx7:9B$,�?;<(H$D! #+nq$f=��?9)1���1C/Qn:�)5 18J+14GH1�8(j;<(F7:9)$'�';l/2�)� $C@A1f$_! #Xn.$_e�1C/Qn:�)5 1fJX14G�1%pd*X1I6 J+#X�B=46'#F! #Xn:1"eI(X! � 1I/2�)5)=46 � 3a=��?9B(D@A106'#X5)=�Jj(+5 J+5)#)pdJv@4u

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R �

~f = g ~k � pX�?9)� ~f = −g ~k p+$xeI(X*X�)=�5 #%=�J+�H#C;HJ+#%h4(�6?(X6,#�@A1�$%=I/21�� 1I5)(�#%=�( z k�5)(X*X� ��1q�?9)�H5)(+5 � G�1Ct4ud��#%=�/d(XJ+�)(4@s/2#=�$�;�6'#�@A�0@A1D5B(+! $'-:1�5_J+1�&)5 #X��b $o6'#4@s(K/2� �)$I@A1^kê=v9B� 6?(_�Xu ¢%tvuD��#%e�� JX� *%(X5�=I/21��

z#%=�1:@A1D*X1I� J+� 6?(j9B5 #K5)(+5 � G�1Xp

Jj(+6'#�;<(�@A1 ~f = g ~ku

�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

��������� ρ

z

x

0

H

p = pa

p = pa + ρgH

£ 9)� 6?("�Xu ¢ L � � �)#X*%(X-�1�5 1C=4J+�)�xn�� *X#%hx7:9)$'�';<(H$D! #Xn.$"=4�?9)1���1I/Qn�� 5 18J+1�G�1

��(H#%*%(+6'#H$%=�*X#4@A1I5 �"6,#%#X�%;B� 5B(+J+5 �K=4�B=4J+1C/Ö!)� #4@A1I6 r �j@A18*%1�6 J+#%�)( ~F5B(>#%=41�6'#X#%�%;B� 5)(XJ+5 #Xh8=��)=4J+1C/d(

x, y, z=4$

fx = fy = 0�fz = g

u ¥ 6'#E=41�J+#EeC(X/21I5 �K$8@A1v;B5B(+& � 5)$o�Xu ��;B#Xg%�j@s(>=41%L

p = p(x, y, z) =

ρ (g dz ) + C

�8(X6,#>=418�B(c;B�"#E/2� � #X*,(+-.$05 1C=4J+�)�xn�� *X#%h�7:9B$,�?;<( p J4@4u�(+6'#:@A1ρ = const.

p?;B#Xg%�j@s(>=41XL

p = ρ g z + CkA�Xu¡���,t

��#%5)=4Jj(+5)Jc(C

=418#C;B�)1��'$C@A1:�)e8h��)(X5 � & 5 � 3f$%=49)#X*,( u���(H6'#X5 6)� 1�Jj(+5K! �)�)/21��F=I(H=49)� 6'1E�Xu ¢HeI(x = z = 0

pp = pa

ph�;B1.@A1

pa(XJX/2#%=I7�1��)=�6 �f!)� � J+�)=I(+6Yu�Ô~(+6'#E=4186'#%5)(+& 5)#x;B#%gX�j@s(H�)(%=4! #C;<149B(�! � � JX�)=46?( L

p = pa + ρ g zks�Xu¡�X�Ct

��eH@A1v;<5)(+& �)5 1akA�Xu¡���,t�eI(+6 n.$'& $C@A1I/2#�;�(�=�$`� eI#Xg%(X�)=46'1"! #X*%� ���l#C;B� 1I�'1�5 1�@A1v;B5B(+& � 5)#%/z = const

p.�l#X5 1! �)1v;<=�Jc(X*+n8(v@A$i3,#X� �)e�#X5)Jc(+9)5 1H�)(+*%5 �YuEÔ~(X6'#j�,1%pYJ+1E� eI#Xg,(+�)=�6,1H! #%*X� ���^�_*X1I6 J+#%�

~g=4$o5 #X�)/d(+9)5 �Yp\��J+#i=I/2#i�

�)(X5 �[email protected](X6?n.$'& � 9)���+*%1�6 J+#%�~g!)#X6?(+eI$I@A1���=�/21I�x5)(4@A*X1�b)1�h�!)#X�)(%=4Jj(�! � �)J+�)=46?( ] *%1�� JX� 6?(j9)5 #�5)(+5 � G�1Xu���e�#%g%(+�B=46?(! #%*X� �ú5)(H6'#4@A#�@\@A1

p = pa5)(+eI� *%(H=41 V+[2{<»B{%¹dU2N^O2{B|BPQ} �,#%e�5)(X&)(+*,(+b 1C/2#�@A1�#%gX� 5 $'J+�B/�J+�)#X$'h¾9)� b 1C/qu

¥ 6,#K=41>J+1I& 5 #,=4J"5)(j9B(Xe��^$F5 1�6'#,/�eI(XJ+*X#%� 1�5)#%/�=�$�;<$<p~��(X6,#"=41>� eI5)(c;a-�1>5)(+9B(+eI�o5 1I6 ��h4(X=�$F6'#%/21H*+9B(j;<(5)(XJ+! � �)J+�)=�(X6

pmk�! �)�)/21�� ] gX#XrB(�5 1�6'#Xhah4(Xe�� �B(+5 #%h�!)� b)(%t4px�)(%=4! #C;B1�9B(�! � �)J+�)=46?(Zb)1`gX� JX��*X�?9)#�=49)� & 5B(

@A14;B5)(+&)� 5 �´kA�%u �,tvp.�)(+e�9)� 6'#X*,(+b 1K=41i=I(X/2#_6'#X5)=�Jc(X5 J+5)(CuZ�·JX#%/ú=v9)$'&)(4@A$`#X5B(^b)1igX� J+�

C = pa + pmu £ (

;B�)$,hI1H=�J+�)(X5 1Xp~(X6'#0�)/d(%/2#f� eI5)(c;_5 � *%#%(DJX1�& 5 #,=4J+��h4(X=�$i6'#%/21H*+9B(j;<(D! #%J+! � �)J+�)=�(X6pvp�*X� 14;B5 #%=�J06'#X5B=4Jj(+5 JX1

Cb 18gX�)J+�

C = pa − pvu

��� �"!$#~f % &�' �"!$#�(*) %+&-, �*.*�*/"�10�# '2&43

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R 5

¯ PQTQS�R,P�� ¼6� �´=4$I;B$f=4185)(+9B(+eI1�=�*X1�J+1I& 5 #%=�J+�K6'#4@A18=4185 18/21I��(v@A$��1I6'(H=418$05 1�6'#%/l=4$I;B$05)(j9B(Xe�1�;B*%1:JX1�& 5 #,=4J+�i6'#�@A1�=�1�5 1x/21��H(v@A$�p h�$,=4J+�)5)(

ρ1

�ρ2

kρ2 > ρ1

t ] =49)� 6?("�Xu û)u

7 7 7 7 7

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

: : : ; ; ; ; < < < < = = = > > >? ? ? ? @ @ @ A A A

B

C

ρ1

ρ2

z1

z2

H1

H2

p = pa

p = pa + ρ1gH1 + ρ2gH2

£ 9)� 6?("�Xu û)L � � �)#X*%(X-�1�5 1C=4J+�)�xn�� *X#%hx7:9)$'�';<(H$D! #Xn.$"=4�?9)1���1I/Qn�� 5 18J+1�G�1

ïd(%=4! #C;B1�9)1�!)� � J+�)=�6?(>$0J+1I& 5 #%=�J+�)/d(E=�$D#C;B� 1I�,1I5 1�� e��B(+e��B/d( LÔY1I& 5 #%=�JEh�$,=4J+� 5)1

ρ1 : p = pa + ρ1gz1, 0 ≤ z ≤ H1

ÔY1�&)5 #%=�JEh�$%=�J+� 5 1ρ2 : p = pa + ρ1gH1 + ρ2gz2, 0 ≤ z ≤ H2

¯ PQTQS�R,P�� ¼6�D� 1I� 1�-:18! � � JX�)=46?(>! #,/2#Xb $Ur)1�*X�

�8(4@ @A1v;B5 #,=4Jj(+*%5 �j@A�K� 5)=�J+� $?/21�5)J0eI(E/21I� 1�-:1fkê�)(+e�9)� 6'1Ct�! � � J+�B=46?(�@A1�JXe�*)uU

r)1�*)uqÔY#�@A1�=�(X*X�j@A1I5)(E=4Jj(X6?9)1�5B(r 1I*0$"6'#4@A#4@�=�1�5)(+9B(+eI�i#C;Bh�#%*%(X�)(v@A$'b)(>/d(+5)#%/21�Jj(X�)=46?(EJ+1I& 5 #,=4J ] 5)(v@A&)1���b)1�=4$"J+#f�?9)�i;<1I=4JX�?9)#X*,(+5)(>*%#C;<(Fkm(+6'#=�1o/21�� 1^/d(j9)��! � �)J+�)=4r)�BtE�?9B��G>�)*%(Zkê(X6'#`=41o/21I� 1F*X1�9)� 6 ��! �)� J+�)=�r �Btvu ¥ 6,#Ö=41i5)(_6 �B(v@A1�*%1Fr 1I*X�Ö;B#X*%1v;B$�)(Xe49)�)& � J+��! � �)J+�)=4r)��/d(X5 #%/21IJj(+�)=�6'(^J+1I& 5 #%=�J_b 1KeI(c$'e�1IJ+��5 1I6 �l5 #X*%�Ö�)(X*X5 #%J+14GE5 �Ö! #+9B#+GH(4@4p�$a#C;B5)#%=4$a5B(=49)$'&)(v@q6?(c;<(�@A185)(>#Xg,(�6)�)(v@s(Hr 1I*X�"gX�?9B(�@A14;B5)(+6?(H*X�)1v;B5 #,=4J>! � �)J+�)=46?()u

«E

ρ

ρ

x

ρm

H

hmI I

£ 9)� 6?("�Xu�ÿ?L � 1��)1�-:18�)(+e�9)� 6'1�! � � J+�B=�(+6?(>! #,/2#Xb $U

r 1�*%�

ý #X� � eI#X5 Jj(+9)5)(i! #%*X� �I− I

@A1�@A1v;B5)(o� e�#%g%(X�)=46?(K! #%*X� �Í$_J+1I& 5 #,=4J+�`hI$%=�J+� 5 1ρm

pdJv@4u�! � � JX�)=4r �Ö5)(iJ+#�@

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R F

3,#X� � eI#X5 Jj(+9)�f$D9)1�*%#%/`�f;B1C=45 #,/�6)�)(+6 $U

r 1�*%�"=�$x@A1v;<5)(+6 �~L

pA + ρgx = pB + ρg(H + x− hm) + ρmghm

⇒ ∆p = (ρm − ρ)ghm + ρgH

 �(X6?9)1Xp /21I� 1�-:1I/l*%�)=4� 5)1hm

/2#XG�1I/2#>#C;B�)1v;B� JX�"�B(+e49B� 6 $D! �)� J+�)=I(+6?(>� eI/21I�'$DJj(+&)(X6'(A

�Bu

G�HJIDH�G KMLON1L�PRQ�Xu w ;B� 14;B� J+�Z! #%6?(+e��)*%(+-:1";B�)7�1I� 1�5)r �j@s(j9B5 #Xho/d(X5 #%/21IJ+�)(�k

Ur 1I*X�Bt0=4! #�@A1�5 #%hF=�(a� 1Ie�1I� *X#,(+� �)/d( ¥ ��ð�u

��#XeI5)(+J+�^=4$F=v9B1v;B1Ib �^!)#C;<(+r �~Lρ1 = 1000 kg/m3

pρ2 = 1200 kg/m3

pρ3 = 800 kg/m3

pρ4 = 900 kg/m3

pρm = 13600 kg/m3

ph1 = 1 m

ph2 = 4 m

ph3 = 2 m

ph4 = 2 m

ppm = 1 kPa

�pv = 2 kPa

u�?u w ;B� 14;B� J+�K� eI�)(+e�eI(H! �)� J+�)=I(+6i$DJj(+&)6 �i�´$f7�$'5 6)r �j@A�K! #XeI5)(+JX�?30*X1�9)� & �)5)( L

hipi = 1, . . . 5

pρm

�ρu

¢ u w ;B� 14;B� J+�´5B(+J+! �)� J+�)=I(+6´*,(+ev;<$%3'(l� e�5)(j;�J+1�&)5 #%=�J+�´$Ù� 1�eI1�� *%#%(+�)$�(+6'#�=�$�! #%6?(+e��)*%(+-�(l*X� ��1�r 1I*X5 #%h/d(+5 #,/21�J+�B(o$a#C;B5)#%=4$�5)(0@A1v;<(X5�� 1C7�1�� 1I5 J+5 �l5 � *%#)L

h1 = 0.9 mph2 = 2 m

ph3 = 0.7 m

ph4 = 1.8 m

pH = 2.5 m

pρ = 1000 kg/m3

pρm = 13600 kg/m3

uû)uq��9B1�*X#�@�6'#%/2#X�)�DeI(XJ+*X#%� 1�5 #%h�� 1Ie�1��)*X#%(X�)(�5)(+9B(+eI1�=41�;B*X1�J+1�&)5 #%=�J+�f6'#�@A1�=41�5 1�/21���(4@A$�p

ρ1 = 1000 kg/m3p

ρ2 = 800 kg/m3p<(>$D;<1I=45)#4@�=41x5B(j9B(Xe��KJ+1�&)5 #%=�JEh�$%=�J+� 5 1

ρ3 = 870 kg/m3u�ð.�)=4�)5 1x5 � *%#%(HJ+1I& 5 #%=�J+�o=�$<L

H1 = 0.8 mpH2 = 0.6 m

�H3 = 1.1 m

pY("� eI5)(c;a5 � *%#%(fJ+1�&)5 #%=�J+�^*X9B(c;�(v@A$i! � �)J+�)=4r)�pv = 0.15 bar

�pm = 0.25 bar

u w ;B�)1v;B� JX�o! #%6'(Xe�� *,(+-:1x/d(+5 #,/21�J+�B(DeI(EG>�)*X#%/�kρm = 13600 kg/m3

t�! � � 6?nq$'& 1�5)#XhH6'(X#��J+#D@A1D! � � 6?(XeI(+5)#i5)(F=v9B� r ��k

H = 0.5 mpα = 45◦

tvuF��(XJ+�)/�#C;B�)1v;B� JX�a! � � J+�B=�(+6a$�! � #,=4J+#%� $�� e�5)(j;J+1�&)5 #%=�J+�Fh�$,=4J+�)5 1

ρ2

! �)�o6'#4@A1C/�b 1x! #%6'(Xe�� *,(+-:1�/d(+5)#%/21�JX�)(Dg%� J+�l = 0

kêeI(+5)1I/d(+�)� J+�F! �)#%/21�5)$"5)� *X#,($D&)(%=4�i/d(+5 #,/21�J+�B(%tvu

ÿ?u���� #Xe.! #%6?9)#X!B(+r�r)�?9)� 5';<� � & 5 #%h�=4$I;<(:*X�)=�� 5 1H

*%1�� JX� 6?(j9)5 #�@A1q! #%=�Jj(+*+n�1�5B(�r 1I*�Jj(+5)6 �?3�eI�';B#X*,(�;<$%GE� 5)1hp,�E6'#4@s(.@A1�3%1��B/21�J+� &)6 �D=4! #�@A1�5)(�=�(8! #%6?9)#%! r 1I/quQ��=�$I;f=41�5)(+9)� *,(�J+1�& 5)#%=4J�h�$%=�J+� 5 1

ρud��#+9)�)6'(:b 1�gX� JX�

*X�)=�� 5)(E5)� *X#,(>J+1�& 5)#%=4JX�F$i=4$I;B$K6?(c;<(E=�1�r 1I*D! #XJX! $'5 #D�)=�! $'5 �FJ+1I& 5 #X��b $ ù ¥ JX/2#,=�7�1I�)=46 �i!)� � J+�)=I(+60@A1pau £ /d(+JX�)(+J+�F;<(�@A1�! �)#%/21�5B(0=4Jj(X-:(0*%(+e4;B$,3,(E$F=�$I;B$K� e�#%J+1��)/d=�6?( u8ð.149)� &)� 5 1

Hphpρ�pa

=�/d(XJ+�)(+JX�! #XeI5)(+JX�)/qu

���2S T�Ã"ÓoÀ ÆFÇKÃKÊ8ÃiÉ�ÏDÀ ÂiÀ ÇKÀ8��ÂKÐÒÆi����ÇVU�8(X#.��J+#�=�/2#�*%�';B1�9)�8$�$'*X#C;<$<pI! #X*%� ���)5)=46'1d=4�?9B1�$�=�Jc(XJ+� r �x7:9B$,�?;<(q! #XJX� & $�=I(X/2#�#C;x! � �)J+�)=46?(xkê5 #X�B/d(j9)5 � 3

5)(X! #X5B(%t��K#C;B� 1��'$C@A$0=�1:!)#%/2#Xb)$D� e��B(+eI(Kks�Xu ¢%tvp?6'#%h4(�5)�j@A185)(+#C;�/21�JH#X*j;<1:!)#X5 #%*X� J+�~L

d~P = −p~ndA ⇒ ~P = −∫

A

p~ndAks�Xu¡�C�%t

w *%#q@A1�5)(4@A#X! ��J+�j@A�E� eI�)(+e ] =4�?9B$�! �)� J+�)=�6'(�5)(85 1I6 $H! � #XeI� *X#Xn:5)$�! #%*X� �´#C;<� 1��'$C@A1I/2#8Jj(+6'#8��J+#�� 5)J+1�hI�)(j9)�B/2#! #Xn:1a!)� � J+�)=�6?(Z! #ZJ+#�@"! #%*X� ���YuW��eK@A14;B5)(+&)� 5 1�ks�Xu¡�C�Xtf*%�';B�)/2#l;<(�@A1a1�9)1I/21I5 Jj(+� 5B(Z=4�?9B(�!)� � J+�)=�6?(

d~P$,=�/21��)1�5)(_=4$'! �)#XJ+5 #a#C;�=�/21I�)(_5 #%�)/d(j9)1K1�9)1I/21I5 Jj(+� 5 1i!)#X*X�)���dA

k�*%1�6 JX#X��5 #X�B/d(j9)1K! #%*X� ���Ö$,=�/21��)1�*%(%/2#$'*X1I6K#C;i!)#X*X�)���Bt4u���#%*X#%/Ö! #%J+! #Xh¾9<(+*+nq$0b)1I/2#E*X�?;B1�J+�i6?(X6,#E=�1x! �)(X6 J+� & 5)#0/2#+GH18#C;B� 1v;<� J+�i! �)(X*%(XrYp)=�/21I�i�

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R X

� 5)J+1�5 eI� J+1IJK=��?9)1H6'#4@A#%/�JX1�& 5 #,=4J"$F=4Jj(+-.$F(+!B=4#+9B$,JX5 #XhH/2� � #X*,(+-�(okm=4�?9B(03?�';B� #,=4Jj(+JX� & 6'#Xh�! � � J+�B=46?(%t�;B1�9)$C@A15)(>5)1�6 $D�B(+*X5)$D! #X*%� �0u

��#,=�/d(+JX�)(v@s/2#E@A14;<(+5�� 1�eI1�� *%#%(+�`=I(F6'#%=��)/�gX#%& 5 �)/�eI�';B#,/�$�6'#,/21f=41f5)(j9<(+e���J+1�& 5)#%=4J^h�$,=4J+� 5)1ρkm=v9)�)6'(

�%u �,tvu2�8(�gX#X&)5 #%/oe��';B$>=41�5B(j9B(Xe��>! �)(X*X#X$'h4(X#X5 ��#XJ+*%#X�Yp%6,#�@A�:@A1�eI(XJ+*X#%� 1�5>5 1I6 �)/�eI(XJ+*%(X�)(+& 1C/qu\��#X9)� 6'#%/�=4�?9B#%/J+1I& 5 #,=4J�;B149B$I@A185B(�Jj([email protected](+J+*,(+�)(X&^k�#%5H@A18! � � & *%� ��b)1�5"� 1�r �B/2#EeI(+*%� Jj(+-�=46 �B/`*X1IeI(%/d(�eC(�eI�';o=4$I;<( ] ;<(Hg%�"=I/2##C;B�)1v;B�?9B�>=��?9)1�#%! J+1��)1�b 1I-:(8$�*X1IeI(X/d()p+/2#X�)(%/2#x=4�)(X& $'5)(+JX�>=��?9)$�! � � J+�B=46?(%t ùK�8(+6'#.@A1�$%=�/21I� 1�5B(�Jc(8=��?9B(,ùfü%;<1=�185)(j9B(Xe��"5)(+!B(c;B5)(�Jj(+& 6?(HJ+1x=��?9)1Cù w ;Bh�#%*X#X�)�05)(>#%*%(�! � Jj(+-�(E=41�5)(+9B(+eI1:$0#X*X#,/`! #%J+! #%h¾9B(+*Xn.$�u

Y�Y�Y�Y�Y�Y�Y�Y�Y�YY�Y�Y�Y�Y�Y�Y�Y�Y�YY�Y�Y�Y�Y�Y�Y�Y�Y�YY�Y�Y�Y�Y�Y�Y�Y�Y�YY�Y�Y�Y�Y�Y�Y�Y�Y�YY�Y�Y�Y�Y�Y�Y�Y�Y�YY�Y�Y�Y�Y�Y�Y�Y�Y�YY�Y�Y�Y�Y�Y�Y�Y�Y�YY�Y�Y�Y�Y�Y�Y�Y�Y�YY�Y�Y�Y�Y�Y�Y�Y�Y�Y

Z�Z�Z�Z�Z�Z�Z�Z�Z�ZZ�Z�Z�Z�Z�Z�Z�Z�Z�ZZ�Z�Z�Z�Z�Z�Z�Z�Z�ZZ�Z�Z�Z�Z�Z�Z�Z�Z�ZZ�Z�Z�Z�Z�Z�Z�Z�Z�ZZ�Z�Z�Z�Z�Z�Z�Z�Z�ZZ�Z�Z�Z�Z�Z�Z�Z�Z�ZZ�Z�Z�Z�Z�Z�Z�Z�Z�ZZ�Z�Z�Z�Z�Z�Z�Z�Z�ZZ�Z�Z�Z�Z�Z�Z�Z�Z�Z[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[�[

\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\�\

]�]�]�]�]�]]�]�]�]�]�]]�]�]�]�]�]]�]�]�]�]�]]�]�]�]�]�]]�]�]�]�]�]^�^�^�^�^�^^�^�^�^�^�^^�^�^�^�^�^^�^�^�^�^�^^�^�^�^�^�^^�^�^�^�^�^

_�_�_�_�_�_�__�_�_�_�_�_�__�_�_�_�_�_�__�_�_�_�_�_�__�_�_�_�_�_�__�_�_�_�_�_�__�_�_�_�_�_�_`�`�`�`�`�`�``�`�`�`�`�`�``�`�`�`�`�`�``�`�`�`�`�`�``�`�`�`�`�`�``�`�`�`�`�`�``�`�`�`�`�`�`

a�a�a�a�a�a�aa�a�a�a�a�a�aa�a�a�a�a�a�aa�a�a�a�a�a�aa�a�a�a�a�a�aa�a�a�a�a�a�aa�a�a�a�a�a�aa�a�a�a�a�a�ab�b�b�b�b�b�bb�b�b�b�b�b�bb�b�b�b�b�b�bb�b�b�b�b�b�bb�b�b�b�b�b�bb�b�b�b�b�b�bb�b�b�b�b�b�b

c�c�c�c�c�c�c�cc�c�c�c�c�c�c�cc�c�c�c�c�c�c�cc�c�c�c�c�c�c�cc�c�c�c�c�c�c�cc�c�c�c�c�c�c�cc�c�c�c�c�c�c�cc�c�c�c�c�c�c�cd�d�d�d�d�d�d�dd�d�d�d�d�d�d�dd�d�d�d�d�d�d�dd�d�d�d�d�d�d�dd�d�d�d�d�d�d�dd�d�d�d�d�d�d�dd�d�d�d�d�d�d�dd�d�d�d�d�d�d�d

e�e�e�e�e�e�e�e�e�ee�e�e�e�e�e�e�e�e�ee�e�e�e�e�e�e�e�e�ee�e�e�e�e�e�e�e�e�ee�e�e�e�e�e�e�e�e�ee�e�e�e�e�e�e�e�e�ee�e�e�e�e�e�e�e�e�ee�e�e�e�e�e�e�e�e�ee�e�e�e�e�e�e�e�e�ee�e�e�e�e�e�e�e�e�e

f�f�f�f�f�f�f�f�f�ff�f�f�f�f�f�f�f�f�ff�f�f�f�f�f�f�f�f�ff�f�f�f�f�f�f�f�f�ff�f�f�f�f�f�f�f�f�ff�f�f�f�f�f�f�f�f�ff�f�f�f�f�f�f�f�f�ff�f�f�f�f�f�f�f�f�ff�f�f�f�f�f�f�f�f�ff�f�f�f�f�f�f�f�f�f

D

C

P

v

ξ

η

u

zzC

ρα

∆vC

vD

vC

�dA

d~P

~n

£ 9)� 6?(K�%u �)L £ � 9B(H! � � J+�B=46?(>5)(H�)(+*%5 �"! #X*%� �

��=v9)14;B1�b �B/¤� 1v;<#X*X�B/d(�=41F;<(v@A1_(j9Bh�#X�)� Jj(X/�eI(�#C;B� 1��'� *,(+-:1F! �)(X*Xr)()p�=�/21I�)( p�� 5 J+1I5 e�� JX1�Jj(l��! #+9B#+GH(4@s(5)(X!)(c;<5 1�Jj(+&)6,1>=4� 9)1�!)� � J+�)=�6?( u� �#f=49)1v;B1Ib �?3F� eI�)(+eC(0=�1H/2#XG�1x9<(+6'#E;B#%b �Yp�! �)�)/21�5 #,/_@A1v;<5)(+& �)5 1"ks�Xu¡�C�%tvp�$'e;B#%gX� #:! #%e�5)(XJ+$8�)(X=�! #C;B1�9)$8! � � JX�)=46?(�eI(�5 1C=4J+� ��n:�)*�7:9B$,�?;>6'#4@A�>/2� � $C@A1�$8! #Xn.$>��1I/Qn�� 5 1qJ+1�G�1%L

p = C+ρgzkê! #Xh¾9)14;<(v@AJX18! � 1v;�(+*%(X-:(��?9)�KJj(v@�;B1I#H$f6 -:� e��K! � #,7>udð�u £ (jn�5 � 6'#X*,(hg�i"j*ilknmojpkqmrkns"tvu�j*ilknm4j1wRxzy{k*|�j+tvu¯ P2N?WQ�?T�}QUQ{a{%¹dP2R{~�TQ|�N?ØfRFVXT~[dRiOQP\TQ�?T2V+WYN�UQN�P2N?|<U\¸ÖOQ{B|BPQ}• £ �?9B(H! � �)J+�)=46?(>5)(H5 1I6 $0�B(+*X5)$D! #X*%� ��;B149B$I@A1�$'*X1I6K$D! �)(X*Xr $D6'#�@A��@A185 #%�)/d(j9<(+5"5)(>J+$0!)#X*X�)�0u• £ �?9)$D!)� � J+�)=�6?(>$%=�/21I�)(+*,(X/2#H#C;KJX1�& 5 #,=4J+�i6?(H! #%*X� ���f6'#4@s(�@A18#X6)*%(+��1�5)(�J+#,/�J+1I& 5 #%��b $�u• £ �?9)$D!)� � J+�)=�6?(>�)(+& $'5)(%/2#>! #%/2#%b $D#Xg%�)(X=�r)( L

P = (pc − pa)AkA�Xu¡�I¢,t

pc ]�N OQV%{ [2¸��?U\T ! � � JX�)=�(X6i$ � R,º_T\}_�?¸ ! #%*X� ���pa ] (+JX/2#,=�7�1I�)=46 �"! � �)J+�)=�(X6A ] ! #X*%� ���)5)(H! #X*%� ���"5)(H6'#4@A$0=�18J+�)(jGE�i=4� 9B(H! � � J+�B=46?(

• ð.1�� J+�)6'(+9)5)(�6'#X#X�%;B�)5)(+Jj(�J+14GE� ��Jc(H6'#4@s(>=�1�/21I� �"#C;i=49)#Xg%#C;B5 1�!)#X*X�)���K=41�#C;B� 1I�,$C@A1:5)(H#%=�5 #X*%$E� e��)(XeI()L

zc =pc − paρg

kA�Xu¡��û't

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R �

• �8(c;<(�! #X*%� �ÙeI(+6 9B(+!)(�5 1�6)��$'h4(+#α=I(�3,#X� �)e�#X5)Jc(+9)#%/ak

180◦ < α < 0◦tvp+�)(%=4! #C;<149B(�! � � J+�B=46?(�5B(�!)#X*X�)���

@A1�5 1I�)(+*%5 #%/21I� 5)(Fkê! � � JX�)=�(X6o9)�)5 1I(X� 5 #D�)(%=4J+1>=�(D! #X*%1�b)(X-�1C/�6,#%#X�%;B� 5B(+J+1ztvp�J+1H=4J+#%h4(0=��?9B(D!)� � J+�)=�6?(

;B149B$I@A1q$8Jj(+&)6 �D6'#�@s(�=�1.5B(+e��)*%(1�ltrs*i"j{���{�lkrilkv�$moj'pc���B(X=4JX#4@s(+-:1q#C;HJ+1�GE� ��Jj( £ ;B#�Jj(+& 6'1

D@A1.#C;<� 1��'1�5)#

� e��B(+e�#,/qL∆vc =

IcξvcA

kA�Xu¡�Cÿ%t

IcξuIu�u J+1�GE� ��5 �i/2#,/21�5 JE� 5)1�� r �+@A18eI(H#%=�$

ξ

vc =zc

sinα

u�u�uv6,#%#X�%;B� 5B(+Jj(�JX14GE� ��Jj(ikê#X*X#H5 �j@A1�5 1�6 �"$'5 � *%1�� eC(j9)5 �K� eI�)(+e ] #%5f*,(jGE�feC(E=v9)$'&)(4@

=I(>=v9)� 6'1E�%u �)uQ��h4(+#α/2#+GH18gX� JX�"eC(c;<(XJ+�"�"5B(x;<� $'h��K5)(+&)� 5 ] � 1�r �B/2# α @A18$'h4(+#

6'#�@A�"6'#,=4�"e��';ieI(X6?9B(+!B(H=I(H*X1��)J+� 6?(j9)#,/qudÔ~(c;�(>b 18$?/21I=4JX#sinα

;<(E=4J+#�@A�cosα

tvu¥ 6'#Z! #%*X� �Î5 �+@A1�=4�)/21IJ+� � &)5)( p�#X5';<(�!)#%=4JX#4@A����! #,/21��)(X-�1a5)(X!)(c;<5 1`Jj(+& 6'1a! #Z6'#%#X�%;B� 5)(XJ+�

uuÔY#

�)(X=�J+#4@s(X-�1�@A1�#C;B� 1I�,1I5 #8� eI�)(+eI#%/∆uc =

IξηucA

uQ��*%1�b �)5 �>eI(c;�(+Jj(+6?(�5)(X��1�! #%*X� ���Hb)1�gX� JX�0=4�)/21IJ+� � &)5 1Xu• � =49)$'&)(v@A$�� {BP\TQM%{BU\�,N'[2U\T � OQ{B|BPQ}_T !)� � J+�)=I(+6l$`=4*%�)/�Jj(+& 6?(%/d(o! #%*X� ���`�)/d(^�)=4JX$`*X� 14;B5 #,=4J p�!B(=4�?9<(�!)� � J+�)=�6?( ¹dR%[2¸v�IRF¸�� R,º_T\}_�?¸�O2{B|BPQ}_T��

�������������������������������������������������������

pmpm pvpv

h1

h1h1h1

h2h2

h2

h2

h3

ρ1 ρ1

ρ1ρ1

ρ2

ρ2ρ2

ρ2 ρ3P P

PP1

P2

∆vc ∆vc∆vc� � ¢ û

C C C2

£ 9B� 6?(f�%u��?Ld�8(+�)(X6 J+1��)�)=4J+�)& 5 �i=v9)$'&)(4@A1�*%�f#C;B�)1��'� *%(X-:(H=4� 9)18! � � JX�)=46?(>5)(H�)(X*X5 $D!)#X*X�)�

¯ PQTQS�R,P���ñ w ;B� 1I�,�)*%(+-:18=4�?9B1:!)� � J+�)=�6?(>eI(HûD=v9)$'&)(4@s(>=�(>=v9)�)6,1>�%u���Xu ¥ !)=�#+9)$'J+5)�"!)� � J+�)=I(+6K$0JX14GE� ��JX$<L

pc = pa − pv + ρ1gh1 + ρ2gh2

£ �?9B(H! � �)J+�)=46?()LP = (pc − pa)A = (−pv + ρ1gh1 + ρ2gh2)A

��#X#%�X;<� 5)(+Jj(�J+14GE� ��Jc()Lzc =

pc − paρ2g

= − pvρ2g

+ρ1

ρ2

h1 + h2

�8(+!)(j;B5)(�Jj(+& 6?(>=4�?9B1P ] r 1I5 Jj(+�K! � � JX�)=46?(ik�$"=4*%�)/�=49)$'&)(v@A1I*X�)/d(�@A1

α = 90◦t

∆vc ≡ ∆zc =IcξzcA

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R z�

¥ 6'#q@A1�*%� 1v;B5)#%=4J8!)#XJ+! �)� J+�)=�6'(pv

*X1�bB(�#C;EeIgX� �B(ρ1gh1 +ρ2gh2

k�$HJ+14GE� ��J+$�!)#X*X�)���E*+9<(c;<(�(+!B=4#+9B$,JX5 �! � � JX�)=�(X6f/d(+-:�"#C;"(+JX/2#,=�7�1I�)=46'#Xh%p%J4@4ud! #XJ+!)� � J+�)=I(+6Bt4p,;B#Xg%� b 1:=41:5 1�h4(XJ+� *X5B(x*X�)1v;B5 #,=4J�eI(�=��?9)$

Pp'��JX#

e�5)(X& ��;<(.@A1�! �)1�J+! #,=4Jj(+*Xn:1I5 �D=�/21I�E=4�?9)1�! #XhI� 1��H(+5 ]~¸o� {<S´V+[Q¸6}2N��I¸_UQRE�?P2R'»BN"S�R,ØfN?�?T`V%S:R,P`VXT~[dR��N'[2T��%N��fM%UQN?WÙS:T\UQ¸BVi�?P2R'»BN_¸�M%TQS�N?�?T�¸Z{B»BMXTQP�WYN,¹�NaVXR^M%N S�R,Ø�¸v�CR^»)P2{"�IU2N�|BP2RX¹2U2{BVX�^M,N_Ø�¸l¸UQR,WQ¸Ö{%¹��CRX¹dUQNn}QTQUQRiP2N?|<UQ{B� R,ºoN_T\[QT�U2R,W\¸lS�{BS�R,U\�?UQ¸��CRX¹dUQNn}QTQU\¸lM%N�U2R%WQ¸�{%¹�{BV%N u�Ô\(+6'#j�'1Xp $#X*%#%/�=49)$'&)(v@A$fb 1x=41:;<#XgX�)J+�K�"5)1�h4(+JX� *X5)(H*%� 1v;B5)#%=4J>eC(

∆vc ] 7�� eI� & 6 �YpBJ+#Ee�5B(+& �f;<(:@A18Jj(+&)6'(D

� e�5)(j;J+14GE� ��Jj( p (j9B�f5 18JX� 1�g,(�! #,/21��)(XJ+�i=4�?9B$D$DJ+$D5B(+!)(j;B5 $DJj(+&)6 $ ] ;B#%*X#+n�5 #�@A1�! � �i=4�)(X& $'5)(+*,(+-.$E$'e�1�JX�f$#Xg%e�� �f5 1�h4(XJ+� *X5)$H*%� 1v;<5 #%=�J�eC(

∆vckê(X6'#H=�1:!)� ��$D/2#,/21�5 J+5)1�@A14;B5)(X& � 5 1ctvu2 �(X6?9)1Xp ¸�|�R,WlV%S:N �?PQN S�{o¹�N

V+T\[2NoO\PQTQ�?TQVXWYNK¹�RX[2¸v�CRK{%¹Z� R{}QUQ{<V+�?TlWYNFO2{B|BPQ}_T�WY{*��¸��CRK{BW\|�N?}aR,UQNi� {<S�� Rl}QU2{)}a×\¸��YT`¹�N��CR� R,UQ�%N?PZOQPQT\�?T2V+WYN_¸�|�R,WZTQVXOQ{%¹�� R,º_T\}_�,N u�?u ¥ !)=�#+9)$'J+5)�"!)� � J+�)=I(+6K$0JX14GE� ��JX$<L

pc = pa + pm + ρ1gh1 + ρ2gh2

£ �?9B(H! � �)J+�)=46?()LP = (pc − pa)A = (pm + ρ1gh1 + ρ2gh2)A

��#X#%�X;<� 5)(+Jj(�J+14GE� ��Jc()Lzc =

pc − paρ2g

=pmρ2g

+ρ1

ρ2

h1 + h2

�8(+!)(j;B5)(�Jj(+& 6?(>=4�?9B1P ] r 1I5 Jj(+�K! � � JX�)=46?(ik

α = 90◦t

∆vc ≡ ∆zc =IcξzcA

�a#X*%#%/"=v9)$'&)(4@A$�5 1C/d(q5 1v;B#%$?/2� r)( ] $:J+14GE� ��J+$:*+9B(j;<(.5)(+J+!)� � J+�)=I(+6Ypc! �)1�J+! #,=4Jj(+*Xn:1I5 �x=�/21I�.@A1��)=4!)�)(+*,(+5Yp�i7�� e�� &)6 �f5B(+!)(j;B5)(HJj(+& 6?(>=4�?9B1q@A18�)=4! #C;iJ+1�G>�)��Jj( u

¢ u ¥ 6'#��)/d(%/2#�=v9B$,&B(v@";<(_@A14;B5 $�! #X*%� �Î6 *,(X=41a*%� ��1�J+1�& 5)#%=4JX�Yp��)/d(Xb 1I/2#�#X5 #X9)� 6'#�=4� 9B(Z6'#+9)�)6,#Z�B/d(J+1�&)5 #%=�J+��6'#4@A1K=4$a$_6'#X5)Jc(X6 J+$`=I(F! #%*X� ���Yu w *j;B1"=410;<(v@A$a� eI�)(+eI�Ö=I(X/2#oeC(o=��?9)$

P2

u���eI�)(+eDeC(^=��?9)$P1

@A18!)#X� ! $'5 #E(X5)(j9)#%h4(+5f6'(X#�$"=v9)$'&)(4@A$F�Xu¥ !)=�#+9)$'J+5)�"!)� � J+�)=I(+6K$0JX14GE� ��JX$<L

pc = pa − pv + ρ1gh1 + ρ2gh2 + ρ3gh3

£ �?9B(H! � �)J+�)=46?()LP = (pc − pa)A = (−pv + ρ1gh1 + ρ2gh2 + ρ3gh3)A2

��#X#%�X;<� 5)(+Jj(�J+14GE� ��Jc()Lzc =

pc − paρ2g

= − pvρ3g

+ρ1

ρ3

h1 +ρ2

ρ3

h2 + h3

�8(+!)(j;B5)(�Jj(+& 6?(>=4�?9B1P ] r 1I5 Jj(+�K! � � JX�)=46?(ik

α = 90◦t

∆vc ≡ ∆zc =IcξzcAw *j;B1�*,(jGE�f�B=4Jj(H! � � &)(>6?(X#�! #C;a�Xu

û)u ¥ !)=�#+9)$'J+5)�"!)� � J+�)=I(+6K$0JX14GE� ��JX$<L

pc = pa + pm + ρ1gh1 + ρ2gh2

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R �

£ �?9B(H! � �)J+�)=46?()LP = (pc − pa)A = (pm + ρ1gh1 + ρ2gh2)A2

��#X#%�X;<� 5)(+Jj(�J+14GE� ��Jc()Lzc =

pc − paρ2g

=pmρ2g

+ρ1

ρ2

h1 + h2

��1�5 Jj(X�f! �)� J+�)=�6'(:@A18$0J+14GE� ��J+$ �  [Q¸6}2N��"WYN,¹�Nq�CR¢¡CN V^¸ZWY{<Sô|B[dN,¹dN`UQN?�?OQPQT\�?T2VXN?W�T\[2T�O2{)�?OQPQT\�?T2VXN?W�¸ZWY{BU\�,N?WQ�?¸�VXN`PQN?|<U2{BSOQ{B|<P\}_T

£¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤££¤£¤£¤£

¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥¥¤¥¤¥¤¥

¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦¦

§§§§§§§§§§§§§§§§

¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨¨¤¨¤¨¤¨¤¨

©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©©¤©¤©¤©¤©

pm

pmpv

pv

pa

P

P1 P2

� �C

£ 9)�)6'("�%u �)L £ � 9B(>! � � JX�)=46?(>$0=49)$'&)(v@A$>;<1A@s=4JX*%(>! � �)J+�)=46?(>6'#4@A�"*+9<(c;<(H$Dh4(%=4$D5)(H�)(X*X5 $0! #X*%� �

• ��� �>#C;B�)1��'� *%(X-�$x=��?9)1�6'#4@A#,/Fh4(%=d;B1�9)$C@A1�5)(:5 1�6)$�! #%*X� �fpj! � 1�JX! #%=�Jc(X*+n8(X/2#�;<(�@A1�! #Xn:1q! � �)J+�)=46?(�$h4(%=4$M� {BS�{�¡XR,UQ{ p+Jv@4u2$�=�*%(+6'#�@~Jj(+& 6 �H$xh4(%=4$8! � �)J+�)=�(X6��B/d(��B=4J+$�*X� 14;B5 #%=�J ] 5)(�=49)� r �"�Xu �:$x=v9)$'&j@s(c$��=�*X$I;<(E$K=4$I;B$�@A1

pmp<Jv@4u�$K=v9)$'&)(4@A$K��@A1H=4*%$�;�(

pvu���#X��J+#E$"=�*%(X6,#�@.Jj(X& 6 �F�)(X*X5 18!)� � J+�)=I(+6F�B/d(

�)=�J+$0*%� 1v;<5 #%=�J p)=4� 9B(H! � � J+�B=46?(�;B149B$I@A18$0J+14GE� ��J+$f!)#X*X�)���Yu•  �#%h�#X*%#X� ] =4�?9)$f$%=I/21�� 1I*%(X/2#H$'*X1I6f#C;ih4(X=I(�6?(H! #X*%� ���~u• ��#%� �)=4JX�)/2#K! #XJ+!)$,5)#"�B=4J+��#Xg%�)(+eC(+ro6?(X#0��$o=v9)$'&)(4@A$FJ+1I& 5 #%=�J+�Yp)@A1v;B5)(X& � 5)(�kA�%u��C¢%t4uH��(K=v9B$,&B(v@A1�*%15)(E=49)� r �f;B#%gX�j@s(>=�1XL

P1 = pmAJv@4u

P2 = −pvAð.�';B�B/2#�;<(:b 1I/2#8$H=v9)$'&)(4@A$H��;B#%gX� JX�H5)1�h4(+JX� *X5 $�*X� 14;B5 #%=�J�eI(8=4� 9)$

P2 ] e�5B(+& ��;<(.@A1�=�J+*%(X� 5 �E=�/21I�;B1s@s=4J+*,(�=��?9)1:=4$'! � #%Jc(X5Yp'Jv@4ud$>#X6'#+9B5 #%/_*,(+ev;<$%3>*+9<(c;<(�*X1�b)�D! � � JX�)=46'#%/`5 1�hI#�$E*%(Xev;B$,3?$H$>=�$I;B$�p,!B(=��?9B(�;B1�9)$C@A185)(HeI(+JX*%(+�B(+&f=I(>=4! #Xn:5 1x=�J+�)(X5 1 �

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ?¼

G�H+ª«H�G KMLON1L�PRQ�Xuq��3'�?;B�)(c$C9)� & 6'# ] ! 5)1�$?/d(+JX=�6'#%/�r)�?9)� 5';<� $^/2#XG�1�;<(K=41E6 � 1Ib 1>6?9)� !_g%1�eHJ+� 1�-�( uD�8(j;<(f5)("6?9)� !^;B1�9)$C@A1

=4�?9<(Fp)6?9)� !i=418=�! $X��Jc()u w ;B� 1v;<� J+�i! #,/21��)(X-�1�6?9)�)!)( p)(X6'#�@A1�! �)#%/21�5B(E=4Jj(+-�(

kê(,tD� eI#XJ+1I�)/d=46?(k�g'tD! #X9)� J+�)#X!)=�6'(��#XeI5)(+J+�K=4$f=v9)14;B1�b �K! #C;�(+r �YL

GpFpDpHpnppa

�ρu

�?u�ï21�eI1�� *%#%(+�:=�(q6'#X5B=4Jj(+5 JX5 �)/f5 � *X#%#%/f$Cn�()pc=�! #4@A1I5.@A1�=�(�3?�';B�)(c$C9)�)& 6'# ] ! 5 1�$?/d(XJX=46 �B/f!)#4@s(+&B(+*%(X& 1I/0! � � JX�)=46?( u��e�5)(j;�$Cn8(�*+9<(c;<(.5)(XJ+! � � JX�)=�(X6

pm0 = 0.2 baru w ;<� 1v;B�)J+�8eI(q6'#+9B� 6'#q=�1d6?9)� !8!)#C;B� h�5)1�(+6'#�5)(+JX! � � J+�B=�(+6

$Cn�(o$��)1�e�1I� *X#,(+� $�5)(X�)(X=�J+1D5)(pm1 = 0.4 bar

u^ðq(Xev;B$,3a/21�-�(^=�Jc(X-�1f� e�#%J+1��B/d=46 �Yuo �(+JX�`! #C;<(Xr �`=4$�LD = 100 mm

pd = 40 mm

pH = 1 m

ph = 120 mm

pG = 50 N

ph0 = 100 mm

ppa = 1020 mbar

�ρ = 900 kg/m3

u¢ u�ï21�eI1�� *%#%(+�8!)� � 6?(+eC(+5H5)(�=v9)� r)�H=I(X=4JX#4@A�>=41q� e�;B*X1.6'#%/2#%� 1XuQ�l9)1�*%#4@\6'#,/2#X� ��5)(j9B(Xe�1q=41qJ+1I& 5 #%=�J+�>h�$,=4J+� 5)1ρ1 = 1000 kg/m3

�ρ3 = 3000 kg/m3

pq;B#X6`@A1^*%� 1v;B5)#%=4Ja5B(+J+! �)� J+�)=�6'(Ö� eI5)(c;�JX1�& 5 #,=4J+��h�$,=4J+� 5)1ρ1

pm = 0.05 baru.�´;B1I=�5 #4@�6'#%/2#%� �H@A1�J+1I& 5 #,=4JDhI$%=4JX� 5 1

ρ2 = 800 kg/m3� eI5)(c;o6'#�@A1x*+9B(j;<(>! #%J+! � � JX�)=�(X6

pv = 0.1 baru��ú! � 1Ih��)(j;B5 #,/�e��?;B$F� eC/21��'$F6'#%/2#%�)(0=�1H5B(j9B(Xe��o!)�)(+*%#X$'h4(+#%5 �F#XJX*X#X�F;B�)/21I5 e��j@s(

2a × bp

6'#4@A�i@A1KeI(XJ+*X#%� 1�5aeC(+J+*,(+�)(X& 1I/�#%gX� JX5 �)/�#%6,#^#%=�106'#4@s(o! � #+9<(+e��Ö6 � #%e0Jj(+& 6)$ w ��$'! �)(X*X5)(D@A1f5)(o�)(+*,(r � J+1�GH()u w ;B� 14;B� J+��/2�)5 �)/d(j9B5 $l=��?9)$

F!)� �Z6'#4@A#4@>b 1ieC(+J+*,(+�)(X&leC(+J+*,(+�)(XJ+�l#XJ+*%#X�Yu��#Xe�5B(+J+�Z=4$�

=v9)14;B1�b)�"!)#C;<(+r �~La = 200 mm

pb = 1 m

ph1 = 1.5 m

ph2 = 1 m

�α = 45◦

uû)u���� 1�hI�)(c;<5 �Ee��';K� eI/21I�'$�;<*X1�6'#,/2#X� 1�$>6'#4@A�B/d(�=41�5)(j9<(+e��0J+1�& 5)#%=4J�h�$%=�J+� 5 1

ρ@A1�#XgX9)� 6?(8! �)(X*X#%$,h4(X#X5 �)6'()p

��� �)� 5 1bk�$'! �)(X*X5 #o5)(F�)(X*%(X5ar � J+1�GH(%t��Ö/2#XG�1"=410#%gX� Jj(XJ+�`#X6'#F#,=41f6,#�@s(F! �)#+9B(Xe��a6 � #%e0Jj(+&)6 $ w �

$'! �)(+*%5)(�@A1H5)(D�B(+*%(X5Fr � JX14GH()u w ;<� 1v;B�)J+�^!)#+9)#XGH(v@�Jj(+&)6,1 w k��)e��)(X& $'5)(+J+�^�)(X=�J+#4@s(X-�1ht�! � �^6'#%/21�b)1

! � 1Ih��)(j;B5 �`eI�';lg%� J+�`$_! #X9)#+G>(v@A$�6'(X#F5)(^=v9)� r)�Yp�(+6'#E@A1H1 = 7 m

�H2 = 3 m

ulk�¬�­�®$¯�°*±$¯³²µ´4¶z´o°o´r·1¸­µ¹�®o°o¹�º¼»*´$½¤¾�¿"¿�À*ÁÃÂ

Ä1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1ÄÄ1Ä1Ä

Å1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1ÅÅ1Å1Å

Æ1Æ1ÆÆ1Æ1ÆÆ1Æ1ÆÆ1Æ1ÆÆ1Æ1ÆÆ1Æ1ÆÆ1Æ1ÆÆ1Æ1ÆÆ1Æ1ÆÆ1Æ1ÆÆ1Æ1ÆÆ1Æ1Æ

Ç1Ç1ÇÇ1Ç1ÇÇ1Ç1ÇÇ1Ç1ÇÇ1Ç1ÇÇ1Ç1ÇÇ1Ç1ÇÇ1Ç1ÇÇ1Ç1ÇÇ1Ç1ÇÇ1Ç1ÇÇ1Ç1Ç

È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1ÈÈ1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1ÈÈ1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1ÈÈ1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1È1ÈÉ1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1ÉÉ1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1ÉÉ1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1ÉÉ1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É1É

h =?

H1

H2

Ê

ρρ

ËÍÌoÎnÌ�Ï�Ì�Ð�Ñ

Ò ¹�ÓÔ¹�ÕÔ¹�Öh =

H2

1 + H1H2 + H2

2

3(H1 − H2)= 2.633 m

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R 'ñ���µ× T�Ã"ÓoÀ ÆFÇKÃKÊ8ÃiÉ�ÏDÀ ÂiÀÎÏFÇKÃi�xÐ Æi���HÇVU

��#,=�/d(+JX�)(v@s/2#�@A1v;B5 $i6)� � *X$K! #%*X� �ô6'#4@s(8@A1H$i6'#X5 Jj(X6 J+$i=�(0J+1�& 5)#X��b)$ih�$%=�J+� 5 1ρkm=v9)�)6'(^�Xu �%t4u8��1�6?(�@A1HJc(

! #%*X� �¤=��)/21�J+�)� & 5)(>$f#C;B5 #%=�$E5)(>�)(X*%(+5x− z

u

Ø�Ø�ØØ�Ø�ØØ�Ø�ØØ�Ø�ØÙ�Ù�ÙÙ�Ù�ÙÙ�Ù�ÙÙ�Ù�Ù

0

d~P

d~P

d~P

dA

z

z

z

x

x

y

A

ρ

dAx

dAz

~n

~n

−αγ

d~Px = −dPx~i

d~Pz

£ 9)� 6?(K�Xu � L £ �?9<(�!)� � J+�)=�6?(>5)(H6 � � *%$0! #%*X� �

�9)1I/21I5 Jj(+� 5)(_=��?9B(_! � �)J+�)=46?(d~P

@A1F#C;B�)1��'1�5)(�� eI�)(+eI#%/ôkA�Xu¡�C�%tvu��Y(^�B(+e49B� 6 $�#C;Z�)(X*X5 �?3l! #X*%� ���~p�6'#C;6 �)� *X�?3x!)#X*X�)����*%1�6 JX#X� �x5)#X�)/d(+9B(�1�9)1I/21I5 Jj(+� 5)�?3x! #X*%� ���

dA5 �)=4$8�B=4J+#%h�! �B(+*XrB( pcJj(+6'#�;�(�b 1.��1�9)1I/21I5 Jj(+� 5)1

=��?9)1dP

gX� JX�f�)(Xe49)�)& � J+#%h8! �)(+*%r)( u��#,=�/d(+JX�)(v@s/2#q5 1I6 $�!)� #X� eI*X#Xn:5 $�149)1C/21�5 Jj(+�)5 $�!)#X*X�)�

dA! #X*%� ���

A& �+@A�q@A1�! #+9B#+GH(4@B#C;B�)1��'1�5�6'#X#%�X;<� 5)(+JX#%/

zudð�1I6 J+#%�

~nJ+18! #%*X� ���"=�1�/2#XG�1�5)(X! �)=�(XJ+�K6?(+#)L

~n =~i cos∠(~n,~i) + ~k cos∠(~n,~k) =~i cosα+ ~k cos γks�Xu¡���%t

h�;B1�=41�$'h¾9)#%*X�α�γ/21��)1�$>! #XeI� J+� *%5 #%/_/d(XJ+1I/d(XJ+� & 6'#%/a=I/21�� $ikm=�/21��f=4$'! � #%Jc(X5D=�/21I� $>#XgX�)Jc(X-:(�6?(XeI(jn�6 �E5)(

=I(+J+$ t.$D#C;B5)#%=4$05)(H! #XeI� J+� *,(+5K=�/21��K#C;BhI#X*,(+�)(4@A$,b)�?3E6,#%#X�%;B� 5B(+J+5 � 30#%=I( u�9)1I/21I5 Jj(+� 5 $�=4� 9)$

d~P6'#�@s(Ö;B149)$C@A1`5B(Z! #%=I/d(+J+�B(+5 $Ù! #X*%� �

dA/2#XG�1I/2#�� e��)(Xe�� JX��! � 1�6'#�-:1�5)1_;B*X1

! �)#4@A1�6)r �j@A1_k�! � �B/21�J+�)/2#o;<(>@A1f! � #�@A1�6 r �+@s(F#X*%10=4� 9)105)(x ] #%=4$�5 1Ih4(+J+� *%5)( p\;B#%6i@A105B(

z ] #%=�$o! #%e�� JX� *X5)()p26'(X#�f;<(>=4$f6'#%=4�)5 $%=��0$'h¾9)#X*,(

α�γ cosα > 0

�cos γ < 0

tvL

d~P = d~Px + d~Pz ≡ −dPx~i+ dPz ~k = −ρ g z (~i cosα+ ~k cos γ)dAkA�%u��$�+t

��e�! #,=v9)14;B-�1q@A14;B5)(X& � 5 1x=49)1v;B�~L

dPx = ρ g z (dA cosα)︸ ︷︷ ︸

dAx

⇒ Px = ρg

Ax

z dAx ⇒ Px = ρg zcxAxkA�Xu¡���,t

h�;B18=�$<LAx ] ! � #�@A1�6 r �+@s(>! #X*%� ���

A$0! �B(+*Xr)$

x ] #%=4185B(��B(+*%(X5 y − z

zcx ] 6'#X#%�X;<� 5)(+Jj(�J+14GE� ��Jj(>! #X*%� ���Ax

/21I� 1�5)(H#C;F=v9)#%gX#C;B5)1:!)#X*X�)���Yu

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ,þ

ð.�)(XJ+�)/2#E=41x=I(c;<(H5)(H� eI�)(+eEkA�%u��$�+t4uÛ��e�-:1�h4(>=v9B1v;B�f;<(:@A18! �)#4@A1�6)r �j@s(dPz

#C;B� 1I�'1�5)(�� e��B(+e�#,/qL

dPz = −ρ g z (dA cos γ)︸ ︷︷ ︸

−dAz

⇒ Pz = ρg

A

zdAz ⇒ Pz = ρgVks�Xu¡���,t

h�;B1�@A1V

eC(+! � 1C/2� 5)(^6'#4@s(o=�1D;<#XgX�+@s(o! �)#4@A1�6)J+#X*,(+-:1I/�6 � �)*X1"! #X*%� ���Ö$a*X1��)J+� 6?(j9)5)#%/�! �)(X*Xr $�;<#^=v9B#XgX#C;<5 1! #%*X� ���Yud�8(+!)(j;B5)(�9)� 5)�j@s(E=4�?9B1

Pz!)� #+9B(Xe��"6 � #%e:JX14GE� ��JX1xeI(X! � 1I/2�)5 1

Vu

�8(X6,#>=41�! �B(+6 J+�)& 5 #E#C;B� 1I�'$I@A$06,#,/2! #X5)1�5 J+1Px

�Pz

g%� b 18! �)� 6?(+eI(X5 #H5)(>=v9)14;B1�b �B/Ö!)� �)/21��)�)/d( u¯ PQTQS�R,PÜ��Ý�6� w ;B�)1v;B� JX�l=��?9)$a! � �)J+�)=46?(^6'#4@s(F;B149)$C@A1"5)(^6 � � *%$a! #X*%� � #%g+9)�)6'(i;B149<(o#,/2#XJj(+&)(^=�7�1I� 10eC(! �)�)/21��ieC(H=49)� 6'1Xu

R

ρ

x

zH

zcx

Ax

Þ6ßzàµá�àãâ Ï"Ì�ä â{àDåãæ{ß{à�ç ä æ Ï æ{å ÐvÌæpè Îzé è�ê Ì�ävëoìJërí àxç ä è ì à Ðvî æ ì à

ï ä è ì à Ðlî æ ìð̤Ð{ä æ{ê*à�çzè�ê ä{ñ æ ëxç ä{Ì ê î{ë

Cx

Dx

Px

£ 9B� 6?(f�%u��"� L w ;B� 1I�'� *%(X-�1x! � #�@A1�6 r �+@A1�=4�?9B1:!)� � J+�)=�6?(>5)(>6 � � *%$D! #%*X� �ú#XgX9)� 6?(�#%/2#XJj(X&)(H! #+9)$,=�7�1I� 1

��� *%#qb 1I/2#�#C;B�)1v;B� JX��3%#X�)� e�#%5 Jj(j9)5 $�6,#,/2! #X5)1�5 J+$:=4�?9B1�! � � JX�)=46?( uQ��#X*%� �Ax

6'#4@s(�=�1Q;B#Xg%�j@s(q! � #�@A1�6 J+#%*%(X-�1C/6 �)� *X1.! #X*%� ���8$

x ] !)�)(+*%r $�! � 1v;�=4Jj(+*Xn�(�@A1v;B5 $��)(X*X5 $�! #%*X� ����=4� 9B(�!)� � J+�)=�6?(�6'#4@s(�;B149)[email protected])(�JX$8�)(+*%5 $�! #X*%� �=�1x/2#+G�1�#C;B�)1v;B� JX�"!)#%/2#Xb)$D� e��B(+eI(Kks�Xu¡���%t��?9B�"�)e��)(XeI("kA�Xu¡�I¢,tvp?Jv@4u

Px = (pcx − pa)Ax ≡ ρg zcxAx

��(H6'#X5)6 � 1�Jj(X5f! �)�)/21��F=I(H=49)� 6'1�Jj(E=4�?9B(�@A18#C;B�)1��'1�5)(�� eI�)(+eI#%/qL

Px = ρ g (H +R)R2π

��� �i#C;<� 1��'� *,(+-.$0*X1I� J+� 6?(+9)5 1x6'#%/2!)#X5 1I5 J+1Xp<! #%*X� � b)1I/2#D! #C;<149)� JX�F5)(H;B*,(�;B149<( p<6'(X#E��J+#x@A1�! � �)6'(XeI(X5 #5)(E=49)� r �_�%u��%�Xu

�*X�';B1I5 J+5 #x@A18;<(0b 1�5)(Dh�#X�)-��K;<1�#0!)#X*X�)���";B1�9)#X*,(+J+�F=��?9B(D6'#�@s(�@A1�$,=�/21��)1�5)(E*%1�� JX� 6?(j9)5 #05)(+*%� ��1%pB�FJX#=��?9)$>b 1I/2#�#Xe�5B(+& � JX�0=�(

Pz1ud�8(:;<#X-:�H;B1�#�! #X*%� ���H;B1�9)$C@A1�=��?9B(86'#4@s(�@A1�$%=I/21�� 1I5)(8*%1�� J+�)6'(+9)5 #x5B(+5 �?GH1Xp,�EJ+$

=��?9)$�b)1I/2#�#XeI5)(+& �)J+�E=�(Pz2

uQï21Ie�$C9)J+$C@A$'b)(:*X1I� J+� 6?(+9)5)(�=4�?9<(�6'#�@s(�;<149)$C@A1�5)(�! #%6?9)#%!)(+r�gX� b 12@A14;B5)(X6'(:�)(Xe49)�)r �#%*X1.;B*%1�=4�?9)1��EgX� b 1�$%=�/21I� 1�5B(�*X1��)J+� 6?(j9)5)#85)(+5 � G�1XpI@A1���@A1�6'(X#8��J+#8b)1I/2#x*%�';B1�JX�Yp%�)5 J+1�5)e�� J+1IJ�=��?9)1

Pz2*X1Ib �

#C;i� 5 J+1I5 e��)J+1�Jj(D=��?9)1Pz1

u

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R n�

R

ρ

x

zH

zcx

Ax

x

x

CxDx

Px Pz1

Pz2

Vz1Vz2

£ 9)�)6'( �Xu¡�X�XL w ;B� 1��'� *,(+-:1Z*X1I� J+� 6?(j9B5 1�! � #�@A1�6 r)�j@A1�=4� 9)1�! �)� J+�)=�6'(´5)(�6 � �)*X$�!)#X*X�)� #Xg+9B� 6?(�#%/2#XJj(X&)(! #X9)$%=I7�1�� 1

��5 JX1�5 eI� J+1�JX�F=4�?9B(Pz1

�Pz2

=418#C;B�)1��'$C@A$D! #%/2#%b $D� eI�)(+eC(KkA�Xu¡���,tvudÔ~(X6'#>=�18eI(>=4� 9)$Pz1

;B#Xg%�j@s( L

Pz1 = ρgVz1 = ρgVz1 = ρ g

[1

2R2π(H +R) − 1

4· 4

3R3π

]

= ρgR2π

[H

2+R

6

]

��(X! � 1C/2� 5)(Vz1

@A18@A1v;B5B(+6?(K�)(+e�9)� r �_eC(+! � 1C/2� 5)(K& 1IJ+*X�)J+� 5 1fr �?9)� 5?;B�)(K! #X9)$'! � 1�&)� 6?(R

�a*X�B=4� 5 1H + R

�& 1IJ+*X�)J+� 5 1�9)#%! J+1�! #X9)$'! � 1�&)� 6?(

Rud�8(+! #,/21�5 �B/2#�@A#%��@A14;B5 #%/`;<(�@A1�#X*%(�=4�?9<(�$,=�/21I� 1�5)(�*X1I� J+� 6?(+9)5 #H5)(+*%� ��1%u

��5 JX1�5 eI� J+1�JD=��?9)1Pz2

@A1XL

Pz2 = ρgVz2 = ρgVz2 = ρ g

[1

2R2π(H +R) +

1

4· 4

3R3π

]

= ρgR2π

[H

2+

5R

6

]

�8(X#���J+#8/2#+G�1C/2#�*X�';B1IJ+�>� 5 J+1I5 e��)J+1�Jx=��?9)1Pz2

@A1q*%1�b �H#C;E=��?9)1Pz1

pX!B(�@A1��)1�e�$C9)JX$I@A$'b)(�=��?9B(Pz

$%=I/21�� 1I5)(*%1�� J+�)6'(+9)5 #>5)(X5 �?G�1��K-:1�5K� 5 J+1I5 e��)J+1�J8@A1%L

Pz = Pz2 − Pz1 = ρgR2π

[H

2+

5R

6

]

− ρgR2π

[H

2+R

6

]

=2

3ρgR3π

£ /21I�D*X1I� J+� 6?(j9B5 1�6'#%/2! #%5 1�5)J+1�=4� 9)1�6'#4@s(�;<149)$C@A$>5)(85 1I6 $>6 � � *%$�!)#X*X�)�´/2#+G�1I5 #x#C;B�)1v;B� JX�D�E5)(�=49)1v;<1�b �Yp@A14;B5 #%=�Jj(+*%(X5f5B(+& � 5~L

¥ 6'#�=�1oeC(+! � 1C/2� 5)(�6,#�@s(`=41i;B#Xg%�j@s(`! �)#4@A1�6)J+#X*,(+-:1I/�6 � � *%1o! #%*X� ���l;B#Ö=49)#Xg%#C;B5 1! #%*X� ����J+1I& 5 #%=�J+�l6,#�@s(D@A1i=I(o-�#,/´$a6'#X5)Jc(X6 J+$�7�#%�)/2� �)(�=�(^5 1I#X6 *,(+��1I5 1f=�J+�)(X5 1J+1I& 5 #%=�J+�Yp #%5';<(�@A1x=�/21I�i=4�?9)18$,=�/21I� 1�5K*X1I� J+� 6?(+9)5 #>5)(+*%� ��1Ekm=4�?9B(

Pz1t4u ¥ 6'#E=41�Jj(

eI(X! � 1C/2� 5)(�7�#%�)/2� �)(�=�(�#X6 *,(+��1I5 1�=�J+�)(+5)1�! #%*X� ���Yp,=I/21��f*X1��)J+� 6?(j9)5)1�6'#%/2! #%5 1�5)J+1=4� 9)1q@A1�$%=�/21I� 1�5K*%1�� J+�)6'(+9)5 #>5)(X5 �?G�1Ekm=4�?9B(

Pz2t4u

 �(X6?9)1Xp�$,6)$,!)5 $�=4�?9B$l!)� � J+�)=�6?(Ö=I/2#Ö�)(Xe49)#XGE�?9)��5)(a;B*X1_/21��'$,=4#Xg%5 #`$'! �)(X*X5 1o6,#,/2! #X5)1�5 J+1%p�6'#4@A1�& � 5)1=�$,&)14n�5 ��=��)=4JX1I/Ù=4�?9B(lò ] J+#H@A1>� eH�)(+e�9)#Xh4(0��J+#K��J+#�@A1>6 �)� *%(f! #X*%� ��=4�)/21IJ+� � &)("$F#C;<5 #%=�$"5B(0*%1�� J+�)6'(+9)5 $x − z

�)(+*,(+5K6'#4@s(E!)� #+9B(Xe��i6 �)#Xex-:1�5 #fJ+14GE� ��J+1v� � {Ö×\R�»)TQ�?T�V+[Q¸6}2N��"¸ZV+|<T2S M,N,¹dN � T2S�N`WY{*��TÙ×\R�»BT\�?TP2NÍ~�R,U\T�¸Ö{BW\|<T\PQ¸Ö{B|�{¼¡fW\¸�P2VXN u

óµ� � '2ôo& 0 ' ��!r#Dõ*� %+&Û'2& .$#µ($# , . & !$�".*� %+&�'�&2ö"÷ø÷ù% &-, .") %6ô #µõ ö � '�& .*#µú��"ûo# '-÷ õ &-ü .*� ' � '2ôo& 0 ' ��!r# 3

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R z5

Px

Pz

P

£ 9B� 6?(f�%u��c�?Ldï21�eI$C9)J+$C@A$'b)(>=4�?9B(H! �)� J+�)=�6'(

Ô\(>� 1�eI$I9BJ+$C@A$,bB(E=4�?9B(H! �)� J+�)=�6'(:@A1q@A14;B5)(X6'()L

P =√

P 2x + P 2

z =

[ρ g (H +R)R2π]2+

[2

3ρgR3π

]2

= ρgR2π

(H +R)2 +4R2

9

¯ PQTQS�R,PÜ��Ý�6� w ;B�)1v;B� JX�l=��?9)$a! � �)J+�)=46?(^6'#4@s(F;B149)$C@A1"5)(^6 � � *%$a! #X*%� � #%g+9)�)6'(i;B149<(o#,/2#XJj(+&)(^=�7�1I� 10eC(! �)�)/21��ieC(H=49)� 6'1Xu

R

ρ

x

z

H

zcxAx

Þ6ßzàµá�àãâ Ï"Ì�ä â{à�åãælßzàDç ä æ Ï æ{å ÐvÌ æpè Îzé è�ê Ì�ävëoìJërí àxç ä è ì à Ðvî æ ì à

Cx

Px1 Px2

4R

3πAx1 = Ax2

Ax1 Ax2

Vz

£ 9B� 6?(f�%u��C¢ L w ;B� 1I�'� *%(X-�1x! � #�@A1�6 r �+@A1�=4�?9B1:!)� � J+�)=�6?(>5)(>6 � � *%$D! #%*X� �ú#XgX9)� 6?(�#%/2#XJj(X&)(H! #+9)$,=�7�1I� 1

��#X*%#%/l! � �)/21I� $�p�! � �?9B� 6'#%/Z#C;B�)1��'� *%(X-:(Px

p�6)� � *X$K! #%*X� �ô! #X*%� �ú;B1�9)�)/2#"5)(>;<*%(>;<149B(0�o#C;B� 1I�'$I@A1C/2##C;BhI#X*,(+�)(4@A$,b)1�! � #�@A1�6 r)�j@A1

Px1�Px2

6'#4@A1:;B1�9)$C@A$D5)(>Jj(�;B*%(�;B1�9B(H6 � � *%18! #X*%� ���~u£ �?9B1

Px1�Px2

=4$D�)=�J+#Xh8�)5 J+1�5)e�� J+1IJj(E�f#%5 1x=4$�p?5)(H#,=45 #%*X$FkA�%u��"�%t�#C;B� 1I�'1�5 1�� eI�)(+eI#%/qL

Px1 = Px2 = ρg

(

H +4R

)R2π

2

�8(X6,#f=4$i#X*%1H=��?9)1>=4$'! � #%J+5 �?3o=I/21�� #%*%(f� 1�eI$C9)J+$C@A$'b)(f=��?9B(f! � � JX�)=46?("$x ] ! �)(X*Xr $ib 1HgX�)J+�0@A14;B5)(X6'(05 $C9)�Yp

Jv@4LPx = Px1 − Px2 = 0

� #%h¾9)��=�/2#^;B#_�)=4JX#XhieI(X6?nq$,&)6'(o;<(o;B#j�'1C/2#��^@A1v;<5 #%=�Jc(X*X5 #,/�(X5)(j9)�)e�#%/quZ�8(X�)/21Xp.=4*%(X6'#4@�1�9)1I/21I5 Jj(+� 5)#4@! #%*X� ���

dA1

=�1�/2#+GH1x! � �?;B� $,GE� J+���F#C;Bh�#%*%(X�)(v@A$'b)(E=��)/21�J+�)� & 5)(f149)1C/21�5 Jj(X� 5)(D!)#X*X�)�dA′

1

6'#4@s(f=41�5B(j9B(Xe��5)(>�B=4J+#�@�3,#X� � eI#X5 Jj(+9)��kê�)=4JX#4@q� e�#%g%(+�B=46'#4@�!)#X*X�)��� � tH5)(H6'#4@A$>;B1�9)$C@A1�=4� 9B(

dPx�B=4J+#%h8� 5 J+1I5 e�� JX1�Jj(E�K! �)(X*Xr)()p

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R nF

(+9)�`=4$'! � #%J+5 #%h0=�/21I�)( u ¥ 6'#o=�$?/2� �)(X/2#Ök�� 5 JX1�h��B(j9)�)/2#'t8! #ir)149)#�@x! #X*%� ���Ap\;B#%gX� b)10=41>;�(H@A10� 1Ie�$C9)J+$C@s(+bB(

=��?9B(>! � �)J+�)=46?(H$x!)�)(+*%r $8@A1v;<5)(+6?(H5 $C9)�Yu

��#,/2! #X5 1I5 Jj(Pz

@A18#C;<� 1��'1�5B(�� eI�)(+eI#%/qL

Pz = ρgVz = ρg

(

R2πH +1

2· 4

3R3π

)

= ρgR2π

(

H +2R

3

)

�8(X6,#�@A1Px = 0

p?�)1�e�$C9)JX$I@A$'b)(E=��?9B(H! � �)J+�)=46?(P@A1P = Pz

u�

G�HJýDH�G KMLON1L�PRQ�Xu���#+9)$C9)#%! Jj(X=�J+��! #X6?9B#X!)(Xr k

R = 1 mtKeI(XJ+*%(X�)(�6 � $,GE5 ��#XJX*X#X��$ÙgX#%& 5 #%/ eI�';B$Ù� 1�eI1�� *%#%(+�B(l=I(Ö;B*X1

J+1�&)5 #%=�J+�`kρ1 = 800 kg/m3

�ρ2 = 1000 kg/m3

t4uqð.�)=�� 5)(>5 �)*X#%(>JX1�& 5 #,=4J+�>@A1H = 2 m

p<(>5)(+JX! � � J+�B=�(+6$D! � #,=4J+#%� $0�)e�5)(j;KJ+1I& 5 #,=4J+�>@A1

pm = 5 kPau w ;B� 14;B� J+�F=��?9)186'#4@A1��)=�J+14GE$f�K=I/2� & $feI(+*%� J+-:186'#4@A�)/d(:@A1

! #X6 9)#X!)(Xrf!)� � & *%� ��b 1I5feC(He��';F� 1Ie�1��)*X#%(X�)( u�?u w ;B� 14;B� J+�E=4� 9)1�6'#4@A1q#X! JX1�� 1Ib $C@A$�r �?9B� 5';B� �)& 5 �E! #%6?9)#X!B(+r ¥ ��� � �)5 1

L�>=I7�1�� 5)��! #%6?9)#%!)(+rH! #+9B$,!)� 1�& ]5 � 6?(

RuQ �(+J+�K=4$0=49)1v;<1�b �K! #C;<(Xr �YL

R = 0.2 mpL = 2 m

ph = 2 m

ppm = 0.1 bar

pρ = 1000 kg/m3

u

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R zX

���ãþ T�Ã"ÓoÀ Æi�xÊ8ÃiÉ�ÏDÀ£ �?9<(K! #XJ+�B=46?(H@A1E$�=�J+*%(X� �a=4�?9<(K! � � J+�B=46?(K6'#4@s(f;B1�9)$C@A1D5)(K5 1I6,#KJ+1�9)#F6'#4@A1�@A1E!)#XJ+! $'5 #o�?9)�^;B149B�)/2� & 5 #

! #%J+#X! n:1I5 #E$D5 1I6,#�@�J+1I& 5 #,=4J+�Yu

z A

V

Pv

D

C

ρG

£ 9)� 6?("�Xu¡��ûBL £ �?9B(H! #XJX�)=46?( ] =4� 9B(H! � � J+�B=46?(>6'#4@s(�;B149B$I@A185B(�!)#XJ+#%!?n:1I5 #EJ+1�9)#

£ �?9<(�! �)� J+�)=�6'(�6'#4@s(8;B149)$C@A1�5)(�! #XJ+#%!?n�1�5 #HJ+149B#�@A1:#C;B� 1I�'1�5)(8;B#Xg%� #�! #%e�5)(XJ+�)/a�)e��)(Xe�#%/�kA�Xu ¢%t4p?=�J+�B/`��JX#@A18$f#X*X#,/`=v9B$,&B(v@A$0!)#X*X�)�

A! #E6'#�@A#4@.=�1�� 5 J+1Ih��)(+9)�K! #+n�18! � � JX�)=46?(>eI(XJ+*X#%� 1�5B( L

~P = −∮

A

p~ndA = −∮

A

(pa + ρgz)~ndAkA�%u �v�%t

�8(X#:��J+#�@A1�!)#Xe�5B(+J+#8� e�J+1�#%� �j@A1�! #Xn�()p%�)5 J+1�hI�)(j9D!)#85 1�6'#4@deC(+J+*%#X� 1I5 #4@2! #X*%� ���E=�1�/2#+G�1�J+�)(+5B=�7�#X�B/2�)=�(XJ+�$f� 5 J+1Ih��)(+9K!)#HeC(+! �)1I/2� 5 �K6'#4@s(�@A18#%h��)(X5 � & 1I5)(�JX#%/Ö! #X*%� ���_k�J+1I#X� 1C/d(Eü?(c$%= ] w =4JX� #XhI�)(c;<=�6'#XhCt4p'Jv@4u

A

(pa + ρgz)~ndA =

V

∇(pa + ρgz) dV = ρ g

V

∇z dV = ρg

V

~k dV = ρgV ~kkA�Xu��?�ct

��#%5)(+& 5)#)pY� e�@A14;B5)(X& � 5)(aks�Xu��r�%t��ZkA�%u � �Ct:=41H;<#XgX�+@s(f�)e��)(XeHeC(i=4�?9B$o! � �)J+�)=46?(K6'#4@s(0;B1�9)$C@A1E5)(K! #%J+#X!?n�1�5)#J+1�9)# ]�V+T\[2¸ÖO2{)�?T2V+WYN��

~P = −ρgV ~k Pv = ρgVks�Xu��X�%t

 �(X6?9)1Xp%5)(x5)1�6'#x! #%J+#X!?n�1�5)#xJ+149B#)pX;B149)$C@A1:=4�?9B(�! � � JX�)=46?(8$,=�/21��)1�5)(8*%1�� JX� 6?(j9)5 #�5)(+*%� ��1��0#X5)(q@A1�@A1v;B5)(X6?(J+1�GE� 5 ��J+1�& 5)#%=4JX�o6'#4@s(x@A1�J+1�9)#%/��)=4J+�B=45 $'Jj( u: �#D#%*X#%hx� 1�eI$C9)Jj(+Jj(x@A1�;B#%��(+# ¥ �,3?�)/21v;"@A#X� $

VII*%1�6 $K!Yu 5Yu 1Xu

£ J+#%h4(>=41�& 1I=�J+#D=418#%5)(�5)(+eI� *%( ¥ �,3?�)/214;B#X*,(�=��?9B(>�?9)�K! #%J+�)=I(+6Yu  T\[2NZO2{B�?TQVXWYNl¹dR%[2¸v�CRÖ¸�� R,º_TQ}��?¸�O2{B� {)O�ÿ"R,U2R`M%N O\P2R'S�TQU2R kêJj(+& 6?(

D5)(F=v9B� r ���%u��Iû,tvu £ ;B� $'hI1

=�J+�)(+5)1:JX14GE� 5)(>J+1�9B(�;B149B$I@A1�$DJX14GE� ��JX$0J+1�9B(Kk�r 1I5 J+� $"/d(X=41 ] Jj(+& 6?( £ t��f$,=�/21I� 1�5)(�@A18*X1I� J+� 6?(+9)5 #H5)(+5 � G�1Xu¥ 6,#>=41��)(c;<�0#�3%#%/2#%h�1I5 #%/�J+1�9)$�p?J+14GE� 5)(�J+1�9B(��"=4�?9<(�! #%J+�)=�6'(�! #XJX! $'5 #�!)#XJ+#%!?n:1I5 #Xh�;B149<(�=�1�;B1�9)$C@A$E$>�)=4J+#�@Jj(+&)6 �Yud�´=v9B$,&B(v@A$05)143%#,/2#XhI1�5 #%h�J+1�9B( p JX1:;B*X1�Jj(+& 6'1x=�1:5)18! #X6?9<(+!)(4@A$<u

Ô\(+6'#j�'1Xp,*X1�#,/d(89B(X6'#�=41�/2#+GH1�!)#X6?(+eC(+J+�okê5 ! �Yud!)� #4@A1I6 J+#X*,(+-:1I/a6)� � *X1�! #X*%� ���f$>*X1��)J+� 6?(j9)5)#%/a! �B(+*Xr)$�;<#=49)#Xg%#C;B5 1.! #X*%� ����J+1�& 5)#%=4JX�Bt\;<(:5)(�;B149B�)/2� & 5 #8!)#XJ+#%!?n:1I5 #�J+149B#:;B149)$C@A1qJj(+6'#j�'1.=��?9B(:! #%J+�)=46?(�6'#4@s(�@A1Q@A1v;B5)(X6?(J+1�GE� 5 �iJ+1�9)#%/Ö�)=4JX�)=45 $'J+18JX1�& 5 #,=4J+�~p)Jv@4L

Pv = ρgVpkA�Xu��+¢,t

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R z�

�K$%=I/21�� 1I5)(�@A18*X1I� J+� 6?(j9B5 #>5)(+*%� ��1%ud�Y(+! � 1C/2� 5)(Vp

@A1�eI(X! � 1I/2�)5)(>! #XJX#X!?n�5 #Xh�;B1�9B(>J+1�9B( u

zAV

Pv

D

ρ ρ

Vp

£ 9B� 6?(K�Xu¡�Cÿ?L £ �?9B(>!)#XJ+�)=�6?(�5B(x;<149)�)/2�)& 5 #E! #%J+#X!?n�1�5)#EJ+149)#

Ô\(+& 6?(H$06'#4@A#�@�;B1�9)$C@A1x=4�?9<(�!)#XJ+�)=�6?(>$D#X*%#%/�=49)$'&)(v@A$05)(j9B(Xe��"$DJ+1�G>�)��J+$"! #XJX#X!?n�1�5 18eC(+! �)1I/2� 5 1%uG�H��DH�G ��������N�L����ÛQ� ��DL���øQ���N�������Q��ÔL����� Q�������øQ��ÛQ� ��DL�øL ���RQ��!�"����#��$

� 5 1�6)�)/´=v9B$,&B(v@A1�*%�)/d( pd&)� -�1I5 � r)(F;<(D@A1f� 1Ie�$C9)J+$C@A$'b)(�=4�?9<(F! � �)J+�)=46?(^6'#4@s(K;<149)$C@A1"5)(F5 1I6'#o! #%J+#X!?n�1�5)#J+1�9)#x$'! �)(X*X#x=��?9B(8! #%J+�)=�6'(�5B(X/_/2#+GH1�e�5)(XJ+5 #x#X9B(+6'�H(+J+�>#C;B� 1I�'� *%(X-�1�=��?9)1�! �)� J+�)=�6'(x5)(85 1I6 $�6)� � *X$H! #%*X� �fu

D

Pv

P ∗N

PP

A1A1

A2

£ 9)�)6'("�%u��"� L w ;B� 1��'� *,(+-:18=4�?9)18!)� � J+�)=�6?(>5)(H6 � � *%$D! #X*%� �¤/21�J+#C;<#%/Ö! #XJ+�B=46?(

��#,=�/d(+JX�)(v@s/2#K6 � �)*X$F! #%*X� �·=I("=49)� 6'1i�Xu¡���)u&%ô1�9)�)/2#";<("#C;B�)1v;B�)/2#K6'#+9B� 6?(�@A1D=��?9B("! � �)J+�)=46?(P

5)("J+$6 �)� *X$f! #X*%� �0u2��J+#,/�r � n.$"/2#+G�1C/2#>! � �)/21I5 � J+�F�K=v9B1v;B1Ib �K! #%=�J+$'!)(+6Yu

¥ 6,#�eI(%/2�)=v9)�B/2#x;<(�6 � � *%#4@�! #%*X� ���>;B#C;<(%/2#�#C;B�)1��'1�5 �0gX� #�@��)(+*%5 �?3E! #X*%� ���~p+;B#%gX� b)1I/2#�@A14;B5 $EeC(+! � 1C/2� 5 $eC(f6'#�@A$o/2#+GH1I/2#i=�/d(XJ+�)(+JX�o;�(f!)� 1v;<=�Jj(+*+n8(�@A1v;B5)#iJ+149)#K6'#4@A1:@A1E! #%J+#X! n:1I5 #K$oJ+1I& 5 #%=�J+�YuD�8(X#f��J+#H@A1H;B#Xg%� #! #%e�5)(XJ+#_� 1�eI$C9)J+$C@A$'b)(a=4�?9<(^!)� � J+�)=�6?(^6'#�@s(F;<149)$C@A1i5)(�Jj(+6'#o;B#Xg%�j@A1�5 #_J+1�9)#"@A1F=4�?9<(^!)#XJ+�)=�6?( u ��5 J+1Ih��)(+9l$� eI�)(+eI$okA�%u �v�%t.=41x/2#XG�1�! #C;B1�9)� J+�K5)(H� 5)J+1�hI�)(j9i! #>! #%*X� ���

A1

�f�)5 J+1�hI�)(j9i! #E!)#X*X�)���A2

L

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼¼�

~Pv = −∮

A

p~ndA = −∫

A1

p~n dA1

︸ ︷︷ ︸

~P

+

(

−∫

A2

p~n dA2

︸ ︷︷ ︸

~P∗

N

)

= ~P + ~P ∗N

��#%��J+#�G�1�9)�)/2#�;<(�#C;B� 1v;<�)/2#E=4�?9)$Pp?�f�)/d(4@A$'b �K�f*%�';B$>;<(�@A1 ~P ∗

N = −~PNp,;B#Xg%�j@s(�=�1�6,#%5)(+&B(+50�)e��)(Xe�eC(

#C;B�)1��'� *%(X-�18=��?9)1�! � � JX�)=46?(>5)(H6 � �)*X$0!)#X*X�,30/21IJ+#C;B#%/Ö! #%J+�)=46?()L~P = ~Pv + ~PN = ~PN − ρ~gV

ks�Xu��jû,th�;B1q@A1

VeC(+! � 1C/2� 5)(H6'#4@s(>=41:;<#XgX�+@s(�eC(+J+*,(+�)(X-�1C/`6 �)� *X1�! #%*X� ���"=I(�#C;<h�#X*,(+�)(4@A$'b �)/ag%� #4@A1C/��)(X*X5 � 3D! #X*%� ���Yu

��(H6'#X5)6 � 1�Jj(X5"=49)$'&)(v@�=�(>=v9B� 6'1Xp?eI(X! � 1C/2� 5)(V@A1x7�#X�)/2�)�)(+5)(>=I(X/2#>$'*X#j�'1�-:1I/K@A14;B5 18�)(X*X5 1�! #%*X� ���Yu

ρ

PvPv

Pv

PNPN

PN

P

P

£ 9)�)6'("�%u��$�'L w ;B� 1��'� *,(+-:18=4�?9)18!)� � J+�)=�6?(>5)(H6 � � *%$D! #X*%� �¤/21�J+#C;<#%/Ö! #XJ+�B=46?(

¥ 6,#�G�1�9)�)/2#�;<(`5)(X! � ��1I/2#F@A14;<(+5�#X! ��J+���)e��)(XeXp�6?(c;�(�=41�eI(X! � 1C/2� 5)(V

;B#%gX�j@s(a$'*%#j�,1I-�1C/n

�)(X*X5 �?3! #%*X� ���Yp?� eI�)(+e�eI(E=��?9)$D!)� � J+�)=�6?(>5)(H6 � � *%$0! #%*X� �Ù@A1XL

~P =

n∑

i=1

~PNi + ~Pv =

n∑

i=1

~PNi − ρ~gVks�Xu��XÿXt

��#X9)� 6'#��)(+*%5 �?3�!)#X*X�)���H� eI(XgX�)(XJ+�>eI(�� 1I��(+*,(+-:1q! � #XgX9)1I/d(.@A1�=�J+*%(X�>9)� & 5)#Xh�� eIgX#X�B( u\ü'1I5 1��B(j9)5 #BpX! #,=4J+#�@A�g%1I=46'#%5)(+&)(X50g%� #[email protected])(+& �)5)(Kk�� eIgX#X�B(��B(+*X5)�?30! #%*X� ���Bt�5)(H6,#�@A�i/2#+G�1IJ+18� 1I��� J+�KeC(c;<(XJ+�f!)� #XgX9)1I/qu

�%1v;B5B(+& � 5B(EkA�Xu��Xÿ%t)@A1�*X1I6 J+#X�B=46?(.@A1v;B5B(+& � 5B(��E! � #4@A1I6 J+#%*%(+-:1I/^5)(�;B*,(�� eI(XgX�)(X5)(:$'! �)(+*%5)(:! �)(X*Xr)(�;B#%gX�j@s(%/2##C;BhI#X*,(+�)(4@A$,b)1�! � #�@A1�6 r)�j@A1Xu

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼6

¯ PQTQS�R,P��� 5 u w ;<� 1v;B�)J+���)5 J+1�5)e�� J+1IJ8=4�?9)1.! � �)J+�)=46?(:5)(�6 � � *%$8! #X*%� ��=I(�=49)� 6'1Xpj#%g+9)�)6'(�& 1�JX*X� J+�)5 1q#%/2#XJj(X&)(r � 9)� 5';B�B(>! #+9)$'! �)1�& � 6?(

R�0;B$,GE� 5 1

Lu�ð.1�9)� & � 5)1

ρ�H

Jj(+6'#j�'18=�/d(+JX�)(+J+�K! #%e�5)(XJ+�)/qu

x

y

R

H

ρ

Pv1 Pv2

P =?

V1

V2

PN1

PN2

PN3

kê(,t(b)

£ 9)�)6'("�%u��"� L w ;B� 1��'� *,(+-:18=4�?9)18!)� � J+�)=�6?(>5)(H6 � � *%$D! #X*%� �¤/21�J+#C;<#%/Ö! #XJ+�B=46?(

��� #%g+9)1C/`b 18g%� J+�i� 1I��1I5"5B(�;B*%(H5)(X& � 5)()p #XeI5)(+& 1I5)(>=�(Kkê(,tq�ak�g't.5)(E=v9)�)r �a�Xu¡���)ud�´=v9)$'&)(4@A$0! #C;o=4� 9)1Dkê(,tPN1

�Pv1

=4$f#C;B� 1I�,1I5 1:� e��)(Xe��)/d()L

PN1 = (pc1 − pa)A1 = ρg

(

H − R

2

)

R√

2L

Pv1 = ρgV1 = ρg

(R2π

4− 1

2R2

)

L = ρgR2

4− 1

2

)

L

��� #�@A1�6 r)�j@A1x=4�?9B1P

5)(>#,=41:6,#%#X�%;B� 5B(+J+5 #%h8=4�)=�J+1I/d(x− y

=4$0#C;B� 1��'1I5 1�� e��)(Xe��)/d()L

Px = PN1 cos 45◦ = PN1

√2

2= ρg

(

H − R

2

)

RL

Py = PN1 sin 45◦ − Pv1 = ρg

(

H − R

2

)

RL− ρgR2

4− 1

2

)

L = ρg

(

H − R

)

RL

��5 JX1�5 eI� J+1�JD=��?9)1P@A1XL

P =√

P 2x + P 2

y = ρgRL

√(

H − R

2

)2

+

(

H − R

)2

�´=49)$'&)(v@A$D!)#C;(b)

� 5 JX1�5 eI� J+1�JX�i#C;Bh�#%*%(+�B(v@A$'b �?3D=4� 9B(E=4$D#C;<� 1��'1�5 �"� eI�)(+eI#%/qL

PN2 = ρg

(

H − R

2

)

RL , PN3 = ρgHRL

Pv2 = ρgV2 = ρg · 1

4R2πL

��� #�@A1�6 r)�j@A1x=4�?9B1P

=4$�LPx = PN2 = ρg

(

H − R

2

)

RL

Py = PN3 − Pv2 = ρgHRL− ρg · 1

4R2πL = ρg

(

H − R

)

RL

�8(X�)(+*%5 #)pX;B#Xg%�j@A1�5)�K=�$D�)=�J+�i� 1�eI$C9)Jj(+J+�Yu�

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼B¼

G�H��DH(' KMLON1L�PRQ�Xu� �*X1�6 $'h¾9)1:@A1v;B5)(X6 �?3o!)� 1�& 5)� 6?( p

d = 1.2 mp��^J+14GE� 5)(

G1 = 12 kN�G2 = 4 kN

p~=4!)#4@A1�5)1H=�$F$,G�1�JX#%/qu¥ 6'#�=�1�#%*X1�6 $'h¾9)1:=�Jj(+*X1�$E*X#C;B$Kk

ρ = 1000 kg/m3t4p,#C;<� 1v;B�)J+�"=��?9)$EeC(+J+1IeI(+-�(x$E$,GH1�J+$

Tp,6?(+#��06'#+9)�)6 �

=41�;B1I#HeC(+! �)1I/2� 5 1�9<(+6'��1�6 $'h¾9)1�5)(+9B(+eI�f� eI5)(c;F5 �)*X#%(H=v9B#XgX#C;<5 1�!)#X*X�)���Yu�?u���$'h¾9)� r)(�k

ρm = 7800 kg/m3t0! #+9B$,!)� 1�& 5)� 6?(

r = 6 mm5)(+9)14GH1^5)(�#XJ+*%#X��! � 1�&)5 � 6?(

d = 10 mmu

w ;B� 14;B� J+�_=4�?9B$F6'#4@A#%/�6 $'h¾9)� r)(D;B1�9)$C@A1H5B("=�1v;B�)��J+1>*X1I5 J+�?9<( p~(X6,#K=4$o! #XeI5)(+J+��=v9)14;B1�b)�^! #C;�(+r �YLF =

100 Npρ = 1000 kg/m3

pD = 30 mm

pH = 500 mm

u

���*) ó¤��ÇKÈ,+0Ãi��À8ÌÍÈ´É�Ã"ÓoÈ�ÆFÇKÃKÊ8ÃiÉ�ÏDÀúÂiÀ¤ÏFÇKÃi�xÐÍÆi����ÇVU �^ÃFÇKÈ�ÏKÊ8Âi��ö ÆFÇKÃ.-ö�È�Âi��ö ó�õ�ÓoÈ�ÇK����È õ~È.�^ÂiÀ8ÅiÃiÂiÈ ÉqÊ�À�Ê8ÃiÏDÈ Ñ�ÓFÐfÃf�^À

% 149B�)/2#x;�(�#C;B�)1v;B�)/2#H=4�?9B$E! � � J+�B=46?(�5B(�6 �)� *X$E! #%*X� ��=�(�=v9)� 6'1H�Xu¡���)uQ��#C;<=�1�J+�B/2#�=41�;<(H=�/2#H! � �?9B� 6'#%/� eI*X#j�'1�-�( w @�9)1I� #X*%1d@A1v;<5)(+& �)5 1�� eC(+g%�)(j9)�E!)#XJ+! $'5 #�! � #%� e�*%#+n�5 $�eC(+! �)1I/2� 5 $E7:9)$'�';�(x�>;<(�=�/2#x6'#%5)=4Jj(+JX#X*%(+9)�;<(�@A1�! #XJX� 1�g%5 #:;<(x=�$'/d(�=4*X� 3H=4� 9B(86'#4@A1.;B1�9)$C@A$�5B(8J+$HeI(+!)� 1I/2� 5)$�g%$�;<1d@A1v;B5B(+6?(�5 $C9)�>;<(xgX�>#X5)(�g%�?9B(8$>=4Jj(X-�$/2� �)#X*%(X-:()u

ρ PG

P

PN

R

£ 9)� 6?("�Xu¡���)L w ;B� 1��'� *,(+-:1:=��?9)18! �)� J+�)=�6'(>5B(�6)� � *X$f! #X*%� ��;B� � 1I6 J+5 #,/Ö!)� �)/21�5)#%/ w @�9)1�� #%*X1.@A1v;<5)(+& �)5 1

��eC(+gX1I� �)/2#>=�(j;<(�JX$0eI(X! � 1C/2� 5 $D6?(+#H5)(>=v9B� r �_�Xu¡���)p,Jj(X6,#�;<(�@A1v;�(+5f;B1�#H! #%*X� ���f6'#4@A��@A18#XhI�)(+5)� &)(+*,(xg%$�;<16 �)� *%(i!)#X*X�)�·5)(i6'#4@A$^J+�)(jGE�)/2#^=4�?9)$�! � �)J+�)=46?()p2(F#%=4Jj(+9)�^;B1�#E@A14;B5)(i� 9)�a*X� ��1D�)(X*X5 �?3�! #X*%� ���~u0 �(X6?9)1%pJ+1I& 5 #,=4J>6'#4@s(>=4185)(+9B(+eI�f$'5 $'Jj(+�"J+18eC(+! � 1C/2� 5 1q@A18$0=�Jj(+-.$f/2� � #%*%(X-:()p?Jv@4u

~PN + ~R︸ ︷︷ ︸/ Ðvë çlâ Ì çzè�ê ä{ñ ælâzå ÐvÌ åãælß Ì + ~PG

︸︷︷︸ËÍÌ ç ä àµá6æ{â{å ÐvÌ åãælß Ì = 0ks�Xu��r�,t

h�;B18=�$<L ~PN ] =4�?9<(>! � � J+�B=46?(>5)(H�)(+*%5 $D! #%*X� �q�~PG = ρ~g V ] J+14GE� 5B(>J+1�& 5)#%=4JX�"6'#�@s(H=�1�5)(j9B(Xe��"$'5 $'Jc(X�KeI(+!)� 1I/2� 5)1v�~R ] =��?9B(H6'#4@A#,/`6 � �)*%(>! #%*X� ��;B149B$I@A185B(>J+1�& 5)#%=4J>$'5 $'Jj(+�"eI(X! � 1I/2�)5 1Xu

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼)ñ

£ �?9<(PG

;<149)$C@A18$"J+14GE� ��J+$KeI(+!)� 1I/2� 5)1Vp?;<#X6F=4� 9B(

PN;B149B$I@A1��)=4! #C;FJ+1�GE� ��Jj(E�)(X*X5 18!)#X*X�)���ak�! #%h¾9)1v;�(v@

! �)� & $0*%1�eC(+5 $DeC(>=4�?9)1�! �)� J+�)=�6'(>5B(>�)(+*%5 1�!)#X*X�)���Bt4u�8(X6,#�@A18! #>J+� 1Ib 1I/10�$'J+5 #%*X#,/`eI(X6'#X5 $�p ~P = −~R p';B#+9<(+e��B/2#x;B#E�)e��)(XeI(�eI(>=4�?9B$0! � �)J+�)=46?()L

~P = ~PN + ~PG = ~PN + ρ~g Vks�Xu��l�+t

�ú#X!)��J+1C/�=v9B$,&B(v@A$�pY6?(c;<("=41HeI(X! � 1C/2� 5)(K=�(%=4J+#�@A�o#C;n�)(+*%5 �?3F! #%*X� ���o��6 � � *%1>! #X*%� ���^5)(06'#�@A$FJ+�)(+G>�B/2#

=��?9)$0!)� � J+�)=�6?( p � eI�)(+e8=I(H=��?9)$f! � � J+�B=46?(:@A1XL

~P =

n∑

i=1

~PN + ~PG =

n∑

i=1

~PN + ρ~g Vks�Xu��r�%t

¯ PQTQS�R,P��ÝF6� w ;B� 14;B� J+�K� 5)J+1�5 eI� J+1IJ0=4�?9B18! � � J+�B=46?(>5)(H6 � � *%$D! #%*X� �¤=�(>=49)� 6'1E�Xu��r�)p,#%g+9)�)6'(>¢32cûH#%/2#XJj(X&)(r � 9)� 5';B�B(>! #+9)$'! �)1�& 5 �)6'(

R�f��� � � 5 1

Lu £ /d(+J+�)(XJ+�K! #XeI5)(+JX�)/�*%149)�)& � 5 1

HpρpR

�Lu

x

y

R

H

ρ

PG1 PG2

P =?

V1

V2

PN1

PN2

PN3

èmä%ì(b)

£ 9)�)6'("�%u �v� L w ;B� 1��'� *,(+-:18=4�?9)18!)� � J+�)=�6?(>5)(H6 � � *%$D! #X*%� �¤/21�J+#C;<#%/Ö! #XJ+�B=46?(

�8(X#���!)� 1�Jj3,#C;B5 ��! � �)/21I�Yp+�H#X*%(4@Yb 1.gX� J+��$'�)(+�'1�5�5)(�;B*%(�5)(+& �)5)( pj$'*X#j�'1I-�1C/K�)(+e�9)� & � JX�?3��)(X*X5 �?3x!)#X*X�)���$fr �?nq$E;B#%gX�j@s(X-:(HeI(X! � 1C/2� 5 1

Vu�Ô\(+6'#:@A18eI(>=v9B$,&B(v@q! #C;akm(%t4L

PN1 = ρg

(

H − R

2

)

R√

2L

PG1 = ρgV1 = ρg

(3

4R2π +

1

2R2

)

L = ρg

(3

4π +

1

2

)

R2L

��� #�@A1�6 r)�j@A1x=4�?9B18! � � J+�B=46?(>5)(H#%=�1:6'#%#X�%;B� 5)(XJ+5 #%h:=��)=4J+1C/d(x− y

L

Px = PN1 cos 45◦ = ρg

(

H − R

2

)

RL

Py = PG1 + PN1 sin 45◦ = ρg

(3

4π +

1

2

)

R2L+ ρg

(

H − R

2

)

RL = ρg

(3R

4π +H

)

RL

��5 JX1�5 eIJ+� J+1IJD� 1�eI$C9)J+$C@A$'b 1x=4�?9B18! � � J+�B=46?(:@A1XL

P =√

P 2x + P 2

y = ρgRL

√(

H − R

2

)2

+

(3R

4π +H

)2

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼ þ

��(>=49)$'&)(v@.! #C;akêg,t.� 5)J+1�5 eI� J+1IJ+�o=4� 9B(PN2

pPN3

�PG2

=4$D#C;<� 1��'1�5)�f� eI�)(+eI�)/d( L

PN2 = ρg

(

H − R

2

)

RL; PN3 = ρgHRL

PG2 = ρgV2 =3

4ρgR2πL

w ;BhI#X*,(+�)(4@A$,b)1�! �)#4@A1�6)r �j@A1x=4� 9)1PL

Px = PN2 = ρg

(

H − R

2

)

RL

Py = PN3 + PG2 = ρg

(3R

4π +H

)

RL

G�H54�H�G KMLON1L�PRQ�Xuq� 6'#%=�#%/Ù! � 1Ih��)(j;B5 #,/�eI�';B$_� eC/21��'$^;B*%(i�)1�e�1I� *X#,(+�)(K5B(j9B(Xe��`=410eI(+JX*%(+�B(+&a#XgX9)� 6?(K! #X9)$C9)#X! JX1Xu¢�%1v;<5)(

-�1Ih�#%*%(�=�J+�)(X5)(�#%6 *%(X��1I5)(q@A1:J+1�& 5)#X��b)$Dh�$,=4J+� 5)1ρp?(8;B�)$,h4(�� e49B#+G�1I5)(x;B1A@s=�J+*X$E6'#X5B=4Jj(+5 J+5)#Xh:! � � J+�B=46?( u

¥ 6'#�@A18! #%6'(Xe�� *,(+-:1:/d(X5 #%/21IJ+�)(h = 134 mm

#C;B� 14;B� J+�i=��?9)18� eIJ+1�eC(+-�(>�K=I/2� r)(X-:(>*%1�e�1 ¥ ] ¥ u' �(+JX�K=�$! #C;<(Xr �YL

h1 = 1.2 mph2 = 2.2 m

pR = 0.4 m

ppv = 8 kPa

pα = 30◦

pρ1 = 900 kg/m3

pρ2 = 1000 kg/m3

�ρm = 13600 kg/m3

u�?u�ï21�eI1�� *%#%(+�a#%g+9)�)6'(o! #+9B$I9B#X! J+1_k

R = 1 mt�! � � & *%� ��b)1�5^@A1f*%1�e�#,/ ¥ ] ¥ eI(^e��';Z6'#4@A�F@A1"5)(Xh�5 $'J�! � 1I/d(

3%#%� � e�#%5 Jj(j9)��eI(�$,h4(X#α = 45◦

uQ��� 1�eI1�� *%#%(X� $8=41q5)(j9<(+e���*X#C;<()pj(�5B(v@A*X�)��#4@\Jc(X& 6 �H� 1�eI1�� *%#%(X�)(�/2#X5)J+� �)(X5@A1�� 5B=4J+� $?/21I5 JfeC(0/21I� 1�-:1�!)� � J+�)=�6?( u w ;B� 1v;<� J+�^=��?9)1H6'#4@A1�#%! J+1I� 1�b $C@A$K*X1Ie�$ ¥ ] ¥ eI(H;B*,(D! #%6?(+e��)*%(+-�(� 5)=�J+� $?/21�5 Jj()L�kê(%t

pm = 10 kPa�ak�g,t

pv = 20 kPau

6 7 ��98°�� ��Î�;:°�� <=8?>"8�@: BA <C8D>FE���>HG I;J�KL>"KM>�� �

��#C;o! �)#%=4JX�)/�r 1�*%#X*%#C;B#%/l=41x!)#C;B�)(+eI$?/21�*%(:@A14;B5)(x@A1v;B� 5B=4J+*%1�5)(0r 1�*f6,#%5)=4Jj(X5 J+5 #%hx! � 1I& 5 � 6?( p<�?9)�or 1�*f6,#�@s(=�1x=�(X=�J+#4@A�"#C;i5 � eI(�;B1I#X5 �)r)(>�)(+e�9)� & �)J+#Xh�! � 1I& 5 � 6?(>6'#4@A18=�1:�)1v;B5 #E5B(c;B#%*X1I=�$C@A$�@A1v;B5)(>5B(x;<� $'h�$�u

½E��� �������N-���É�Âi����ÂiÈ ÆFÇKÈ�Ê8Æi��ÉqÊ�À8��ÏDÈ£ �?G�/2143'(+5 � 6'187:9)$'�';<(H& �)5 1�J+� �>@A1v;<5)(+& �)5 186'#4@A1�#X! �B=4$C@A$D#%=�5 #X*%5 1�eI(X6,#%5 1�7�� e�� 6'1%L• ��(+6'#%50#>#C;B�?G>(+-.$0/d(X=�1 ] @A14;B5)(+&)� 5)(H6'#X5 J+�)5 $'� J+1�Jj(•  �� $'h��O0�$'J+5 #X*>eC(+6'#X5 ] @A1v;B5B(+& � 5B(>6,#X9)� & �)5 1�6 � 1�Jj(X-:(• ��(+6'#%50#>#C;B�?G>(+-.$D1�5 1I� h��+@A1 ] ! � *%�f! �)� 5 r � !iJX1��)/2#C;B�)5)(X/2� 6'1� 5B(v@A#X!)��J+�j@A1C/·=v9)$'&)(4@A$�px�)(c;<�´=41`#�!B(+� r �+@s(j9)5 �B/ô;B�)7�1��)1�5 r �+@s(j9)5 �B/�@A1v;B5)(X& �)/d(Z!)#�J+� �´!)� #%=�J+#X� 5)1

6'#X#%�%;B� 5)(XJ+18�i*X� 1C/21�5 $�p 6'#�@A1x5 �j@A1�/2#%h�$'b 18� 1I��� JX�Yu���(+J+#D=�18! � �i� 1I��(X*%(+-.$0#C;<� 1��'1�5)�?3f! �)#Xg+9B1I/d(E=4JX� $C@s(+-�(/2#%h�$_;B1C7�� 5 �)=I(+J+�l! � 1IJ+! #,=4Jj(+*%6,1"6'#4@A1"b 1Ke�5)(XJ+5 #_#+9<(+6'��(XJ+��� 1I��(X*%(+-:106'#%5 6 � 1IJ+5 #XhK! �)#Xg+9B1I/d( ul�8(+�)(X*X5 #Bp/2#%�)(H! #%=�J+#4@s(XJ+�K�"#%! �)(X*j;<(X5 �0�)(Xe49)#%h�eI(H$'*X#j�'1�-:1�JX�?3f! �)1�J+! #,=4Jj(+*%6 �Yu

��� �0! � #X$'&)(X*%(X-�$E=4JX� $C@s(+-�(xJ+1I& 5 #%=�J+�"�Dh4(X=�#X*%(�*%1�#,/d(x*%(+GE5 #�/21I=�J+#HeI(c$'e��)/d(x�0#XgX9B(X=�J85)(c$'6'1�#�=�J+� $C@s(+-.$! #C;05B(+e��)*X#%/F;B� 5B(X/2� 6?(q@A14;B5 #C;B�)/21I5 e��+@s=46 �?3D=�J+� $C@s(+-�( udÔY#H=4$>=4J+�)$I@s(X-:(x6,#C;06'#�@A�?3H=�1�=4*%1�7�� e��)& 6'1�*X1�9)� & � 5)1eC(+*X�B=41�=�(%/2#:#C;�@A1v;<5 1�! � #,=4J+#%� 5 1�6'#X#%�X;<� 5)(+JX1.$,=�/21I� 1�5 1�$�! �)(+*%r $�=�J+� $C@s(+-�( p%�H$�5 1�6'#,/o5)(4@A#X! ��J+�j@A1C/^=49)$'&)(v@A$�a#C;a*X� 1C/21�5)()p~Jv@4u

f = f(t, l)ufÔ.� ! � &B(+5a! � �B/21��K@A1v;<5 #C;B�)/21I5 e��j@s=�6'#XhD=4JX� $C@s(+-�(H@A10=4J+�)$I@s(X-�106 � #XeE=�J+� $C@A5 #

*X9B(+6 5)# � @A1I�f=�1�$>J+#,/�=v9B$,&B(v@A$>! � #,/21�5 1�7�� e��)& 6 �?3D*%149)� &)� 5)(x!)#�! #%! � 1�&)5 #%/a!)� 1I=�1�6 $D/2#%h�$>eI(X5 1I/d(X� � J+�fe�gX#%h-:1�hI#X*X� 3i;B�)/21I5 e��j@s()u ¥ 6'#>@A1D=�J+� $C@s(+-:1E6 � #%eD=4J+� $C@A5 #i*X9B(+6)5 #i=4Jj(+r �)#X5)(X� 5 #)p~Jv@4u0(+6'#i7�� eI� & 6'1>*X149B� & � 5 1E5)1eC(+*X�B=41>#C;_*%� 1I/21I5)( p\*X1Ib�=I(X/2#K#C;_!)� #%=�J+#X� 5)1E6,#%#X�%;B� 5B(+J+1%pY#%=�5 #X*%5 1�@A14;B5)(X& � 5 1D6'#�@A�)/d(K=41E#X!)�)=4$C@A1E=4J+�)$I@s(X-�17:9)$'�';�(Db 1>=41>=4*%1I=4JX�F5)("(j9)hI1�g%(X�)=46'1�@A14;B5)(X& � 5 1%u w *%(v@�/2#C;B1�9�=�J+� $C@s(+-�(f=�1�! �)�)/21�-.$C@A1E�o5)(f=4J+� $C@s(X-:(f6 � #%er 1I*X�Yu

½E�¾½ óôÉ�Âi����ÂiÈúõ~È.�^ÂiÀ8ÅiÃiÂiÈ�8(X#.��J+#2@A1�� 1I& 1�5)#�$�$'*X#C;B$�pI! � �8! � #%$'&)(+*,(+-.$�=4J+�)$I@s(X-:(�J+1�& 5)#%=4JX�86 � #Xedr)1�*X�8=�1d$,*%#C;B1d=v9)14;B1�b)1�! � 1IJ+! #%=�Jj(+*X6'1%L• =4J+�)$I@s(X-�1q@A185)1I=4JX� �xn�� *X#Fkêh�$,=4J+� 5B(H7:9B$,�?;<(>=4185 18/21�-�( p

ρ = constt

• =4J+�)$I@s(X-�1q@A1x=�Jj(+r � #%5)(+�)5 #"km7�� e��)& 6'18*X149B� & � 5 18=�185 1x/21�-�(v@A$fJ+#X6'#%/�*X� 1C/21�5)(,t• =4J+�)$I@s(X-�1Y@A1�@A1v;B5 #C;<�)/21�eI�j@s=46'#�km7�� e�� &)6,1d*%149)�)& � 5 1deC(+*X�B=41�=�(%/2#q#C;q@A14;B5 1�!)� #%=�J+#X� 5)1d6,#%#X�%;B� 5B(+J+1d$,=�/21��)1�5 1;B$,Gô#,=41�r 1�*%�Bt

ü,9B(X*X5 1i7�� eI� & 6'1F*%149)� &)� 5 1i6'#4@A1F=�1F#C;B� 1I�,$C@A$`!)� �?9)� 6'#,/�!)� #X�)(X& $'5)(a=4$Ö! � � JX�)=�(X6Z�ZgX� eI� 5)(a=4JX� $C@s(+-�(kêJc(X& 5 �j@A1D=�� 1v;<-:(Kg%� e�� 5B("=�J+� $C@s(+-�( � tvuD��("-:�?3%#%*X#"#C;B� 1I�,�)*%(+-:1H5)(X/�5)("�)(X=�! #+9<(+h4(+-.$F=4JX#4@A1H;B*%1>(+9)h�1Ig%(X�)=46'1@A14;B5)(+&)� 5 1Xp�6,#�@A1f=�1E;B#%gX�j@s(4@A$^� ef#%=�5 #X*%5 �?3"@A1v;B5)(X& � 5)(^$'e0hI#X� 1D5B(+*X14;B1�5)10! � 1IJ+! #,=4Jj(+*%6,1%pd(FJ+#�=4$"@A1v;B5B(+& � 5B(6'#X5)J+� 5 $'� J+1IJj(>�OP.1I� 5 $C9)�j@A1I*%(�@A1v;B5B(+& � 5B(ik�;B#%gX�j@s(>=41�� eq@A14;B5)(X& � 5 186'#+9B� & � 5 186)� 1�Jj(+-�(%t4u� � ôRQ"÷Ã% .")Dûµ!r# ö .*) % &�'2ôRQ"÷Ã% .$#�/ & ûD�*.TSÛ�".*� ôo& ú��*0�#ã!$.")VU�( Q*& õ*."� ö # dA 3

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼¼5'DH�'DH�G WX� N��øL�YRQ��øL �Z�����ÛQ���� Q�������L

��=49)$'&)(v@A$>=4Jj(Xr � #X5B(+� 5 #%h��E5 1I=�J+� ��n:� *%#Xh�=�J+� $C@s(+-�(8#%! ��J+�0#Xg+9B� 6:@A1v;B5)(X& � 5 1�6'#X5 J+�)5 $'� J+1�Jj(x=�1�=4*%#C;B�>5)( L

V̇ = constkm� u��ct

h�;B1�@A1f=I(V̇

#%e�5)(X& 1�5 M,N?OQPQR'S:T\U2V+WQTúO\P2{B� {)W 6 � #%eDr 1�*%#X*%#C;\uo��(+!)� 1I/2� 5B=46 �`!)� #XJ+#%6`6 � #%e05 1I6 $_! #X*%� ��=�1;B1C7�� 5 � ��186?(+#>7:9)$'6'=:*%1�6 J+#%�)(HgX� eI� 5 1�6 � #%e8J+$0!)#X*X�)�0p'Jv@4u

V̇ =

A

(~v, ~n) dAk �?u��Xt

��#%6'(+GE�)/2#F=�(j;<(i5)(i��Jj(ib 1f=�10=4*%1I=�J+�a#X*,(v@8� e��B(+eD$_=v9)$'&)(4@A$a=4J+� $C@s(X-:(i6)� #Xefr 1�*BpQ�a��Jj(>@A10J+#�=4� 14;B-�(g%� e�� 5B(H=�J+� $C@s(+-�( u

�8(X#0��JX#�@A1>! #%e�5)(XJ+#)p�! #,=4J+#�@A1�;B*%(0#%=45)#X*X5B(D� 1�G>�B/d("=�J+� $C@s(+-�( pY9B(X/2� 5B(+�)(X5o��J+$'� gX$C9)1I5 Jj(+5Yp���5)("=v9)� r)�� u��8=4$0! � � 6?(+eC(+5 �"! � #,7��?9)�KgX�)e�� 5 1�! �)�?9)� 6'#%/l=4JX� $C@s(+-�(>6 � #Xe�r 1I*>$0Jj(�;B*%(>=v9B$,&B(v@s( u

[ ä%ë2æ ã)äXå ã ßEÞ4â+å)á]\sä_^�î ` á'åbaXáCà)îIã â+ã ßfÞ4â+å ác\sä_^�î d�å ß,í�æ?à"Þ�å îVef^:îga%åbh�æ ã)îrrr

zzz

v = v(r, z)v = v(r, z) v = v(z)

£ 9)�)6'(Ö�?u¡�XL_��� #%7��?9B��=4J+� $C@s(X-:(�! � �?9)�)6,#,/�9B(X/2�)5)(+� 5)#XhF��JX$,�)gX$C9)1�5)J+5 #Xh�=4J+�)$I@s(X-:(�6 � #%eFr 1�*���! � #,7��?9=�� 1v;B-:18gX�)e�� 5 1

�8(�#%=�5 #X*%$H�)e�h¾9)14;<(H! � #%7�� 9B(�g%� e��)5)(�!)� �09<(X/2� 5)(X� 5 #,/`�"J+$'� gX$C9)1I5 J+5 #,/Ö=�J+� $C@s(+-.$�p)/2#+GH1�=�1�eI(+6 n.$'& � JX�=49)1v;B1Ib 1XL

• $D#Xg,(��)14GE�)/d(E=4JX� $C@s(+-�(>gX� eI� 5)(>=41x/21I-:(H! #E! #%! � 1I& 5 #%/�! � 1C=41�6)$• *X1I6 J+#X�"gX�)e�� 5 1�$"=4*%(X6'#4@�Jj(+&)6 �K! #X! �)1�& 5 #%h�! � 1I=�1�6?(�@A18#X� JX#Xh�#%5)(j9<(+505B(�!)� 1I=�1�6��=v9)$'&)(4@A$D;<(8@A1��)=�! $'-�1I5 #E;<(8@A1�*X1I6 J+#%�FgX� eI� 5 1x$i=�*%(+6'#�@.Jj(+&)6 �F! #%! � 1�&)5 #Xhx!)� 1I=�1�6?(�@A1�#%� J+#%h�#X5B(j9B(X5"5B(

! �)1I=41I6i6?(jG�1C/2#H;<(E=4JX� $C@s(+-:1��)/d(�@A14;B5 #C;B�B/21�5 eI�j@s=46 �i6?(X�)(+6 JX1��YpB�i$fJ+#,/�=v9)$'&)(4@A$x@A1�/2#%h�$'b 18$'*X1C=4J+�i!)#4@s/2#X*%1J+eI*)u V+P2R+¹dØ0T � |BP2RX¹2U2{BVX�?T�ÕKT\M%T�}QWQT � |<R%[2T�}QTQUQN �i5)(>Jj(v@q5)(X& � 5K! � �)/21I5 � J+�D@A1v;B5 #C;<�)/21�5 eI�j@s=46)�F/2#C;B149~u��#X*%#%/a=49)$'&)(v@A$D=�$E*X1�6)J+#X� �

~v�~n6'#X9)� 5 1C(+� 5 �f$D=4*,(+6'#4@�Jj(+&)6 �f!)#X! � 1I& 5 #%h�! �)1I=41I6?( p'!)(H=41:� e��)(XeHk �?u��Xt�=4*%#C;B�

5)()LV̇ =

A

v dAkm�?u ¢%t

£ � 14;B-�(�g%� e��)5)(>=41:;B1C7�� 5 � ��1:5B(E=v9)14;B1�b �"5)(X& � 5YL£ � 1v;<-:(�gX� eI� 5)([email protected]�� 6 J+� *%5)( pc6'#X5B=4Jj(+5 JX5)(�g%� e�� 5B(�! #�! #%! � 1I& 5 #%/K!)� 1I=�1�6 $�6'#4@s(�#,=4J+*,(+� $C@A1�)=4JX�"eC(+! �)1I/2� 5)=�6 �K! � #%J+#X6"6?(+#E=4J+*,(+� 5)�f! �)#%7��?9igX�)e�� 5)()u

vs A =

A

v dA =⇒ vs =1

A

A

v dAkm�?u û,t

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼�F

�¤=4*%�)/Z! �)#X�)(X& $'5 �)/d(fb 1I/2#"6'#X� �)=�J+� J+�a=�� 1v;B-.$Fg%� e�� 5)$<p���� 5';<1�6?=sb)1�g%� J+��� e�#,=4Jj(+*Xn:1I5Yp�#C;B5 #,=45 #"(+6'#

=�186'(+G�1�;<(:@A18gX�)e�� 5)(>=�J+� $C@s(+-�(H$05 1I6,#,/`r 1I*X#%*X#C;B$v = 1 m/s

p)/2�B=v9)�i=4185B(H=�� 1v;<-�$"gX� eI� 5 $0=�J+� $C@s(+-�( ��Í=49)$'&)(v@A$Ö! � #%/21I5 1F! #%! � 1I& 5 #XhK! �)1I=41I6?(_r 1�*%�Yp�!)� #%/21I5 � b 1o=�1i�ZgX� eI� 5)(_=4JX� $C@s(+-�( u���g+9)� eI� 5 �Z/21I=�Jj(h�;B1�=41�/21I-:(E!)� 1I=�1�6Yp�=4J+�)$I@s(X-�1�5 1�b)1x�)/d(+JX�E@A1v;B5)#C;B�)/21�5)e��j@s=�6 �F6?(+�)(X6 J+1I�Yp)!)(8@A1�$"J+�)/�! � 1C=41Ir �)/d(E5 1C/2#XhI$,b)1;B1C7�� 5 �)=I(+J+�D=�� 1v;<-�$HgX� eI� 5 $H=4J+� $C@s(X-:()u £ JX#Xh4(�=41�/2#%�)(v@A$�$%=�*X#4@A�)J+�>! � 1I=�1�r �E6'#�@A�>=�$�5)(�;<#X*X#Xn:5)#%/^�)(X=�J+#4@s(X-�$#C;Ù/21C=4Jj(a! �)#%/21�5)1^! �)1I=41I6?( p�Jj(+6'#_;<(Ö=4J+�)$I@s(X-�1^$�-��)/d(Ö�)/d(F@A1v;B5 #C;<�)/21�5 eI�j@s=46)��6?(+�)(X6 J+1��~p�!)(i@A1�/2#XhI$,b)16'#X�)�)=4J+�)J+�F=4� 14;B-:18gX� eI� 5 1�$DJ+�)/l! �)1I=41Ir �)/d()u

i�jlknmporqtsgu&v�u_wxqyo*uTz�vn{pz�|�wxupv�}T|lo*~c��|���qtknqt��j_upk��

v1 v2

A1

A2

£ 9)� 6?(E�?u��?Ld��� #,/21�5)(H! #%! � 1I& 5 #Xh�! �)1I=41I6?(�r)1�*X#%*X#C;<(

�%1v;B5B(+& � 5B(�6'#%5 J+� 5 $'� JX1�Jj(>eI(>=v9B$,&B(v@.=I(>=v9)� 6'1�� u �H=418=�*X#C;B�"5)( L

V̇ = const =⇒ v1A1 = v2A2

h�;B18=�$v1

�v2 V+P2RX¹2Ø"RF»)PQMXTQU2R $0! �)1I=41Ir �)/d(H! #%*X� ��� 5)(

A1

�A2

u'DH�'DH(' �����������¤Q��t���ÔL1�t��N��RL�YøQ��RL

��#,=�/2J+�)(4@s/2#0@A14;B5 $`r)1�*o6)� #XeK6'#4@A$a=4JX� $C@A�lJX1�& 5 #,=4J���$%#X& �)/2#o;B*,(F! � 1C=41I6'(Ö� ] �0��� ] �?p�Jj(X6,#F;<(^$a-��)/d(=�J+� $C@s(+-:1��)/d(8@A1v;B5 #C;<�)/21�5 eI�j@s=46)�F6?(+�)(X6 J+1��akê#X*X#�@A1�*X�?9)#0*%(jGE5 # ] eI(E! �)1I=41I6'18eI(E6'#4@A1x=�1x! � ��1�P.1�� 5)$I9B�j@A1�*,(@A14;B5)(+&)� 5)(�J+#�/2#%�)(qgX�)J+�8eI(j;B#X*%#+n�1�5 # � t4u���e�$'*X14;B1�5)1Q!)� 1�J+!)#%=4Jj(X*X6'1XpCeI(q! �)1I=41I6'1��d�8�.*,(jGE�8=v9)14;B1�bB(Q@A1v;B5)(X& � 5)()L

Y1 = Y2 + Yg1−2

km� u ÿ%th�;B1.@A1%L

Yi =piρ

+ gzi + αiv2i

2

km� u �,tYi ] $'6 $'! 5)(H1�5)1�� hI�j@s(H! #�@A1v;B�)5 � r �i/d(X=�186'#4@A$07:9)$'�';i! #,=41v;<$I@A18$

i ] J+#,/`! � 1C=41I6 $0r 1I*X�Ypi = 1, 2

upiρ

] ! � �)J+�)=45B(>1�5 1I� h��j@s(>!)#�@A14;B� 5 �)r �F/d(X=�1 ] 1I5 Jj(j9)!)�j@s(gzi ] ! #+9B#+GH(4@A5)(H1�5 1I� h��+@s(�!)#�@A14;B� 5 �)r �F/d(X=�1αiv2i

2] 6 � 5)1�J+� r 3,(>1�5)1�� hI�j@s(>! #:@A1v;B�)5 � r �F/d(%=41

ð.149B� & � 5)(αi

=�1�5B(+e��)*%(0® {BPQTQ{)[QT2VX{B|lWY{BR'ÕKT � Tn�CR%UQ� �?9B� WY{BPQR,W � T2{)UQT�WY{BR'ÕKT � Tn�CR,U\�`WQTQUQR,�?T�}\WYRR,U2R%P�¡+Tn�CR u w *%�)/�6'#X1I7��)r �j@A1�5)J+#%/�=�106'#X� �)h�$C@A1fhI� 1���6?(i6'#4@s(o=�10& �)5 �Ö(X6,#�=41f6 � 5 1IJ+� & 6?(^1�5 1I� h��j@s(^�)(X& $'5)(! �)1�6'#�=4�)1v;B-:1FgX�)e�� 5 1o6'(X#

v2i /2

pQ@A1I��7�� e��)& 6 �ZgX�)e�� 5)(a/21I-:(`! #`! #%! � 1�&)5 #%/�! � 1C=41I6 $Ùkm=4� 14;B-�(agX� eI� 5)(K@A1

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼¼X

eC(X/2� ��n:1I5)(>gX� eI� 5)(,tvudÔ~(X6,#Eb)1:#%*%(4@�6'#X1C7�� r �j@A1I5 J>eI(+*%�)=4�)J+�f#C;i� 1�G>�B/d(E=4J+� $C@s(X-:()L

α =

{

1.058,JX$,�)gX$C9)1�5)J+5 #D=4JX� $C@s(+-:1

2,9B(X/2�)5)(+� 5)#>=�J+� $C@s(+-:1 km� u��Xt

ð.�';B�B/2#0;�(x@A1>6'#C;^J+$'� g%$I9B1�5 J+5)#XhE=4J+�)$I@s(X-:(α ≈ 1

k�$'*%1�6ob)1I/2#K$%=�*%(v@s(XJ+�α = 1

eI(fJ+$'� g%$I9B1�5 J+5)#K=�J+� $C@s(+-:1Ct4p;B#%6�@A1`6'#C;´9B(X/2�)5)(+� 5)#Xh`=4JX� $C@s(+-�(

α = 2u  �(�@A1

αlam > αtur/2#+G�1C/2#l;<(ZeC(+6?nq$'& �)/2#���5B(�#%=45)#X*X$

� eIh¾9)1v;�(�!)� #%7��?9<(�g%� e��)5 1�eI(:#X*,(�;B*%(:� 1�G>�B/d( ] 5)(+�)/21%pX6'#C;EJ+$'� gX$C9)1I5 J+5 #%h�=4J+� $C@s(X-:(�$�*X1�b)1I/F;B149)$�! #X*%� ���)5 1! #%! � 1I& 5 #Xh.! � 1C=41I6'(�r 1I*X�x�)/d(%/2#8=v9)$'& $C@�;<(�@A1qgX� eI� 5)(�! � � gX9)�?GE5 #�6'#X5)=�Jc(X5 J+5)(�k�J+$'� g%$C9)1�5 JX5 #.@A1�e�hI� #,t4p+Jj(+6'#�;�(@A1�! � #%7�� 9Kg%� e��)5 1x6'#C;FJ+$'� gX$C9)1I5 J+5 #%h�=4JX� $C@s(+-�(D=v9)�)& 5 �j@A�F!)� #%7��?9B$f=4�)1v;B-:1�g%� e�� 5)1Xp)5)1�h�#0��J+#8@A1�J+#0=v9B$,&B(v@eC(K9B(X/2�)5)(+� 5)�`! � #,7��?9~u ¥ 6'#^$^eC(c;<(XJ+6 $�5 �j@A1"5)(+h¾9B(X��1I5��)14GE�)/�=4JX� $C@s(+-�( pd! #C;B�B(+e�$?/21I*%(i=41E;<(F=41f�)(c;<�a#J+$'� g%$C9)1�5 JX5 #D=4J+�)$I@s(X-�$ �

ä àV��à ä àãâ Ï â{æ�â{æ{ê"è

���b��� ��� �V������ �

v1

v2

z1

z2

£ 9)�)6'(E� u ¢)L~ �1I#�r)1�*X#%*X#C;<(�$06'#%/218=�J+� $C@A�F7:9)$'�';

��#,=v9)14;B-��8& 9B(+585)(q;B1I=�5 #4@Y=4JX�)(+5 ��@A1v;B5)(X& � 5 1�km�?u�ÿXt�! � 1v;�=4Jj(+*Xn�( ¡X¸Y»)TQ�?WYR�VX�?P\¸v�IU2R�R,UQR,P�¡+Tn�IRxO\PQT~[2TQWY{BSVX�?P\¸v�CN?Ø"No� R{}QUQ{<V+�?TZ{%¹�O\P2R'VXR,WYN�"WYNFOQP2R,V%R,W\¸l¼ p'�f#%5x@A1�@A1v;B5B(+6"e�gX�)� $>9)#X6?(+9)5 �?3Eh�$'g%� Jj(+6?(x1I5 1�� hI�j@A1��hI$,g%� J+6 $f1�5 1I� h��+@A18$%=v9B1v;iJ+� 1�-�(>! � � 9)� 6'#%/l=4J+�)$I@s(X-:(>JX1�& 5 #,=4J+�K#C;i! � 1C=41�6?("��6?(�!)� 1I=�1�6 $K�?u

Yg1−2=∑

Yglok1−2+∑

Ygtr1−2

km� u �,t­ {BWYN'[2U\T ¡X¸Y»)T � T�R,U2R,P�¡+Tn�CR

�8#%6'(+9)5 �EhI$,g%� r �D1I5 1��)h��j@A1�=41�;B1��H(+*,(v@A$H5)(x5 1I6'#%/a#C;B�)1��'1�5 #,/_/21I=4JX$D$>r 1�*%#X#C;B$Hh�;B1�=�1�#C;<� 1��'1�5B(81�5 1I� h��+@s(7:9)$'�';�(i�+#C;B$'eI/21C�+#C;`7:9)$'�';<(K�a#%5)(i=41E6'#X�)�)=4J+�aeC(i=4J+*,(+�)(X-�1E5)1�6'#XhD=�1�6 $'5';<(X� 5 #%hD=4J+� $C@s(X-:( ] J+#F=4$_/21I=�Jj(h�;B10r 1�*F/21I-:(i!)�)(+*,(+ra=4*%#XhE! � $,G>(+-�(`k�6'#+9B1�5 #Bp~6 � �)*X� 5)(,tvpd/21I=�Jj(ih�;B1H;B#X9B(+eI��;B#F=4$,G�1I-:(F�?9)�a! � #%��� �)1�-�(! #%! � 1I& 5 #Xh0! � 1I=�1�6?( pd/21C=4Jj(i$_r 1�*%#X*%#C;B$F5)(i6'#�@A�)/d(o=�1D5)(j9<(+e�10*X1�5)J+�?9)�Ypd�)J�;Qu £ *X�a#%*X�ah�$'g%� r �a1�5)1�� hI�j@A1f=�1#C;B�)1��'$C@A$D5)(H#%=�5 #X*%$Dð.(4@s=4g,(j3%#%*X#%h�#Xg%�)(X=�r)( L

Yglok= ζ

v2

2

km� u �,t

h�;B1�@A1�#C;BhI#X*%(X�)(v@A$'b �ζ6,#%1I7�� r)�j@A1�5 JD9)#X6?(j9B5 #Xh8hI$,g%� J+6?(E1I5 1�� hI�j@A1�� 9)�F6'#X1C7�� r �j@A1I5 JD#XJX! #X�)()u�ð.� 1v;<5 #%=�J+�

#%*X�?3`6'#X1C7�� r �j@A1I5 Jj(ob)1"$Ö=4*X�B/´eI(j;<(+r)�)/d(og%� J+��$'5)(+!)� 1v;�! #XeI5)(+J+1KeC(o6'#X5)6 � 1�Jj(X5Ör 1I*X#%*X#C;\u�Ô\(+6'#j�'1Xp�=�J+*%(X�

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼¼�

;B#%h�#%*X#X�B(H@A1D;<(D@A1 »BPQMXTQUQN v »)PQMXTQUQN�T\M,N�[d{BWYN'[QU2{�¡Ö{B�?OQ{BPQN u_Ô\(+6'#ob)1f� 1Ir �)/2#^eI(o=�$,G�1�-:10!)#X! � 1I& 5 #%h! �)1I=41I6?("km=v9)�)6'(D� u �%t�h�$'gX� Jj(X601�5)1�� hI�j@A18gX� JX�"#C;<� 1��'1�5"� eI�)(+eI#%/qL

Yg = ζsuv22

2

 �(X6?9)1Xp�1�5 1I� h��j@s(^$a� eI5 #%=�$ζsu

v22

2

@A1K�+#C;<$,eI1�Jj(X�X#C;Ö7:9)$'�';B5 1K=4JX� $C@A1"eC(o=�J+*%(X�)(+-:1f*%� JC9)#Xh4(o5)(�/21I=4JX$`h�;B1�@A1;B#%�x9)#�;B#>! �)#%/21�5)18! � 1I=�1�6?( u

�%1v;B�)5 ��9B#X6?(j9)5)��#XJX! #X�l6'#C;l6'#Xh4(^b 1K5 1K6'#X� �)=�J+��ð.(4@s=4g,(j3%#%*F#Xg%�)(+eC(+ri@A1"9)#%6?(j9)5 �l#XJX! #X�Ö$%=v9B1v;Z5)(+h¾9)#%h! �)#X���)� 1�-�(>r 1�*%#X*%#C;<( p'=v9)�)6'(E� u ûBu

v1 v2

A1

A2

£ 9)� 6?(E�?u û)Ld�8(+h¾9B#�! �)#X���)� 1�-:18! #X!)� 1�& 5)#Xh�! � 1C=41�6?(

ü~$,g%� Jj(+6"1�5 1I� h��+@A18$%=v9B1v;i5)(+h¾9)#%h�!)� #X��� � 1�-�(E=41�#C;B�)1��'$C@A1:6'#%� � ��b)1�-:1I/�P�#%�%;<( ] �8(+� 5)#X#X*%1�7�#X�B/2$I9B1XL

Ygnp=

(v1 − v2)2

2

km�?u¡���,t

� ¸Y»)T � T�R%U2R,P�¡XTz�CRK¸BV+[2RX¹��?P2R,ØfNw *%��h�$'g%� r �H1�5 1I� h��j@A1�=�$x! #,=v9)14;B� r)(:J+� 1I-:(��)eI/21��'$xJX1�& 5 #,=4J+�>�>e��?;<(:r 1�*%�YuQ���)�?9)� 6'#%/^=4J+�)$I@s(X-:(:g%� e�� 5)#%/

v6)� #Xe�r 1I*H;B$,G>�)5 1

L�K! #%*X� ��� 5 1�! #%! � 1�&)5 #Xh�! � 1C=41I6'(

A#%5"=�1:#C;<� 1��'$C@A1�5)(>#%=�5 #X*%$H �(+�)=��j@A1�*%#Xh:#XgX�B(X=4rB( L

Ygtr= λ

L

4Rh

v2

2

km�?u¡�X�ct

ð.149B� & � 5)(Rh

=4185B(+e��)*%(�� TY¹dPQN%¸'[2T�}QW\T�PQN,¹dTn��¸<V �K#X5K=41:;<1I7�� 5 �)��1�6?(+#>#C;B5 #,=�! �)#XJ+#%& 5 18! #%*X� ��� 5 1��#%6 *%(X��1�5)#Xh:#XgX�B/d( p?Jv@4u

Rh = A/Ou�Ô\(+6'#:@A18eI(H6 � $,GE5 $0r)1�*>! � 1I& 5 � 6?(

D3'�?;B�)(c$C9)� & 6)�"�B(c;B�j@A$,=.@A1v;B5B(+6YL

Rh =A

O=D2π

4Dπ=D

4

!)(f=41� �(+�)=��j@A1�*,(D7�#X�B/2$I9<(DeI(f=v9)$'&)(4@�=4J+�)$I@s(X-:(f6 � #%e�r 1I*X��kê��J+#"b 1�gX� J+��=v9)$'&)(4@�$F=4*%�)/�eI(j;<(+r �B/d( � tF=4*%#C;B�5)()L

Ygtr= λ

L

D

v2

2

km�?u¡�C�%tð.149B� & � 5)(

λ=�1�5)(Xe�� *,( WY{<R,ÕKT � Tz�CR,U\�"�?P2R%Ø"N �0#%5D=41�$H#%! ��JX1I/`=49)$'&)(v@A$HeI(X*X�)=��D#C;fï21A@A5 #X9?;<=�#X*X#%h�gX� #�@s(

�a� 1�9B(+JX� *X5 1D3'�B(+!)(X*X#%=�J+��r 1�*%�kuf� =49)$'&)(v@A$^9B(%/2� 5)(+�)5 #XhD=�J+� $C@s(+-�(ir 1�*%�a=41E! #%5)(+�H(v@A$o6?(X#K3'�?;B�)(c$C9)� & 6)�

h¾9B(XJ+6'1Xp�6'#%1I7�� r �+@A1�5 J`JX� 1�-�(`eC(+*X�B=4��=�(%/2#`#C;Re

��Jj(aeI(X*X�)=�5 #%=�J`=41i;<#XgX�+@s(a� eoJj(X& 5 #Xho� 1���1�-�(��8(+*%�j@A1 ]

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ��

£ J+#%6?=4#X*%�?3x@A14;B5)(+&)� 5)(HeI(>=v9B$,&B(v@�=4J+� $C@s(X-:(>$Dr)1�*X�B/d( p?Jv@4L

λ =64

Re− 9B(X/2� 5B(+� 5 #D=�J+� $C@s(+-:1 km�?u¡�I¢,t

�`=49)$'&)(v@A$:J+$'� gX$C9)1I5 J+#%h�=4J+� $C@s(X-:()pCg%1�ed#%gXe��)�)(�;<(.9)�8=41�r)1�*�! #%5)(+�H(�6?(X#�3?�';<�)(c$C9)� & 6 �8h¾9<(+J+6?( p�3?�';B�)(c$C9)� &)6 �3?�)(X!)(+*,(F�?9)�Ö3?�';B�)(c$C9)�)& 6 �l! #XJ+!)$,5)#o3?�)(X!)(+*,( pQ;B#%gX�j@s(X-�1KJj(+& 5)1feC(+*X�B=45 #,=4J+�

λ(Re, k)5 �j@A1i/2#%h�$'b 1Xp~@A1���=41

5 1��)(X=�! #+9B(+G�1a! #%J+� 1IgX5 �)/ Jj(+&)5 �)/�� 1I��1�-:1I/·ï21s@A5 #+9 ;<=4#%*X�?3`@A14;B5)(X& � 5)()u �Y(+J+#�=�1`5 1I! #XJ+!)$,5)��J+1�#%� �j@s=46)�� 1Ie�$C9)Jj(+JX�Fï21A@A5 #C;<=�#X*%�?38@A1v;B5B(+& � 5B(D/2#X�)(4@A$E;B#%! $'5 � J+�K#C;BhI#X*,(+�)(4@A$,b)�)/`1I6 ! 1��)�)/21�5 Jj(+9)5 �)/l! #C;<(Xr �)/d()u��8(>Jj(v@5)(X& � 5i=41�;B#%�x9)#�;B#E7�#%�)/2$C9B(>6'#4@A18=4$>;�(+J+18$DJj(XgX149B�Yu

ï�u gX� #�@ ��(X*X�)=�5 #%=�Jλ = λ(Re, k)

w g+9B(%=4JH! � �)/21I5 1 ¥ $'J+#%�

�Xuλ =

64

ReRe < Rek = 2320 ]

�?uλ = 0.0025Re0.333 2320 < Re < 4000

��(v@A& 1I5 6'#

¢ uλ = (1.8 logRe− 1.5)−2 4000 < Re < 3 · 106

��#X5)(X6'#X*

û)uλ = 0.3164Re−0.25 4000 < Re < 105

P�9<(+e��+@A$%=

ÿ?u 1√λ

= −2 log

(k

3.71+

2.51

Re√λ

) ��#X9)gX�)$,6 ] ðq(v@AJ23

k< Re <

560

k

� uλ = 0.1

(

1.46k +100

Re

) ¥ 9)J+��$C9

�'uλ =

(

1.74 + 2 log1

k

)−2 ���B(+5 JC9 ] ��� 6 $'�)(c;<e�1Re >

560

k

� uλ = 0.11k0.25 � �)7�� � 5B=4#X5

� uλ = a+ bRe−c

a = 0.0094 k0.225 + 0.53 k Re < 104ð.$I;

b = 88 k0.44 10−5 < k < 0.04

c = 1.62 k0.34

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ�

Ô\(+6'#j�'1XpC! #X� 14;x� e��)(XeI(�;<(XJ+�?3�$�Jj(XgX149B�YpI6'#%1I7�� r �+@A1�5 J�J+�)1�-�(�=41�/2#XG�1d#C;B�)1v;B� JX�8�86,#%� � ��b 1�-:1I/ � $I;B�j@A1I*X#%h;B�+@s(+h��B(X/d(Kk�*%�';B�K! � 14;<(+*,(+-�(%tvu

��eI(j;<(+r)�)/d(Hb 1�*X� 14;B5 #%=�J�6'#%1I7�� r �+@A1�5 Jj(�J+� 1�-�(>$'5)(+!)� 1v;igX� JX�feC(c;<(XJc()p'�?9)�Kb 1�gX� JX�"eC(c;<(XJj(�7�#%�)/2$C9B(�5B(#,=45 #%*X$E6'#4@A1x=�1:#%5"/2#XG�1x=��)(+& $'5)(XJ+�Yu¯ ¸YS:O2No¸ � R,|�{)|�{%¹d¸

�úh�#%J+#X*%#f=�*X�)/�eI(c;�(+r �)/d()p~(fJc(X6'#j�,1H��$F! �)(X6'=��Yp�$Fr 1I*X#%*X#C;B$Kb 1>gX�)J+�^$'hI�)(+�'1�5B(0! $?/2!)()u���J+� r)(4@:!)$'/2!)15)(0=4J+� $C@s(X-�1�b 1xgX�)J+�i#Xg%$%3?*,(+b 1I5f! �)1�6'# �CRX¹2TQU\T�}QUQ{�¡KPQN,¹�NaOQ¸YS:OQR p Yp [J/kg]

6'#4@A�F=41�/2#+G�1�=v3?*%(XJ+� J+�i6?(+#1I5 1�� hI�j@s(�! #�@A14;B� 5 �)r �"/d(%=41�6,#�@A$E! $?/2!)(�! � 1v;�(v@A1�7:9)$'�';B$�u#�x�)e�� & 6)�f! #,=�/d(XJ+�)(+5)#)p,! $?/2!)(��X#Xg%#Xh4(+b)$I@A1C�X7:9)$'�';! �)� J+�)=�5 #%/l1�5)1�� hI�j@A#%/qu

�n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n�

H

Yp

��

��

£ 9B� 6?(>� u ÿ L £ J+� $C@A5)(E/d(+��� 5)(ikê! $?/2!)(%t.$Dr)1�*X#%*X#C;B$

� [dN?U Yp V%RF¸���R,PQU\¸'[2Tn�IR,|�{"�ø�IRX¹dUQNÍ}\TQU\T�¸�|<R,W�OQTQ}_RoVXN�[dR%|�RFVX�?PQN?U2R [email protected](E=v9)$'&)(4@�=�(E=v9B� 6'1��?u�ÿkm=�/21��i=�J+� $C@s(+-�(:@A18#C;K!)� 1I=�1�6?("�:6?(H! � 1C=41I6 $"�%t

Y1 + Yp = Y2 + Yg1−2⇐⇒ p1

ρ+ gz1 + α1

v21

2+ Yp = Yi =

p2

ρ+ gz2 + α2

v22

2+ Yg1−2

km�?u¡��û't�8(X6,#�@A1

z2 − z1 = Hpc@A1v;B5B(+& � 5B(ikm�?u¡��û't.=418=4*%#C;B�f5B( L

p1

ρ+ α1

v21

2+ Yp = Yi =

p2

ρ+ gH + α2

v22

2+ Yg1−2

km�?u¡�Cÿ%t��#%� �)=45B(��?9B�f3?�';<�)(c$C9)� & 6?(>=45)(Xh4(H! $?/2! 1x=41�#C;B� 1I�'$I@A1�5)(H#,=45 #%*X$E� eI�)(+eC( L

Pk ≡ Ph = ρ V̇ Ypkm�?u¡���,t

£ 5)(Xh4(86'#4@s(q@A1:! #%J+� 1�g%5)(8eC(x! #%h�#X5E!)$'/2!)1Hkm=45)(Xh4(86'#4@A$�@A1�! #%J+� 1IgX5 #x$C9)#XGE� J+�E;�(xgX�f=�/2#8;<#XgX� 9)�0=45B(+h�$Pk

tvL

P =Pkηp

=ρ V̇ Ypηp

km�?u¡�*�Xt

h�;B1.@A1ηp

kê/21�3,(+5)� & 6 �Btq=�J+1�!)1�5K6'#X� �)=�5 #%=�J+�"! $?/2! 1Xu

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ<¼

'DH�'DH I KMLON1L�PRQ�Xu���(�! � #%=�J�r)1�*X#%*X#C;H=I(�=v9)� 6'1.#C;B� 14;B� J+�E/d(+6?=��)/d(j9)5)$8*X� 14;B5 #%=�J�5B(+J+! �)� J+�)=�6'(

pm! � �>6'#4@A#4@\b 1�=4J+� $C@s(X-�1�$

r 1�*%#X*%#C;B$xgX�)J+�8@A#X��$'*X1�6>9<(X/2� 5)(X� 5 #BuQ��� �)J+� & 5 �"ï21A@A5 #X9?;<=�#X*8gX�)#4@deI(x=�J+� $C@s(+-:1:$�r 1I*X�8@A1Rekr = 2320

u �(+J+�E=�$��D=v9B1v;B1Ib �>! #C;<(+r)�YL

ρ = 900 kg/m3pν = 4 ·10−5 m2/s

pD1 = 80 mm

pD2 = 100 mm

pL1 = 18 m

pL2 = 10 m

pζu = 0.1

pζv = 3.5

pζk = 0.2

pH = 4 m

u�?u��8(�� 5)=4Jj(+9B(+r �+@A�Z5)(+! $'-:1�5 #�@E*X#C;B#,/´� eI*X� ��1�5�@A1F1�6?=�! 1�� �B/21�5 J`$Ö6'#%/21F=41o! #%/2#%b $a;B�B7�1�� 1I5 r �j@s(+9)5 �?3

/d(+5 #,/21�Jj(+�B(f=I(DGE*X#,/�/21I� 1>!)(c;B#%*X�^! � � J+�B=�(+6?()u��8(f#%=45)#X*X$i!)#X6?(+eI� *%(X-:(f/d(+5 #,/21�Jj(+�)(f#C;B� 14;B� J+��6'#X1 ]7�� r �j@A1I5 J"9)#X6?(+9)5 #Xh�#XJ+!)#X�)("*X1I5 J+�?9<(ζv0

6'(X#f�_=45B(+h�$i$'hI�)(+�'1�5)1H!)$'/2!)1Xu £ J+� $C@s(+-:1�@A1>J+$'� gX$C9)1I5 J+5 #i$3?�';B�)(c$C9)� &)6 �ih¾9B(+JX6 �)/�r 1I*X�)/d()u. �(+J+�F=�$"=49)1v;B1Ib �F! #C;�(+r �YL

l = 3 mpl0 = 5 m

pL = 25 m

pD = 50 mm

ph = 11.8 mm

ph0 = 100 mm

pH = 2 m

pζu = ζk = 0.5

pζv = 2

pρ = 1000 kg/m3

pρm = 13600 kg/m3

ν = 1.006 · 10−6 m2/s�ηp = 0.8

u¢ u���(�r 1�*%#X*X#C;H5)(�=v9)� r)��#C;B�)1v;B� JX��! �)#XJ+#%6�*%#C;B1�6'#�@s(�=�1q=49)� *,(�� eq� 1Ie�1I� *X#,(+�)( ¥ $�� 1�eI1�� *%#%(+�Hð:uX��(�6'#X9)� 6'#

=41�! #%*X1�bB(�! �)#XJ+#%6f6 �)#Xe�r 1�*%#X*%#C;K(+6'#>=41�$D-:1�h4(H$'h��)(j;B�"! $?/2!)(H& �j@s(:@A186?(+�)(X6 J+1I� �)=4JX� 6?(x;�(+Jj(��)e��)(Xe�#%/qL

Yp = Y0

1 −(

V̇0

)2

��#XeI5)(+J+�H=4$�=49)1v;<1�b �>! #C;<(Xr �YLY0 = 60 J/kg

pV̇0 = 40 l/s

pH = 4 m

pL1 = 12 m

pL2 = 15 m

pD1 = 80 mm

pD2 = 100 m

pλ1 = 0.025

pλ2 = 0.027

pζu = 0.1

pζv = 7.5

pζk = 0.2

ppm = 0.2 bar

�pv = 0.1 bar

u

� � 8D>F8°�@:°  A El�=>L¡ J¢A;>£G I;J�KL>"KM>�� �

òD��� ¤ ÇKÀ�Ê�À8ÏÄÊ�È���ÇKÃ�õ~É�Ïià Ð0�������#C;Ö!)� 1�J+!)#%=4Jj(X*X6'#%/�;<(�=41"�)(j;B��#_=4Jj(+r)� #X5)(X� 5 #,/qp�@A1v;B5 #C;<�)/21�5 eI�j@s=46'#,/qp�5 1C=4J+� ��n:�)*X#%/�=4J+�)$I@s(X-�$�k

ρ =

consttvp)! � �K! � #%�)(+& $'5 $f=v9)#XG�1�5)#Xh�r 1�*%#X*%#C;<(�=�1:6'#%� �)=4JX1:;B*%(HJ+� !B(H(+9)h�1Ig%(X�)=46 � 3x@A1v;B5B(+& � 5B( L

• �X14;B5)(X& � 5)(H6'#X5 JX� 5 $'� J+1IJc(• P.1�� 5 $C9)�+@A1�*%(�@A14;B5)(+&)� 5)(

IDHÃG�H�G WX� N��øL�YRQ��øL �Z�����ÛQ���� Q�������L�%1v;B5B(+& � 5B(i6'#X5 J+�)5 $'� J+1�Jj(o! � �`! �)#X�)(X& $'5 $�=49)#+GH1�5 #%hDr 1I*X#X*%#C;<(K=41f#XgX�)& 5 #i! � ��10eC(F/21I=�Jc(o$_6,#�@A�)/d(F=�1

r 1I*X5 1�;B1I#X5 �)r 18h��)(X5)(v@A$0�?9)�i=4!)(4@s(v@A$ ] J+#E=4$0&)*X#X�)#X*X�"/2� 14GH1Xp � 9)�K�)(+& *%1Xu

V̇1

V̇2

V̇n

V̇n+1

V̇n+2

V̇n+m

R

£ 9)� 6?(E¢)u��%L�¥�*%#X� 5)#H/21C=4J+#>$f/2�)14GE�

�%1v;B5B(+& � 5B(�6'#%5 J+� 5 $'� JX1�Jj(>5)(+!)�)=�(X5)(�eC(H& *X#%�"=I(H=49)� 6'1x¢ u¡��h¾9B(X=��YL

n∑

i=1

V̇i =

m+n∑

j=n+1

V̇jkê¢)u��ct

Jv@4u VX¸YS:N_OQP2{)� {BWYN_WY{"�IT�¸'[dN?M,Ro¸ }Q|<{BP�¦�P2Nn}Q|B¸�§ �CR �IRX¹dUQN WYNaV+¸YS:T�OQPQ{B� {BWYN_WY{"�IT�TQM+[dN?M%R^TQM }\|�{BPQN u��eC(c;<(Xr �)/d(x=�1�& 1I=�J+#�/2#XG�1�=�� 1I=�J+�D�D=v9B$,&B(v@d6?(c;<(�=41�5 � *X#�$�� 1Ie�1��)*X#X�)$�#C;B�?G>(+*,(�6'#%5)=4Jj(+5)J+5 �)/_eC(j3?*%(+n.$C@A$'b �=�Jc(+9)5 #,/a! � #%J+� r)(X-�$"7:9)$'�';<(�6 � #Xe:-:1�h4("kê�X/d(+9)�)�X� 1�eI1�� *%#%(+�<t ] $E#X*%#%/`=49)$'&)(v@A$EJj(+6'#j�'1�*,(jGE�0�B=4J+�"! � � 5)r � !YL/2#%�)(HgX� J+�"eI(j;B#X*%#+n�1�5)(�@A1v;B5B(+6'#%=�J�$'6 $'! 5 � 3E;B#XJX#X6?(�$0� 1�eI1�� *%#%(+�0�"$'6 $'! 5)�?30! � #%J+#X6?(�6'#4@A�"� e49B(Xe�1:� e�� 1�eI1�� ]*%#%(+�B( u

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ)þIDHÃG�H(' �����������¤Q��t���ÔL1�t��N��RL�YøQ��RLP.1��)5 $C9)�j@A1�*,(Q@A1v;B5B(+& � 5B(�=41�!)� ��1�5B(�! #XJX! $'5 #��)=�J+�85)(+&)� 586?(+#�$�=v9)$'&)(4@A$�! �)#X�)(X& $'5)(�=v9)#XG�1I5 #Xh2r 1I*X#X*%#C;<( pI�

/2#XG�1q=41q! #%=�Jj(+*X�)J+��eC(�g%�?9)#�6'#�@s(�;B*,(�!)� 1I=�1�6?(�$86'#�@A�)/d(�@A1qeC(c;B#%*X#+n�1�5�@A1v;B5 #C;<�)/21�5 eI�j@s=46)��6?(+�B(+6 J+1I�H=�J+� $C@s(+-�( u��(��)(+e�9)� 6 $�#C;�!)� #X�)(X& $'5)(�! �)#%=4JX#Xh�r)1�*X#%*X#C;<()pI#%*j;B1Y@A1�5 1I#X!?3,#C;B5 #BpI! #%� 1v;�@A1v;B5)(X& � 5)(�6'#X5 JX� 5 $'� J+1IJc(�! #%=�Jj(+*X�)J+���@A#X� 5B(v@s/d(+-:1o;B*X1¨P�1I� 5 $C9)�j@A1I*X1f@A1v;B5B(+& � 5)1XuMP.� #[email protected]� 5 $C9)�j@A1I*X�?3_@A1v;<5)(+& �)5)(�6'#�@A1D@A1_/2#Xh�$'b 1^! #%=�Jj(+*X�)J+�eC(+*X�B=4�Z#C;�gX�)#4@s(_�)(X& *X�Z��g%� #4@s(_r)1�*X#%*X#C;<(�6'#4@A��=41o=�!)(v@s(4@s(v@A$�k�h��B(+5)(4@A$Ö$l�)(+&)*X�Bt4u���� �)/21I5)(©P.1�� 5)$I9B�j@A1�*%1@A14;B5)(+&)� 5 1o=�1F5)(v@Ag%#+n�1F/2#+GH1F�?9)$,=4J+� #%*%(XJ+��@A1v;B5 #,=4Jj(+*%5 �)/�! � �)/21I� #%/ú6?(j;<(a=4$ÖJ+� ��� 1Ie�1I� *X#,(+�)(_/21��'$,=4#Xg%5 #! #%*X1IeI(+5B(:=49)#+GH1�5 �B/or 1I*X#%*X#C;B#,/`kê=49)� 6?(x�Xt4u\��#%*X#,/F6'#X5 6)� 1�J+5)#%/^! � �)/21I� $�JX1�& 5 #,=4Jx=�1�=v9)� *,(�� e�� 1Ie�1��)*X#%(X�)(��6?(H� 1�eI1�� *%#%(X� �)/d(����K¢Kkê=�/21I� #X*%�"=�J+� $C@s(+-�(H$>;B1I#X5 � rB(X/d(>=4$D!)#Xe�5B(+J+�Bt4uÔ�8/d(v@A$'b �"$D*X�?;B$H;<(>=41yP.1�� 5)$I9B�j@A1�*,(@A14;B5)(+&)� 5)( ¸�|<R,W ! � ��1^$�=I/21�� $�=�J+� $C@s(+-�( p:=�(Ö! � �)/21I�Ù=I(�=49)� 6'10@A1a/2#%h�$'b 1o!)#%=4Jj(X*X� J+��;B*X1�P.1�� 5)$I9B�j@A1�*%1@A14;B5)(+&)� 5 1Xp)�"JX#i� ] �x�_� ] ¢ u

R

p1

p2

p3

v1

v2

v3

¼¼

ññ

H

h

£ 9)� 6?(E¢ u��?Ld��� �)/21I�i=v9)#XG�1�5)#Xh:r 1�*%#X*X#C;�(

P.1��)5 $C9)�j@A1�*,(:@A1v;B5B(+& � 5B(�eC(H5 � *X#%1�$0� 1Ie�1��)*X#%(X� �)/d(f���i� L

Y1 = Y2 + Yg1−2 ⇐⇒ Y1 = Y2 + Yg1−R + YgR−2

p1

ρ+ gH =

p2

ρ+ C1

v21

2+ C2

v22

2

kê¢)u �%tP.1��)5 $C9)�j@A1�*,(:@A1v;B5B(+& � 5B(�eC(H5 � *X#%1�$0� 1Ie�1��)*X#%(X� �)/d(f���K¢)L

Y1 = Y3 + Yg1−3 ⇐⇒ Y1 = Y3 + Yg1−R + YgR−3

p1

ρ+ gh =

p3

ρ+ C1

v21

2+ C2

v23

2

kê¢)u ¢,tw *%�)/i@A14;B5)(+&)� 5)(X/d(HJ+�)1�g%(>!)� �';B#C;<(XJ+�"��@A14;B5)(X& � 5 $f6,#%5 J+� 5)$,�)J+1�Jj(>eI(H�)(X& *X$�L

V̇1 = V̇2 + V̇3 ⇐⇒ v1D21 = v2D

22 + v3D

23

kê¢)u û't��e�@A1v;B5)(X& � 5)(^kA�%u �%tvp�kA�%u ¢,t��lkA�Xu û,t�=41>/2#XhI$"#C;<� 1v;B�)J+�og%� e�� 5)1�=4JX� $C@s(+-�(0$K! #�@A1v;B� 5)�)/�;B1I#X5 �)r)(X/d()p�(X6'#0=4$

eC(c;<(XJ+1�*X�)=�� 5 1Hphp 6?(+#H�K1�6 *%� *%(+9)1�5 JX5 �f6'#%1I7�� r �+@A1�5 J+�K#%J+! #X�B(�eC(Hr 1�*%#X*X#C;<1��%p)���i¢)u

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ��

V̇1V̇1

V̇2V̇2

V̇3

V̇3

ζR2ζR2

ζR3

£ 9)� 6?(>¢ u ¢ L £ J+� $C@s(+-:18$0�B(+& *%�

��� �H=4J+�)$I@s(X-�$x7:9B$,�?;<(�6)� #Xe��)(+& *%$:;B#+9<(+e��8;B#�� eI*X1C=45 #%h�h�$'g%� J+6?(�=�J+� $C@A5 1.1�5 1I� h��+@A1q7:9B$,�?;<( pc�HJc(4@~h�$'g%� Jj(+6=�18$,eI�)/d(H$0#%gXe��)�0! � 1I6'#E6,#%1I7�� r)�j@A1�5 Jj(�9)#X6?(+9)5 #Xh�hI$,g%� J+6?(H1�5 1I� h��+@A1Dk 9)#%6'(+9)5 #%h�#%J+! #X�B(%t�$0�)(X& *X�

ζRu

��#%1I7�� r �+@A1�5 JζR

=�1 ¸�|<R,W *X1�eI$C@A1ieI(ag%� e�� 5)1F6'#4@A1o�X� e49B(Xe�1C�+� eK�)(X& *X1%u�Ô\(+6'#`eC(^;B*%(a6?(X�)(+6 JX1�� �)=�J+� & 5B(! �)�)/21��)(>eC(>=v9)� 6'1x¢)Lkm(%t��8/d(X/2#xhI$'gX� J+6'1�1I5 1��)h��j@A1

YgR2 = ζR2

v22

2

�YgR3 = ζR2

v23

2

u ¥ 6,#�@A1�$>eI(j;<(+J+6)$�eC(c;<(XJ+#�=�(%/2#ζR

p?(8�)/d(%/2##X*,(v@�=49)$'&)(v@q#X5';�(

ζR*X1Ie�$C@A1I/2#>�KeI(Hg%� e�� 5)$

v1�v2p J4@4u

YgR2 = ζRv22

2

�YgR3 = ζR

v23

2

ukêg,tf��#X*%#%/`=49)$'&)(v@A$f�)/d(X/2#>h�$'g%� Jj(+6"1�5 1I� h��j@A1

YgR = ζR2

v22

2

òD�¾½ ª�À��^Ày«iÃ

�� ��$?/2! #%/Ù=41D!)#XJ+�)=�6 $C@A1D*X#C;�(f�)eD*X1�9)� 6'#Xh>eI(XJ+*X#%� 1�5)#XhE� 1Ie�1��)*X#%(X�)( ¥ $^*%149)�)6 �a#XJ+*%#X� 1I5 �_� 1�eI1�� *%#%(X��ð��/Q9B(Xe�5 � r)$ £ uQÔ~1�#X�)�j@s=46?(^*X�)=�� 5)(o/Q9<(+eI(o*X#C;B105)(F� e�9B(+eI$_� ef/Q9B(Xe�5 �)6'(o� e�5)#%=4�

ht = 5 muo �(+J+�Ö=4$

=v9)14;B1�b)�^! #C;�(+r �YLV̇B = 6 l/s

pd = 60 mm

pD = 150 mm

pl = 100 m

pl2 = 60 m

pH1 = 2 m

pH2 = 3 m

pλ = 0.03

kêeI(K=4*%1>r 1�*%�Btvpζk = 0.3

pζv = 3

pζv1 = 10

pζu = 0.5

pζm = 0.05

pζR = 0.25

ppv = 0.2 bar

�ηP = 0.75

u(,t w ;B�)1v;B� JX�i*X�)=�� 5 $

Hu

g't w ;B�)1v;B� JX�F=45)(Xh�$D6'#4@s(�@A18! #%J+� 1�g%5)(HeI(H! #%h�#X5"! $?/2! 1%u

¼6� ��$?/2! 1 P1

�P2

J+�)(X5)=4!)#X� J+$C@A$:*X#C;B$�� e��)1�e�1I� *X#,(+�)( ¥ $�� 1Ie�1I� *X#,(+� 1B�CuÛ�8=4J+#%*X� 1C/21�5 #Bp�� e�� 1�eI1�� *%#%(X�)(

B*X#C;<(l=41a*%�)(+bB(Ö!)#X*X�B(+J+5 #,/ hI�)(+5)#%/ $�� 1�eI1�� *%#%(X�

Cpx(�� ea� 1Ie�1I� *X#,(+�)(

C=�J+� $C@A��6?(�! #XJX� #X�H(+& $

6'#X5)=�Jc(X5 J+5 �B/_! � #XJX#X6'#%/V̇ = 3.5 l/s

udï21�e�1I� *X#,(+� ¥ @A1�*%149)� 6)�Yp'!)(�@A1:5 � *%#x$E-�1C/2$>6,#%5)=4Jj(X5 Jj(+5Yp'(x5 �)*X#X�*X#C;B1�$"� 1�eI1�� *%#%(X� �)/d(

B�C

=41x#C;<�?GH(X*%(v@A$06,#%5)=4Jj(X5 J+5 �)/l! �)#XJ+� rB(+-:1I/�6 � #%e8-�� 3�u w ;B�)1v;B� JX�F$'6 $'! 5 $=45)(Xh�$"6'#4@s(�@A1�! #XJ+�)1�gX5B(DeI(D! #XhI#X5i! $?/2! �Yu:��#XeI5)(+J+�^=4$K=v9)14;B1�b �o! #C;<(+r)�YL

l0 = l1 = 4 mpl2 = l3 = 4 m

pd0 = 60 mm

pd1 = 50 mm

pd2 = 30 mm

pd3 = 40 mm

ph1 = 5 m

ph2 = 3 m

pζs = 1

pζu = 0.5

pζk = 0.4

pζv1 = 2

pζv2 = 3

pζv3 = 4

pηP1 = ηP2 = 0.8

pρ = 1000 kg/m3

pλ = 0.028

u

ñ�� ��$?/2!)(�J+�)(X5)=4!)#X� J+$C@A1.*X#C;B$�� e��)1�e�1I� *X#,(+�)( ¥ 6 � #%e�� 1Ie�1��)*X#%(X�B! �)1I/d(�!)#XJ+� #%��(X& $

Pu\��r 1I*X�x! �)1�& 5 �)6'(

d = 50 mmp�6'#4@s(0*X#C;B�o6'(f! #%J+� #X�H(+& $K$'h��B(+�'1�5)(8@A1�gX9)1�5';�(D6?(+�)(X6 J+1I� �)=4JX� 6'1

K = 0.01242 m3/(s m1/2)6'#4@s(�@A1E! #X*%1�eC(+5)(f=�(f;B�)7�1I� 1�5)r �j@s(j9B5 �)/Ù/d(+5 #,/21�J+� #,/�$o6,#,/21�@A1>GE� *%()u>���)#XJ+#%6^6 �)#Xe>� 5)=�Jc(+9B(+r)�j@A$^=�1� 1�hI$C9)� ��1i!)#%/2#Xb)$�#X!)J+#X& 5)#XhK*X#C;<(lk�hI�)(+5B(a=�(_*X1I5 J+�?9B#%/

v2t4u w ;B� 14;B� J+��=�5)(+hI$`6'#4@s("@A1i! #%J+� 1IgX5)(�eI(

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ�5

! #XhI#X5F!)$'/2!)1Xu���#%e�5)(XJ+�^=�$F=v9)14;B1�b �^! #C;<(Xr �YLpm = 3 kPa

ppv = 800 Pa

pl0 = 4 m

pl1 = 2 m

pl2 = 3 m

pd0 = 65 mm

pd1 = 80 mm

pd2 = 50 mm

pH = 1 m

ph = 100 mm

pζu = 0.5

pζk = 0.5

pζk = 0.5

pζR0 = 0.5

pζR1 = 1

pζR2 = 0.5

pζv1 = ζv2 = 1.5

pλ = 0.025

�ηP = 0.8

u¬ j{�T­Íu�trs$j¯® ¥ 6,#�@A1�6'#X1C7�� r �j@A1I5 J06?(X�)(+6 JX1�� �)=�J+� 6'1�gX9)1�5?;B1

KpB! �)#XJ+#%6o7:9)$'�';<(06 � #Xe�r 1�*%#X*X#C;o$K6'#4@A�D@A1

$'h��)(X�,1I5)(�g+9)1I5';<(�@A18#C;B� 1I�,1I5f�)e��)(Xe�#%/V̇ = K

√hu

þO� ��$'! &)(%=4Jj(�! $?/2!)(Z� 1Ih�$C9)�)=I(+5)(�#X! J+#%& 5 �)/�*%#C;B#%/qp8!)� 1�g,(+r $C@A1`g%1�5 eI� 5�� e�#%J+*X#%� 1�5)#Xha� 1Ie�1��)*X#%(X�)( ¥ $#XJ+*%#X� 1I5 �x� 1Ie�1I� *X#,(+�xð:u ¥ 6'#:=�$�! #XeI5)(+J+�H=v9)14;B1�b)��! #C;<(Xr �YL

V̇P = 4 l/spH = 7.5 m

pLu = 8 m

pLp = 12 m

pd = 50 mm

pζu = 0.5

pζv1 = 2

pρ = 750 kg/m3

pηP = 0.7

pd0 = 30 mm

pL0 = 2 m

pζv0 = 20

pζk = 0.5

�λ = 0.02

p'#C;B� 14;B� J+�YLkê(,t> �#%J+#X6"gX1I5 e�� 5B(�$f� 1�eI1�� *%#%(X�

Bu

k�g't0ü~�)(+5 �)& 5 $>*X� 14;B5 #,=4Jx6'#X1C7�� r �j@A1I5 Jj(x#%J+! #X�B(8*X1I5 J+�?9B(ζv0

$H#%! J+#%& 5 #%/_*%#C;B$H! � �D6'#%/21�5 1I/d(8;B#XJX#X6?($f� 1�eI1�� *%#%(+�

Bp 6?(X#H�K=45B(+h�$0! $?/2! 18$0J+#,/Ö=49)$'&)(v@A$�u

k�*'t £ 5)(Xh�$D!)$'/2!)18$f=v9B$,&B(v@A$D6?(c;�(�@A18*%1�5 JX�?9i$D#X! JX#X& 5 #,/�*%#C;B$EeI(XJ+*X#%� 1�5~u

�6� ï21�eI1�� *%#%(+�)� ¥ �ið�=4$"=4! #�@A1�5 �i=v9B#+G�1I5 �)/�r 1�*%#X*X#C;<#%/qu2��#Xe�5B(+J+�K=4$0=49)1v;<1�b �K! #C;<(Xr �YL• 1I6 *X� *,(j9)1I5 J+5 �K#%J+! #X�)�D;B1I#X5 �)r)( L

C1 = 20pC2 = 12

pC3 = 10

�C4 = 15

• ! �)1�& 5 �)r �";B1�#%5 � r)()Ld1 = 50 mm

pd2 = d3 = d4 = d = 40 mm

• �)(Xe49)�)6'(H� eC/21��'$05)� *X#,(�$f� 1�eI1�� *%#%(+�)�)/d(H = 5 m

• pv = 0.1 bar�pm = 0.6 bar

w ;B� 14;B� J+�´eC(+! �)1I/2� 5)=�6'1`! � #%J+#X6'1�6)� #Xe�=�*X1_;<1�#X5)� r 1Xp�6?(+#Z��=�5)(+hI$�! $?/2! 1ÙkηP = 0.8

tK6'#�@A$�J+�)1�g%(! #%=�Jj(+*X�)J+�f$E;B1�#%5 � r $^��;�(�g%�i=41�! � #%J+#X6K6 � #%e�J+$E;B1I#X5 �)r $D$I;B*X#,=4J+�)$,&)� #)u

5�� �8(�=49)� r ��@A1�! �)� 6?(+eI(X50=v9B#+G�1I50r 1I*X#%*X#C;0$>6'#%/21:! $?/2!)(�=�5)(+gC;<1�*%(�*%#C;B#%/_�)1�e�1I� *X#,(+� £ � e��)1�e�1I� *X#,(+�)( ¥�Fð:uB��#XeI5)(+JX�F=4$K=v9)14;B1�b �o! #C;<(+r)�YL

L1 = 20 mpL2 = 15 m

pL3 = 20 m

pλ = 0.03

pH1 = 2 m

pH2 = 4 m�

pa = 1 baru ¥ 6'#f@A1i!)� #XJ+#%6

V̇1 = 7 l/s�l!)#X6?(+eI� *%(X-�1"#XJ+*%#X� 1I5 1

Ur 1�*%��=I(^G>�)*X#%/

h = 165 mmkH = 1 m

pρm = 13600 kg/m3

t.� eI�)(+& $'5)(XJ+�K6'#X� �)=�5 $f=�5)(+hI$D! $?/2! 1%p)6?(+#>�K! � � JX�)=�(X6i6,#�@A�K*+9B(j;<(H� e�5)(j;5 � *%#%(�J+1�& 5)#%=4JX�i$D� 1�eI1�� *%#%(X� $ ¥ uB�Y(+5 1C/d(+� � JX�K=�*X1:9)#%6'(+9)5 1�#XJX! #X� 1%u

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ�FòD��ò ° É�ÆiÃKÊ�Âià Á�À��^Ày«iÃ

�� ��! #,=4J+�)#4@A1�-.$E! � �)6'(XeI(X5 #%/�5)(x=49)� r �^�Xu ¢8! $?/2!)(�J+�)(+5B=4! #%� J+$C@A1�*%#C;B$��)e�� 1Ie�1��)*X#%(X�)(A�

B$>� 1Ie�1��)*X#%(X�

C6 � #Xe^r 1�*%#X*%#C;B1_�Xp��?p�¢ p�û`��ÿ u´��� *X#�*X#C;B1i$�� 1Ie�1��)*X#%(X� $ ¥ =41F#C;<�?GH(X*%(_6'#X5)=�Jj(+5 J+5)�)/ú=4Jj(j9)5)�)/;B#XJX� r)(+-:1I/�6,#%5)=4Jj(X5 J+5 #%h�! �)#XJ+#%6'(

V̇ = 20 l/s$K-:1�h4( p<;<#X6^=�$K� 1�eI1�� *%#%(+�)�

B�

C*%149)� 6)�Yu ��#%e�5)(XJ+�o=�$=v9)14;B1�b)��!)#C;<(+r �~L

pm = 10 kPappv = 50 kPa

pd1 = d2 = 100 mm

pd3 = 120 mm

pd4 = d5 = 150 mm

pL1 = L2 = 2 m

pL3 = 1 m

pL4 = 24 m

pl5 = 15 m

pλ = 0.025

kêeI(D=�*X1xr 1I*X�Bt4pH = 5 m

ph = 2 m

pζs = 4

pζv1

= 5pζv2

= 8pζv4

= 4pζv5

= 10�ζk = 0.8

u w ;B� 1v;<� J+�0g%� e��)5 1�=�J+� $C@s(+-�(�$E=�*X�)/^;B1I#X5 �)r)(X/d(8�"=45)(Xh�$! $?/2! 1Xu

±r±r±r±±r±r±r±±r±r±r±±r±r±r±±r±r±r±²r²r²r²²r²r²r²²r²r²r²²r²r²r²²r²r²r²

³r³³r³´r´´r´ µrµrµµrµrµ¶r¶¶r¶

v4

5

ss

k k

k kv

v2v1 2

4 5

3

1

u

H

h

mp

pv

V.

C

B

A

i�·¸|n�fqº¹3»�¼ª R,}aR,ØfN6� v1 = 1.354 m/s

pv2 = 1.192 m/s

pv3 = 1.768 m/s

pv4 = 2.262 m/s

pv5 = 1.13 m/s

pYP =

143.4 J/kg�P = 7.263 kW

¼6� ��! #,=4J+�)#4@A1�-.$D! �)� 6?(+eI(X5 #%/�5)(�=v9)�)r �o�%u û�! $?/2!)(x! #%J+�)=�6 $C@A1�*%#C;B$��)e�� 1Ie�1��)*X#%(X�)( ¥ �0ð�$>� 1�eI1�� *%#%(X� 1 £�Du���#%e�5)(XJ+�i=4$K=v9)14;B1�b)�F! #C;<(Xr �YL

H = 5 mppm = 10 kPa

ppv = 5 kPa

pηP = 0.8

pζu = 0.5

pζR = 0.5

pλ = 0.03

k�eI(o=4*X1fr 1�*%�Bt��`!)#C;<(+r �a;<(+J+��$�Jj(XgX149B�Yu ¥ 6'#E@A1f$^;B1�#%5 � r �ÖÿF� eI/21I� 1�5B(igX� eI� 5)(^=4J+� $C@s(X-:(v5 = 1.8 m/s

p�#C;<� 1v;B�)J+��! �)#XJ+#%6,1"6 � #%ef=�*X10r)1�*X5)1D;B1I#X5 �)r 1Xp�6?(X#F�l=�5)(+hI$_6,#�@s(E@A1"! #XJX� 1�g%5)(FeC(o! #%h�#%5! $?/2! 1Xu

i Li [ m] di [ mm] ζvi� ¢ �"�v� ���� ¢)u � �"�v� �¢ ÿ �v�v� ¢û û �c�r� ûÿ ÿ �cÿr� ÿ

v

v v

uu

4 5

3

v2v1

v3

R

R

pv

B

DC

A1

4

3

5

2

H

pm

i,·¸|���q�¹3»¾½ª R,}aR,ØfN6�• P.� eI� 5 1x=4JX� $C@s(+-�( L v1 = 1.377 m/s

pv2 = 0.532 m/s

pv3 = 1.546 m/s

pv4 = 1.483 m/s

pv5 = 1.8 m/s

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ�X

• ��(X! � 1C/2� 5)=46)�Z! � #XJX#Xr �YLV̇1 = 35.04 l/s

pV̇2 = 13.54 l/s

pV̇3 = 45.58 l/s

pV̇4 = 16.77 l/s

pV̇5 =

31.81 l/s

• �%1v;B�)5 � & 5 �i�B(c;i�K=�5)(+h4(�! $?/2! 1%LYP = 65.3 J/kg

�P = 3.965 kW

uñ�� �8(Z=v9)�)r �¤�Xu�ÿ^@A1`! � �)6'(XeI(X5�=49)#+GH1�5�r 1I*X#%*X#C;Ù$Ù6'#%/21`!)$'/2!B( px6 �)#Xear 1I*X5 1_;<1�#X5)� r 1Z�XpH�?px¢ pHÿl� �J+�)(X5)=4! #%� J+$C@A1�*X#C;B$l� eo*%149)�)6,#%ho#%J+*X#%� 1�5 #%hF� 1Ie�1��)*X#%(X�)( ¥ $l�)1�e�1I� *X#%� 1

Bp

C�

Du �8=�J+#X*%� 1I/21I5 #)pr 1�*%#X*%#C;B#%/iû8*%#C;<(�=41�=49)� *,(��)e�� 1�eI1�� *%#%(X�)(�ð`$��)1�e�1I� *X#,(+� £ p+#C;B5)#%=45)#:�)e�r 1�*%#X*%#%/ ���)e�� 1�eI1�� *%#%(X�)( £ $� 1�eI1�� *%#%(X�

Dud��#XeI5)(+J+��=4$x=v9B1v;B1Ib ��! #C;�(+r �YL

H = 5 mpH1 = 1 m

pH2 = 2 m

ppm = 0.2 bar

ppv = 0.5 bar

pλ = 0.03

kêeI(D=�*X18r 1I*X�Bt4pηP = 0.8

p 6?(X#E�i! #C;<(+r)�0;�(+J+�F$"Jj(+g%149)�Yu w ;B� 14;B� J+�F=�5)(+hI$f!)#XJ+� 1IgX5 $feI(E!)#Xh�#%5! $?/2! 1Xp\$'6 $'!)(+5_eI(X! � 1C/2� 5)=46)��!)� #XJ+#%6_6,#�@A�^;B#XJ+�)& 1E$o� 1Ie�1I� *X#,(+�D

��6'#%1I7�� r �+@A1�5 J"9)#%6'(+9)5 #%hH#%J+! #%�)(*X1I5 J+�?9B(ζ v7

$>;B1I#X5 � r)� �?u¬ j{�T­Íu�trs$j'L.��� *%#X�F*%#C;B1x$K� 1Ie�1��)*X#%(X� �)/d(Dð:p £ �

D=41H#C;B�?GH(X*%(4@A$06'#X5)=�Jj(+J+5 �B/Z=4Jj(j9)5)�)/Z! � #%J+� r)(X-�1C/6 � #%e:-:�?3�u £ *X1:9)#%6?(j9)5 1�#%J+! #X�)1Xp #,=4�)/�*X1�5)J+�?9B(>eC(+5 1C/d(+� � JX�Yu

Li [ m] Di [ mm] ζ vi �"�v� ¢v�l� ÿ¼ �v� �r�l� ÿñ �v� �r�l� ���þ �v� �r�l� ���� �v� �r�l� ÿ5 �v� �r�l� �F �v�v� �r�l� ù

1

H

H

v v

v

v

v

v

v

v

p

1 1

2

4

6

73

5

H2

1

2

3

4

5

6

7A

B

C

D

v

pm

V.

i�·¸|n�fqº¹3»�¿ª R,}aR,ØfN6� v1 = 1.717 m/s

pv2 = v4 = 1.771 m/s

pv3 = 2.092 m/s

pv5 = 1.091 m/s

pv6 = 2.862 m/s

pv7 = 1.001 m/s

pV̇ = 0.121 m3/s

pYP = 132.56 J/kg

pP = 20 kW

�ζv7 = 125.6

u

À Á E����;IÎ�� J ��JÃ:ÄA;>"EZ��� �C8D>£Å > �ÆKM>"8?J �AÎ�@Go� �ÆK¤�BJ

 �1IJc(+n�(X5K#X! �)=�!)� #XgX9)1I/d(XJ+� 6'18� ex#%g+9B(%=4J+�K�)=�J+� r)(X-:(EJX1�& 5 #,=4J+�i6)� #Xe8#%J+*X#%� 18�i5)(+h¾9<(+*X6'1.@A1�;<(XJE$06 -:� eI�ÇOÈ�ÉnÊÌËyÈfÍlÊ�ÎRÏÌÐcÉfÑTÐcÈ3ËgÏRɯÒÓÈ]ÎRÔRÍlÈ3ÕÖÔR×�ØnÙÚÎRÊÓÛ,ÊOܸÝnÙlÈfÐTÊi#C;_! � #%7>u>ð�9B(c;�(+5)(ßÞE#X�'�'1�*%� b)()uHÔ~1�#X�)�j@s=46 �a$'*%#C;o;<(XJf$=49)1v;B1Ib �)/Ö� 1v;<#X*X�B/d(H=�18g%(Xe�� �)(�5)(H! #�@A1v;B� 5)�)/�;B1�Jj(jn��)/d(>� e�J+186)-�� hI1Xu

S"��� T�Ê�Ày«iÃi��ÂiÀxÇKÂiÀ ÃiÉqÊ8Ãà«iÀ8ÌÍÀ��#,=�/d(+JX�)(v@s/2#2@A14;<(+58*%149)� 6)�x� 1�eI1�� *,(+#%��$:& �j@A1C/i=41�g%#X& 5)#%/fe��';B$:5)(+9B(+eI�x/d(j9)�8#%J+*X#%�:#%��J+�)�?3:�)*X� r)(�kê=v9B� 6?(

ûBu��ct�6 � #%e�6'#4@A�xJX1�& 5 #,=4J��)=4J+�)& 1�$8(XJX/2#%=I7�1�� $�uY��=49)1v;>� 5 1��)r �j@A1Xpj!)#4@A1v;<� 5 ��7:9B$,�?;B5 ��;B1�9)� b ��$:gX9)� eI� 5 �x#%J+*X#%�)(

á_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_áá_á_á_á

â_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_ââ_â_â

ã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ãã_ã_ã_ã

ä_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ää_ä_ä

å_å_å_åå_å_å_åå_å_å_åå_å_å_åå_å_å_åå_å_å_åå_å_å_åå_å_å_åå_å_å_åå_å_å_åå_å_å_åå_å_å_åå_å_å_å

æ_æ_ææ_æ_ææ_æ_ææ_æ_ææ_æ_ææ_æ_ææ_æ_ææ_æ_ææ_æ_ææ_æ_ææ_æ_ææ_æ_ææ_æ_æ

ççççççççççççç

è_è_è_èè_è_è_èè_è_è_èè_è_è_èè_è_è_èè_è_è_èè_è_è_èè_è_è_èè_è_è_èè_è_è_èè_è_è_èè_è_è_èè_è_è_è

é_é_éé_é_éé_é_éé_é_éé_é_éé_é_éé_é_éé_é_éé_é_éé_é_éé_é_éé_é_éé_é_é

H

k

k

£ 9)� 6?(>û)u¡�XL��8=4JX� r)(+-:1�6 � #%e8#XJ+*%#X�"#X��JX� �?30� *%� r)(

b 1�=41x6 �)1�Jj(+J+�o! #EeI(XJ+*X#%� 1�5 �B/Ö6'#%5 *X1I� h�1I5 J+5 �)/l! $'Jj(+-�(X/d( ] /Q9B(Xe�=41x=�$%G>(+*,(E=4*X1:;<#D! � 1C=41�6?(>#%gX149B14G�1I5 #Xh8=I(k − k

5)(Ö=49)� r �Ùû)u¡�Xu Ô\(v@K! � 1C=41�6Ù=41�5)(+eI� *%( WY{)UQ�?P2N?WY{)|�N?UQT�OQPQR'V%R%W ��@A1v;B�)5 #�$Z-�1C/2$�=4J+�)$I@s(X-�1_�)/d(@A14;B5 #C;B�)/21I5 e��+@s=46 �i6?(+�B(+6 J+1I� � w *%(4@�! � 1C=41I6�@A18$C@A14;B5 #>�K5)(v@A$,GE�K! � 1I=�1�6K/Q9B(+eC( u

A − ! #X*%� ���)5)(H#XJ+*%#X�)(

Ak − ! #X*%� ��� 5B(�6'#X5)J+�)(+6'#%*%(+5)#Xh�!)� 1I=�1�6?(ð.149B� & � 5)(^6'#4@s(o! � 1v;�=4Jj(+*Xn�(^#C;B5 #,=D! #%*X� ��� 5 106'#%5 J+�)(X6,#%*%(X5 #XhD!)� 1I=�1�6?(F�l! #X*%� ���)5 10#%J+*X#%�)(o=�1f5)(Xe�� *,(

WY{<R'ÕKT � Tz�CR,UQ�FWY{)UQ�?P2N?W � Tn�IR Lψ =

AkA

< 1k�û)u¡�Ct

��#�@s(+*%(.6'#X5)J+�)(+6)r �j@A1�/Q9<(+eI(d@A1�� eI�)(jGH1�5 �j@s(�6,#C;�#%J+*X#%�)(q6'#4@A��=�1�5)(+9B(+eI1�g+9)� G�1�=49)#Xg%#C;B5 #4@�! #%*X� ���8J+1�& 5)#%=4JX�$f� 1�eI1�� *%#%(+�)$<p'Jv@4u�*X� 14;B5 #%=�J�6'#%1I7�� r �+@A1�5 Jj(H6'#X5 J+�B(+6 r �+@A18#X!)(j;<(�=I(H! #X*%1�b)(X-�1C/`*X�B=4� 5 1

Hu

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R þ��ª«HÃG�H�G ê¢ ��ÛQRPøL��1�¢�����!ë�ì Lí�RQ����������F��$@���øQ�î Q��ÔQøPRL

�ar � n.$�#C;B� 1I�'� *%(X-:(�eI(X! � 1I/2�)5)=46'#Xhd!)� #XJ+#%6?(q6,#�@A���)=4J+�)& 1�� e��)1�e�1I� *X#,(+�)(�6 � #%ed#XJ+*%#X��! #%*X� ��� 5 1d! #%! � 1�&)5 #Xh! �)1I=41I6?(

Ap ! #,=4Jj(+*Xn�(>=41yP.1��)5 $C9)�j@A1�*,(�@A1v;B5)(X& � 5)(>eC(H5 � *X#>$D�)1�e�1I� *X#,(+� $0�f6'#X5)J+�)(+6'#%*%(+5)�0! � 1C=41I6"6 ] 6YL

paρ

+ g H =paρ

+v2

2+ ζ

v2

2

k�ûBu �%t��#%1I7�� r �+@A1�5 J

ζ@A1�6'#%1I7�� r �+@A1�5 Jq9)#%6?(j9)5 #%h2h�$'gX� JX6'(.1I5 1�� hI�j@A1�6'#4@A��� 1I! � 1Ie�1�5)J+$C@A1�1�5)1�� hI�j@A$�! #%J+� 1IgX5 $�eI(�=4JX*%(+�B(+-:1

/d(X6 � #X*%� JC9)#Xh4(�6'#4@A�K5)(%=4Jj(v@A$D�)eI/21��'$D!)� #XJ+#%& 5 18eI#X5 1��Ke��';�(��)1�e�1I� *X#,(+�)()u���e.@A14;B5)(+&)� 5 1Dk�ûBu �%tq=41�;B#%gX�j@s(

v =1√

1 + ζ

2gH = ϕ√

2gH,h�;B1.@A1

ϕ =1√

1 + ζ

k�ûBu ¢,tð.149B� & � 5)(

ϕ=41H5)(+eI� *%( WY{BR'ÕKT � Tn�CR%UQ�a»)PQMXTQU2R u��ú=v9)$'&)(4@A$f;<(0=�1�h�$'g%� r �F$i� 1Ie�1I� *X#,(+� $K/2#XhI$"eC(+5 1C/d(+� �)J+�

kζ = 0

tvp?gX�)e�� 5)(H�)=�J+� r)(X-:(>$fJ+#%/l=v9B$,&B(v@A$8@A1 T~¹dR'N'[2U2N^»BP\M%T\U2N k�5)1I/d(>h�$'g%� Jj(+6?(�1I5 1��)h��j@A1ctvL

vi =√

2gHk�û)u û,t

£ J+*,(+� 5B(8gX� eI� 5)(Dkê� e��B(+e�ûBu �%t<@A1�$'*%1�6D/d(+-�(x#C;0�?;B1I(+9)5 1�g%� e�� 5)1XpI@A1��x@A1ϕ < 1

u2��� #XJ+#%6E6 � #%e�#XJX*X#X��@A1�#C;B� 1I�,1I5� eI�)(+eI#%/�k�gX�)e�� 5)(

v@A18g%� e��)5)(H$06'#X5 JX�)(+6'#X*,(+5 #,/_! � 1C=41�6)$?t4L

V̇ = v Ak = ϕAk√

2gHk�û)u�ÿXt

¥ 6,#`=�1F! #%*X� ��� 5)(Ak

� eI�)(+eI�Z6?(+#Ak = ψA

p�h�;B1>@A1A

! #X*%� ���)5)(_#XJ+*%#X�)()p�;<#XgX�+@s(a=41F=v9B1v;B1Ib �Z� e��B(+eieI(eC(+! � 1C/2� 5)=�6 �K! � #XJX#X6K6 � #%e:#%J+*X#%�YL

V̇ = µA√

2gHk�ûBu �,t

ð.149B� & � 5)(µ = ϕψ

=41_5)(Xe�� *,( WY{<R'ÕKT � Tz�CR,UQ��OQP2{)� {BWYN uÍ��#X1C7�� r �j@A1I5 J�! � #%J+#X6?(^@A1_$,*%1�6�/d(+-:��#C;@A14;B� 5 � r)1"�Ö#X5�=�1feC(F5 1I6 ��#%J+*X#%��/2#+GH10*%1�#%/d(D@A1v;B5)#%=4Jj(X*X5 #o/2#XG�1"#C;B� 14;B� J+�Ö1�6?=4! 1I� �)/21I5 Jj(j9)5 #BuaÔ��)! � & 5)(*%� 1v;B5)#%=4J>eC(

µ$f=49)$'&)(v@A$D6)� $,G>5)#Xh8#XJX*X#X�B( p'6?(c;<(�@A186'#X5 JX�)(+6 r)�j@s(H! #XJ+!)$,5B(:@A1

µ ≈ 0.62u

��;B1C(j9)5)�f! �)#XJ+#%6f6 �)#Xe8#XJX*X#X�"gX�"gX�)#�@A14;B5)(X6"!)� #X� eI*X#C;B$E!)#X*X�)��� 5 1�#%J+*X#%�)(��K�';<1I(j9B5 1�gX� eI� 5 1Xp)Jv@4u

V̇i = A√

2gHk�û)u �+t

� #XG�1�=41qeC(+6?nq$'& � J+�H;<(�@A1�=�J+*%(X� 5 �H! � #XJX#X6H$�� 1�9B(+J+�)*X5 #�*X1�9)� 6'#4@2/21�� �E/d(X-��>#C;E�';B1I(+9)5 #%h�! �)#XJ+#%6'(>kê5 ! �YueC(�6)� $,G>5)�"#%J+*X#%�"=�J+*%(X� 5 �K! � #%J+#X6�@A1

62%#C;K�?;B1I(+9)5 #Xh�! �)#XJ+#%6'(,tvu

ª«HÃG�H(' ïð��N��#�6Ny��� Q� ��ÛQøPRL�����´#%*X#,/Ö=49)$'&)(v@A$"�)=4JX� r)(+-:1�=�1x5 1x*%� ���i$K(+JX/2#,=�7�1I� $�p *X1Ibi$f! �)#%=4JX#X�i6'#4@A�>@A1x�B=4! $'-:1�5i�)=�J+#%/�J+1I& 5 #%��b $

6'#4@s(E=�185)(j9<(+e��K$"=4Jj(+-.$"/2�)� #X*,(+-�( u £ J+�)$,6)J+$'�)(D=4JX� $C@s(+-�(Eb 18! �)(X6 J+� &)5 #D#%=�Jj(+J+�K�)=�Jc(>6?(X#E�"!)� �i�)=4JX� r)(+-.$$"(+JX/2#,=�7�1��)$<p?J4@4u�! #4@s(X*%(H6'#X5 JX�)(+6 r)�j@A1x/Q9B(+eC(Hb 18gX� JX��@s(%=45 #>� eI�)(jGH1�5)()u�P.1�� 5 $C9)�+@A1�*%(�@A14;B5)(+&)� 5)(>eI(H5 �)*X#X1�$� 1Ie�1I� *X#,(+� �)/d(�h¾9B(%=4�YL

g (H1 −H2) = (ζ + 1)v2

2��#%1I7�� r �+@A1�5 Jζ� 1I! � 1�eI1�5 JX$I@A1oh�$'gX� Jj(X6l=�J+� $C@A5 1F1I5 1�� hI�j@A1i�)=�! � 1v;�#%J+*X#%�)( p�;B#X6�@A1ih�$'g%� Jj(+6l1�5 1I� h��j@A1i!)#%=v9B1

#%J+*X#%�)(H! � 1v;�=4Jj(+*Xn:1I5�@A14;B� 5 � r)#%/qp)J4@4ud&?9B(+5)#%/v2/2

5)(�;B1I=�5 #4@�=4J+�B(+5 �H@A1v;<5)(+& �)5 1Xu

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R þO

��e�! � 1IJj3%#C;<5 1.@A1v;B5)(X& � 5 18g%� e��)5)(>=418/2#+GH18� e��)(Xe�� JX�"6?(X#

v =1√

1 + ζ

2g(H1 −H2) = ϕ√

2g(H1 −H2),h�;B1.@A1

ϕ =1√

1 + ζ

k�ûBu �,t

ñbñbñbñòbòbòbò óbóbóbóôbôbô õbõbõbõöböbö ÷b÷b÷b÷øbøbøbø

ùbùbùbùúbúbúbú ûbûbûbûübübübü ýbýbýbýþbþbþÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿÿbÿbÿ

������������������������������������������������������������������������������������������

������������������������������������������������������������������������������������������

������������������������������������������������������������������������������������������

�����������������������������������������������������������������

�����������������������������������������������������������������

�����������������������������������������������������������������

�����������������������������������������������������������������

H1

H2

k

k

£ 9B� 6?(�ûBu � Ld��#C;B*X#C;B5)#��)=�J+� r)(X-�186 �)#Xe8#XJX*X#X�f#X��J+� �?3"� *X� rB(��� #%J+#X6"6 � #%e�#XJ+*%#X��@A1:;<(jn�1.@A1v;B5)(X6

V̇ = v Ak = ϕAk√

2g(H1 −H2) = ϕψA√

2g(H1 −H2)k�ûBu �,t

��� #%� e�*%#C;"6'#X1C7�� r �j@A1I5 Jj(H6,#%5 J+�)(X6 r �j@A18�K6'#X1C7�� r �j@A1I5 Jj(HgX� eI� 5 1.@A186'#X1I7��)r �j@A1�5)JH!)� #XJ+#%6?( pµ = ϕψ

p !)(:@A16'#X5B(+& 5 #>! �)#XJ+#%6f6 �)#Xe8#XJX*X#X�f#C;B� 1I�,1I5f�)e��)(Xe�#%/qL

V̇ = µA√

2g(H1 −H2)k�û)u¡���,t

Ú�6 ! 1��)�)/21�5 JX�f! #%6?(+e�$C@A$�;<(�6,#%1I7�� r)�j@A1�5 JH! � #%J+#X6?(µeI(�5 1�6 �K/d(j9B�0#XJX*X#X�0! � �"! #C;B*X#C;<5 #%/��)=�J+� r)(X-�$D�B/d(

! �)� g+9)� G>5)#E�)=4J+$f*X� 14;B5 #,=4JH6'(X#>�K=49)$'&)(v@A$0�)=4J+�)r)(+-�(>$0(+J%/2#%=�7�1I� $�uª«HÃG�H I ê¢ ��ÛQRPøL��1�¢����Y����# ��ÛQ"������ëÃ�����RQ��RQ����������F��$@�í�RQ�î Q��ÔQøPRL

��#C;E*X1�9)� 6 �)/^#XJX*X#X�)#%/o=�1q! #C;B�)(Xe�$?/21�*,(�#%J+*X#%�H&)�j@A1�*X1I� J+� 6?(j9B5 1.;B�)/21I5 e��j@A1�5)�)=4$�$�e�5B(+J+5 #�@2/21I� �E/d(+-:1�#C;5 1I6'1�6'(X�)(+6)J+1�� �B=4J+� &)5 1q;B$'gX�)5 1q5)(�6'#4@A#4@d=41�#%J+*X#%��5B(j9B(Xe��Kk�#Xg%� & 5 #.@A1�Jj(�6?(+�B(+6 J+1I� �)=�J+� & 5)(�;<$,g%� 5)(:�)(%=4J+#�@s(+-:1#C;K5 � *X#,(�JX1�& 5 #,=4J+�f;B#HJ+1�G>�)��Jj(>#XJ+*%#X�)(,tvu2��#C;"Jj(+6)*X�?3D#%J+*X#%�)(�!)� #%/21I5 1�gX� eI� 5 1�$E! � 1C=41�6)$

k − k=�1�5 18/2#%h�$

eC(+5 1C/d(+� � JX�Yu��#,=�/d(+JX�)(�=�1q*X149B� 6 �H#XJ+*%#X��&)�j@A1�=41.J+14GE� ��J+1�5)(+9B(+eI��5)(�;B$'g%� 5 �

H$x#C;B5)#%=4$85B(�5)� *X#:$x� 1Ie�1��)*X#X�)$Dkê=v9B� 6?(

¢)u ¢,tvuD��� 1IJ+! #%=�Jj(+*X�)b 1I/2#f;<(�@A1E#XgX9)� 6�! � 1C=41�6?(k − k

hI1�#,/21�J+� �+@s=46 �`=49)� &)(X5�!)� 1I=�1�6 $F#%J+*X#%�)( up�9)1I/21I5 Jj(+� 5)(! #%*X� ��� 5)(E#%J+*X#%�)(D=�1�� eC(+gX�)�)(E$K*X�';<$f! �B(+*X#%$'h4(+#X5)� 6?(x;<�)/21�5 eI�j@s(

bk(z)�

dzu w *,(E149B1I/21�5)Jc(X� 5)(0! #X*%� ��� 5B(

=�1H/2#XG�1>=�/d(+JX�)(+J+��/d(j9)�B/Z#XJ+*%#X� #,/qp�!)(x@A1HgX�)e�� 5)(0$F=4*%�)/Z-:1�5 �)/�Jj(+& 6?(X/d(0#C;B� 1I�,1I5)(D�)e��)(Xe�#%/v = ϕ

√2gz

u�9)1I/21I5 Jj(+� 5 �K! �)#XJ+#%6f6 �)#Xe8#X*,(v@�#%J+*X#%��@A1%L

dV̇ = v dAk = ϕ√

2gz bk(z) dz

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R þ�¼

��6 $'! 5 �K! � #%J+#X6V̇

=41�;B#%gX�j@s(H� 5)J+1�hI�)(+r �+@A#%/Ö! #Er 149B#4@�!)#X*X�)��� 5 �"! � 1C=41I6'(k − k

L

V̇ = ϕ√

2g

H+hk2∫

H−hk1

bk(z)√z dz

kêû)u¡�X�ct

��� ������ ��� ��������

����������������������������������������������������������������������������

����������������������������������������������������������������������������

����������������������������������������������������������������������������

����������������������������������������������������������������������������

��������������������������������

��������������������������������

��������

��������

��������������������������������

��������������������������������

hk1

hk2

z

dz

bkH

k − k

C

£ 9)�)6'(HûBu ¢)LÛ�8=�J+� r)(X-�186)� #Xe8*%149)� 6)�f#%J+*X#%�f#%��J+�?3f� *X� rB(w *%#%/�! � � 9)� 6'#%/`@A1H!)� 1�J+!)#%=4Jj(X*+n�1�5 #0;<(

ϕ5 1>eI(X*X�)=��F#C;

zu�ü'1I#%/21�JX� �j@s=46'1>*X1�9)� & �)5 1�$o! � 1I=�1�6 $

k − k�

#C;BhI#X*,(+�)(4@A$,b)1�*X1�9)� & �)5 18$0! �)1I=41I6 $D#%J+*X#%�)(�=�1x/2#XhI$D! #X*%1�eC(+J+�"! � 1I6,#>6'#X1C7�� r �j@A1I5 Jj(H6,#%5 J+�)(X6 r �j@A1%p Jv@4u

bk(z) =√

ψ · b(z), hk1 =√

ψ · h1, hk2 =√

ψ · h2

!)(:@A1�6'#X5B(+& 5 #>! �)#XJ+#%6�@A14;B5)(+6V̇ = ϕ

ψ√

2g

H+√ψ h2∫

H−√ψ h1

b(z)√z dz

kêû)u¡�C�%t

��! �B(+6 J+�)& 5 �)/l! � #%�)(+&)$,5)�)/d(>=41�& 1I=�J+#E6'#X� �B=4J+�K! � �)g+9)�?GE5 �K� eI�)(+e%L

V̇ = µ√

2g

H+h2∫

H−h1

b(z)√z dz

k�û)u¡�I¢,t

ð.149B� & � 5)(µ = ϕ

√ψ=41�5)(Xe�� *,( WY{<R'ÕKT � Tn�IR,UQ�FOQPQ{B� {)WYN�|�R%[QTQWY{¼¡"{)�?|�{BPQN u

ª«HÃG�H ª KMLON1L�PRQ

�� � � 1Ie�1��)*X#%(X� ¥ =4Jj(j9B5 #";B#XJX� & 1V̇ = 20 l/s

*%#C;B1XuKï21�eI1�� *%#%(+�)�A

�B

�B/d(v@A$�6,#%5)=4Jj(X5 J+5 1D5)� *X#%1E*X#C;B1�l=�! #4@A1I5 ��=4$`/d(j9B�)/´#%J+*X#%� #%/�$�! � 1�hI�)(c;<5 #%/�e��';B$�u w ;B� 1v;<� J+��! � #%J+#X6'1"6 � #Xe"� e�9B(+eI5 1K5)(+h¾9B(X*X6'10�)e� 1�eI1�� *%#%(X�)( u~ �(+J+�"=4$0=49)1v;<1�b �"! #C;<(Xr �YL

d = 50 mmpζu = 0.5

pζv = 1.2

pλ = 0.03

pL = 1 m

�µ0 = 0.62

u¼6� w ;B� 14;B� J+�a3,#X� �)e�#X5)Jc(+9)5 $_=4J+�B(+5 � r)$

b*%149)�)6,#%hD#XJX*X#X�B(K#Xg+9B� 6?(i! �)(X*X#X$'h¾9)#%h>J+� #X$'h¾9B(oJj(+6'#";<(F!)� #XJ+#%6

6 � #%e:-:1�h4(Hg%$�;<1V̇ = 2 m3/s

u w =4Jj(+9)�f!)#Xe�5B(+J+�K! #C;<(Xr �K=4$�LH = 1 m

pa = 2 m

�µ = 0.62

u

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R þ<ñS"�¾½ ¤ ��À�Á�ÃiÉqÊ�Ày«iÃi��ÂiÀxÇKÂiÀ ÃiÉqÊ8Ãà«iÀ8ÌÍÀ

��#C;86 *%(Xe��)=�Jj(+r � #%5)(+�)5 �)/0=�J+� $C@s(+-:1I/i=�1d! #C;B�)(Xe�$?/21�*,(q=4J+� $C@s(X-�1�6'#C;86'#Xh4(q=4$�! � #%/21I5 1�7�� eI� & 6 � 3�*X1�9)� & �)5)($�*X� 1C/21�5 $:� 149<(+J+� *%5 #�=4!)#X� 1%p�Jj(+6'#.;<(�=�1�/2#+G�1�=I/d(+J+�B(+J+�:;<(q$�=4*,(+6'#%/0J+� 1I5 $'J+6 $:*%(jGH1<@A1v;B5)(X& � 5 1�eI(�=4Jj(Xr � #X5B(+� 5 #=�J+� $C@s(+-:1XuKÔ.� ! � &)(X5a! � �)/21I�a#X*,(+6 *%#Xh>=4J+� $C@s(X-:(K! �)1v;<=�Jc(X*+n8(K! �)(+G>-:1�-:1D�?9B��!)$,-:1�-:1D� 1�9B(+JX� *X5 #i*%149)�)6,#%h� 1Ie�1I� *X#,(+�)(�6 � #%e:#%J+*X#%�f� 9)�Kr 1�*>& �j@A1�=4$>;B�B/21�5 eI�j@A1xe�5B(+J+5 #E/d(X-�18#C;i! � 1C=41�6?(H� 1Ie�1��)*X#%(X�)("kê=v9B� 6?(�ûBu û'tvu

z

dz

H

V̇ (z)

A(z) t

t = 0

t = T

£ 9)� 6?(Hû)u û)L��8=�J+� r)(X-�18J+1I& 5 #,=4J+�K� e8� 1Ie�1I� *X#,(+�)(

��! #%& 1�J+5)#%/`JX� 1�5 $'J+6)$^kt = 0

t�� 1�eI1�� *%#%(+�06,#�@A��@A1�g%� #�5B(+! $'-:1�50;B#>*X�)=�� 5 1H

! #%& � -:1�;<(>=41:! �)(+eI5 �"6 � #Xe/d(+9)�_#XJ+*%#X��5)(i=4*%#%/�;<5 $�u0��1�6?(�@A1DeI(j;<(+J"6'#X1C7�� r �j@A1I5 JK! � #XJX#X6?("#XJ+*%#X�)(

µ0

pQ6?(+#"�a-�1Ih�#%*%(f! #X*%� ��� 5B(A0

uÔ.� 1Ig%(�#C;B�)1v;B� JX�:*%� 1I/21

TeC(q6,#�@A1db 1�=41�� 1�eI1�� *%#%(+�:�)=�! �)(+eI5 � J+�~u ¥ 6'#��)=�J+� r)(X-�1�=�/d(+JX�)(X/2#�6 *,(+eI�)=4Jj(+r)� #X5)(X� 5 �)/qp

! �)#XJ+#%6"$0! � #X�)e�*X#Xn:5)#%/`JX� 1�5 $'J+6)$f*%� 1I/21I5)(�@A18#C;B� 1��'1I5f� eI�)(+eI#%/qL

V̇ = µA0

2gzkêû)u¡��û,t

!)(�b 1�eI(a*%� 1I/21dt

� e�� 1�eI1�� *%#X�)(`�)=4J+1Ib ��eC(+! �)1I/2� 5)(V̇ dt

u �8=4J+#%*X� 1C/21�5 #Bp�5 � *%#�J+1I& 5 #%=�J+��$Z=�$�;<$Zb 1^=�1=I/d(+-:� J+�aeI(

dzp\Jc(X6'#0;�(H@A1D=�;B� $'h�1D=�J+�)(+5)1E6,#X9)� & �)5)(KJ+1�&)5 #%=�J+�a6'#4@s("�)=4JX1�6 5 10� eE� 1�eI1�� *%#%(X�)(0eC(K*X� 1C/21

dt@A14;B5)(+6?(A(z) dz

u���� 1C/d(�JX#%/21Xp?;B#+9B(Xe��K=41�;B# ¹dTQÕiR,P2R%U � Tz�CN'[2UQR��IRX¹dUQNÍ}\TQUQRFO\P2N?º_ØfR,ØfN^P2R%M,R,P\|�{BN PQN6�V̇ dt = −A(z) dz

k�ûBu��cÿXt��5B(+6i/2� 5 $,=:=I(�;B1I=�5 1x=4JX�)(+5 1q@A14;B5)(X& � 5 18! #%J+� & 18#C;i& � -:1�5)� r 1�;<(>=�186,#%#X�%;B� 5B(+Jj(

z=I/d(+-.$C@A1xJ+#X6'#,/`*X�)1I/21�5B(

kê�)/d(K5 1�h4(XJ+� *,(+5�! � � �)(X��Jj(v@ t4p~Jv@4udz < 0

u ¥ 6,#K6'#X#%�%;B� 5)(XJc(z�)(%=4J+1>J+#%6,#,/�*%� 1I/21I5)( p~#X5';<(i=I("b 1D=�(0;B1C=45 1

=�J+�)(+5)1q@A1v;B5)(X& � 5 1�5 1�b 1x=�Jj(v@s(+JX�"eI5)(+6i/2� 5)$%=+u.ü'1�5 1I�)(j9B5 #)p?! �)�i! $'-�1I5 $0�?9B�i! �)(jGE-:1�-.$f�)1�e�1I� *X#,(+�)()p'$0r �?nq$#C;B�)1��'� *%(X-:(�*X� 1C/21�5)(H6'#4@A18=�1�! #%J+� 1IgX5 #�;<(H=41�5 � *%#>J+1�& 5)#%=4JX�f$D-:1I/2$feI(�5 1�6)$D*X� 14;B5 #,=4JH*X�)=�� 5 1Xp?6'#%� �)=4JX�"=�1;B�B7�1�� 1I5 r �j@s(+9)5)(:@A1v;<5)(+& �)5)(H#Xg+9B� 6?( L

V̇ dt = A(z) | dz | k�û)u¡���,t! �)�"&)1I/2$x@A1%L

| dz | =

{

dz,6?(c;�(�@A1

dz > 0 (6'#%#X�%;B� 5)(XJj(

z�)(%=4J+1�J+#%6'#%/Ö*X� 1C/21�5)(

)

−dz,6?(c;<(�@A1

dz < 0 (6'#%#X�%;B� 5)(XJj(

z=418=I/d(+-.$I@A18JX#X6'#%/Ö*X� 1C/21�5)(

)

¥ 6,#�=41`*%�)(+J+�B/2#�5)(�6'#%5 6 � 1IJc(X5�! � �)/21I�Yp�eC(X/21I5 #%/�@A1v;<5)(+& �)5 1�kêû)u¡��û,ti$�kêû)u¡���%t4pxJv@4u�k�û)u¡�Cÿ%t0;B#Xg%�j@s(�=�1

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R þBþ

;B�B7�1�� 1I5 r �j@s(+9)5)(:@A1v;<5)(+& �)5)(H6,#�@s(H�)(+e4;B*%(4@s(�! �)#%/21�-:� *%1Xp Jv@4u

µA0

2gz dt = −A(z) dz =⇒ dt = − A(z) dz

µA0

√2gz

k�û)u¡�*�Xt��exk�#X!)�)(+*+;<(+5 $ t\! � 1IJ+! #,=4Jj(+*%6 $8;<(.@A1

µ = const.JX#X6'#%/o! �)(jGE-:1�-�(8� 1Ie�1��)*X#%(X�)( pj�)5 J+1�hI�)(jn�1�-:1I/^9)1I*X1���;B1C=45 1

=�J+�)(+5)1q@A1v;B5)(X& � 5 1%LT∫

0

dt = − 1

µA0

√2g

0∫

H

A(z) dz√z

��#%5)(+& 5)#�*%� 1I/2186'#4@A1.@A18! #%J+� 1IgX5 #�;<(>=418�)1�e�1I� *X#,(+�f�)=4! �B(+e�5)��@A1%L

T =1

µA0

√2g

H∫

0

A(z) dz√z

k�û)u¡���,t

¥ 6,#�@A18�)1�e�1I� *X#,(+�"#XgX9)� 6?(Hr �?9)� 5?;B�)(E�?9B�"!)� � eI/21%p)#%5';<(:@A1A(z) = A = const

pB!)(>=41q@A14;B5)(+&)� 5)(Fkêû)u¡���%t.=�*X#C;B�5)()L

T =2A

√H

µA0

√2g

k�û)u¡���,t

ª«H�'DH�G KMLON1L�PRQ�Xu��8(+b �0*X� 1C/21�6'#4@A1�@A1�! #%J+� 1IgX5 #x;<(�=�1�r �?9B� 5';B� �)& 5 �"� 1�eI1�� *%#%(+�E!)� � 6?(+eC(+5D5)(�=v9)�)r �0�)=�! �)(Xe�5 �Yu���#%e�5)(XJ+�

=4$0=49)1v;<1�b �K! #C;<(Xr �YLD = 1 m

pd = 20 mm

pH = 0.5 m

ph = 1 m

pµ = 0.62

u�?u w J+*X#%� 1�5_! �B(+eI(X5ar �?9)�)5';B� � &)5 �Ö=�$�;�=41D! $'5 �a*%#C;B#%/Ù� e0h�#X�)-�1IhD*X1�9)� 6'#XhD#%J+*X#%� 1�5)#XhE� 1Ie�1��)*X#%(X�)("6 � #%e

;B*X1qr 1I*X��@A1v;B5B(+6 �?3�! � 1�&)5 � 6?(:��;<$%GE� 5B( p+6?(+#:�:@A1v;B5B(+6 �?3�6,#%1I7�� r)�j@A1�5)(XJj(�9)#%6?(j9)5 � 3�h�$'gX�)Jc(X6?(�1�5)1�� hI�j@A1Xuw ;B� 14;B� J+��*X� 1C/21>eI(f6,#�@A1�b)1>=�1H=�$I;_5)(X! $'5 � J+�_*%#C;B#%/qu� �(XJ+�^=�$F=v9)14;B1�b)�^! #C;�(+r �YL

ζu = 0.2pζv = 0.5

pζk = 0.5

pλ = 0.03

pl = 4 m

pd = 50 mm

pH = 5 mm

ph = 2 m

�D = 1 m

u¢ u�ï21�eI1�� *%#%(+�:! � �)6'(XeI(X5:5B(�=v9)�)r �x=41��B=4J+#%*X� 1C/21�5 #�!)$,5)�86,#%5)=4Jj(X5 J+5 �)/K!)� #XJ+#%6'#%/

V̇0

�x! �)(Xe�5 �86 �)#Xe�=v9B#+G�1I5r 1�*%#X*%#C;\u���! #%& 1�J+5)#%/ZJ+�)1�5 $'J+6 $o� 1�eI1�� *%#%(+�>@A1>5)(X! $'-�1I5i;B#f*X�)=�� 5 1

Hu8��#XeI5)(+J+��=4$i=v9B1v;B1Ib �^!)#C;<(+r �~L

H = 2 mpD = 0.6 m

ph = 3 m

pl = l1 = 4 m

pl2 = 1 m

pd = 30 m

pd1 = 20 mm

pd2 = 15 mm

pζv = 4

pζv1 = 2

pζk = 0.5

pζu = 0.7

pλ = 0.02

�V̇0 = 2 l/s

u w ;B� 14;B� J+�Z!)#+9)#XGH(v@>�)(X*X5 #%J+14GE5 #%hi5 � *X#,(_$� 1�eI1�� *%#%(X� $�p,6?(X#��K*X�)1I/2186'#4@A1.@A18! #%J+� 1�g%5 #�;<(>=418! #%& 1�JX5 �K5 � *X#ik

H = 2 mt.! � #%/21I5 �KeI(

h0 = 1 mu

û)u ��e�hI#X� -:1�h>#%J+*X#%� 1�5 #%h�r)�?9)� 5';<� � & 5 #%hD=4$I;<()pY! $'5 #Xh>*%#C;B1"km=v9)�)6'(K¢,tvpY! �)1�& 5 �)6'(d = 1 m

�_*X�B=4� 5 1h =

1.5 mpd! � 1�Jj(X& 1"=�1f*%#C;<(F$^;B#%-��lr �?9B� 5';B� �)& 5 ��=4$I;Ö;<�)/21�5 eI�j@s(

D = H = 2 m$a6'#4@A1C/��)/d(^*X#C;B1D;B#

! #+9B#X*X�)5 1�*%�)=4�)5 1 ] 6)� #Xe>r 1I*f! �)1�& 5 �)6'(d0 = 25 mm

�^;B$,GE� 5 1L = 2.5 m

p�! #%e�5)(XJ+�?3o6'#X1C7�� r �j@A1I5)(+Jj(#XJ+!)#X�)()L

ζu = 0.5�ζv = 1.5

�"6'#X1I7��)r �j@A1�5)Jc(HJ+�)1�-�(λ = 0.03

u2ð.�)=�� 5)(�;B5)(HhI#X� -:1�h8=4$I;<(H�"5 � *%#%(�*X#C;B1$F;<#X-:1I/�=4$I;B$"@A1

a = 2 mu ��e��)(X& $'5)(+JX�a*X� 1C/21DeI(i6'#�@A1Db 1f=�1Dh�#%� -:��=4$I;�)=�! �)(Xe�5 � JX�Yuo�8(F=49)� r �i@A1

! � � 6?(XeI(+5"! #X9)#+G>(v@�$fJ+� 1�5)$,JX6 $t = 0 s

uÿ?u £ $I;B#X*%� ¥ �oð�=4!)#4@A1�5)�o=�(Er)1�*X�o! � 1�&)5 � 6?(

du��8(+6'#%5i#XJ+*,(+�)(X-:(>*%1�5 J+� 9B(D*X#C;�(>! #X& �)-�1�;<(E! �)1�J+� &)1�� e

=4$I;<( ¥ $"=�$�;^ð��F! �)�iJ+#%/21x*,(+e4;B$,3f$C9B(+eI�i$"=�$I; ¥ 6 � #%e�/d(j9B�i#XJ+*%#X�K! � 1I& 5 � 6?(d0

6'#4@A�F=�1x5)(+9B(+eI�"5B(! #X6 9)#X! r)$F=4$I;<( u w ;<� 1v;B�)J+�^*%� 1I/21HeI(f6'#4@A1Hb 1H5 � *X#%�o$F=�$�;<#X*X�B/d(D�?GH1v;B5)(X& � J+�~u���#C;<(Xr �YL

d = 50 mmp

d0 = 2 mmpl = 5 m

pH = 2 m

pA1 = 1 m2

pA2 = 2 m2

pA2 = 2 m2

pρ0 = 1.2 kg/m3

pζu = 0.5

pζv = 0.5

pλ = 0.02

pµ0 = 0.62

u £ J+� $C@s(X-�18*,(+ev;<$%3'(�=I/d(+J+�B(+J+�K5 1C=4J+� ��n:�)*X�)/qu

¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬x­F¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R þ �

¬ O\¸�� VX�?|<{`M%N^PQR,}aN?|�N?ØfRKM,N,¹dN �%N?WYNaTQMKW\|�N?M%TQVX�%N � TQ{BU2N?P\UQT � TQVX�?T � N?ØfN• ��� eI� & 6 �F=I(+h¾9)14;<(+JX�f! �)#Xg+9B1I/ ] ;<(�9)�F/2#+GH;<(H! #,=4J+#�@A�K*X� ��1x78(+eI(�$0JX#X6 $f! �)(jGE-:1�-�(kê! $'-�1I-:(,t�� 1Ie�1��)*X#%(X�)( uEÔ~#F=4$^=49)$'&)(v@A1I*X��6?(c;<(K=41%p~� 1Ir �)/2#i! �B(jGE-�1I-�1^k�! $'-:1�-:1Ct�� 1 ]eI1�� *%#%(X�)(f;B#F5 1I6'#XhD*%� 1I/21I5)=46'#Xh0J+� 1�5)$,JX6'(o#Xg,(+*+n8(i=�(F5 1�6 �B/�! � #XJX#X6'#%/

V̇1

pd(F#C;�JX#XhJX� 1�5 $'J+6?(H;<#H5)1�6'#Xhx=49)1v;B1Ib 1�h85)1�6 �)/l! �)#XJ+#%6,#,/

V̇2 6= V̇1

p)�KJj(+6'#E=4*%1:;B#X6i=�18� 1�eI1�� *%#%(+�5)1E�)=4! �B(+e�5)�lkê5)(+! $'5 �<tvuDÔ~(X6'#j�,1%p~(+6'#�@A1E� 1�eI1�� *%#%(X�Yp�5)("! � �)/21I�_#Xg+9B� 6?(0r � 9)� 5';B�B(i=�(=I7�1�� 5)�)/�;B5 #,/qpd! #%=�J+#4@s(Xb 1E;B*%1f78(Xe�1D! �B(jGE-�1I-:(o� 1�eI1�� *%#%(X�)( ] $_J+10;B*%1f78(+eI1Db 1"=41�B(+e49B� 6'#X*%(XJ+�

A(z) ] #C;)9)� &B(+5"! � �)/21I�"eC(c;<(XJj(+6���=I(>(c$�;<� J+#X�)5 �?30*%14GEgX�Yu• £ 6)� r � �)(XJ+�Ù! � #X�)e�*X#Xn:5)��! #X9)#+G>(v@DeC(Ö=�*%(+6)$Z#C;�78(+eC( p�$%=4*%#4@A� JX��6'#X#%�X;<� 5)(+JX$

zk�5)1

!)#%=4JX#4@A��! �)(+*%�?9)#KeI("6'#4@A$o�)(+*,(+5o*%1�eC(+J+��6'#X#X�%;B�)5)(+J+$z�~JX#>@A1>=�J+*%(X�o9B� & 5 #%hE� e�g%#X�)(,tvu

��#,=4Jj(+*%� J+��;B#%*X#+n8(+5:gX�)#4@�P.1�� 5 $C9)�+@A1�*X� 3�@A1v;<5)(+& �)5)( pc6?(+#��.@A1v;<5)(+& �)5)(�6'#X5 JX� 5 $'� J+1IJc(�eI(Jj(4@.! �)#X� eI*X#+n�5 �"! #+9B#+GH(4@4p'Jc(X6'#�;�(�@A1x/2#%h�$'b 1�#C;B� 14;B� J+� ¸�WQ¸�OQN U ! � #%J+#X6K$07�$'5 6)r �j@A�6'#%#X�%;B� 5)(XJ+1

z6'#�@A�)/o=�1.�)1�e�1I� *X#,(+�x!)$,5)���?9B��! �B(+e�5)�"kê$x5 1�6)�)/FeC(c;<(Xr �)/d(:/2#+GH;<(:J+� 1Ig%(

#C;<� 1v;B�)J+�i;<(09)��=41Hb 1�=�1�5 �)*X#0$i�)1�e�1I� *X#%� $K! #C;B� eC(+J+�o�?9)�_=4! $%��Jj(+J+� ] eC(c;<(XJj(+6o¢f=�((c$I;B�)J+#X� 5)�?3f*%14GEgX�Bt4u

• ��#,=4Jj(+*%� J+�Í;B�)7�1��)1�5 r �+@s(j9)5 $ú@A14;B5)(X& � 5 $Í�B=4J+� rB(+-�(�kê! $'-�1I-:(,tvp^@A1v;B5B(+& � 5B( k �?u¡���%t4u��5)J+1�hI�)(j9)�)J+��9B1�*X$Z��;B1C=45 $�=�J+�)(+5)$a@A1v;B5B(+& � 5)1Xp:$,e_! �B(+*X� 9)5 #l#C;B� 1��'1I5 1ohI�)(+5 �)r 1_$6'#�@A�)/d(>=41x/21�-�(v@A$f! � #,/21�-:� *X1

t�z ] eC(c;<(Xr �"=I(>(c$�;<� J+#X�)5 �?3f*%14GEgX�~u