De Motu Fluidorum from L. Euler

7/23/2019 De Motu Fluidorum from L. Euler

1/65


2/65


3/65


4/65


5/65


6/65


7/65

AL= x Ll= y.

lm = u,

ln= v =

uu + vv =

u2 +v2

=v/u

du = Ldx+ ldy,

dv = Mdx+ mdy,

dL/dy= dl/dx

dM/dy = dm/dx dL/dy

dl/dx

/x


8/65

dL/dy, dl/dx, dM/dy, dm/dx

L= du/dx, l= du/dy, M=dv/dx,

m= dv/dy;

du= Ldx + ldy dv= M dx+mdy,

=dx,

= dy,

=u + Ldx + ldy;

=v + Mdx + mdy. dt

=dt(u + Ldx + ldy)

=dt(v+ mdx+ mdy).


9/65

lm = dx ln = dy;

x+dx x

y + dy.

APAL = udt,P p Ll = vdt.

AQ AM = (u+ Ldx)dt,QqMm = (v+ Mdx)dt.


10/65

ARAL = (u + ldy)dt,Rr Ln = (v+ mdy)dt.

u = 0

= 12

dxdy,


11/65

pqr = P prR + RrqQ PpqQ.

PprR = 1

2P R(P p + Rr),

RrqQ = 1

2RQ(Rr+ Qq),

PpqQ = 1

2P Q(P p + Qq).

pqr = 12

P Q Rr 12

RQ P p12

P R Qq.


12/65

AQ = AP+ Q

AR = AP+ R

Qq = P p + q

Rr = P p + r

P Q= Q, P R= R, RQ= Q R.

pqr = 12

Q(P p+r) 12

(QR)P p 12

R(P p+q) pqr = 12

Qr 12

Rq.

Q = dx+ Ldxdt; q = 0 + Mdxdt;

R = 0 + ldydt; r = dy+ mdydt.

pqr = 12

dxdy(1 + Ldt)(1 + mdt)12

Mldxdydt2,

pqr = 12

dxdy(1 + Ldt + mdt + Lmdt2 Mldt2)


13/65

= 12

dxdy,

Ldt + mdt +Lmdt2 Mldt2 = 0,

L + m +LmdtMldt = 0.

Lmdt Mldt

L

m,

L + m= 0.

du = Ldx + ldy,

dv = Mdx+ mdy

L+m = 0. L = du/dx m = dv/dy,

du/dx+ dv/dy = 0,

du/dx+dv/dy = 0.


14/65

AL= x, Ll = y

l= z; l

lL


15/65

.

o

= u, = v, o = w

,

x, y z.

du = Ldx+ ldy+ dz

dv = Mdx+ mdy+ dz

dw = Ndx+ ndy+ dz


16/65

L, l, , M, m, , N, n,

x, y z

dL/dy= dl/dx; dL/dz= d/dx; dl/dz= d/dy

dM/dy= dm/dx; dM/dz= d/dx; dm/dz= d/dy

dN/dy= dn/dx; dN/dz= d/dx; dn/dz= d/dy

dt

udt vdt wdt

V,

V =

(uu + vv+ ww)

dt =V dt

dt.


17/65

, , , o

o

x + dx x x

y y+ dy y

z z z+ dz

=lmn= 12dxdy,

o= dz, = 16

dxdydz

, , , o

dt.

u, v, w

o

u u + Ldx u+ ldy u + dz

v v+ Mdx v+ mdy v+ dz

w w+ Ndx w+ ndy w+ dz

, , , o

dt

, , ,

,,


18/65

,

APAL = udt,

P p Ll = vdt,

p l = wdt;

AQAM = (u+ Ldx)dt,

QqMm = (v+ Mdx)dt,

q m = (w+ Ndx)dt,

ARAL = (u + ldy)dt,Rr Ln = (v+ mdy)dt,

r n = (w+ ndy)dt,

o

AS

AL = (u + dz)dt,

Ss Ll = (v+ dz)dt,

s lo = (w+ dz)dt.


19/65

AP = x+ udt,

P p = y+ vdt,

p = z+ wdt;

AQ = x+ dx + (u+ Ldx)dt,

Qq = y+ (v+ Mdy)dt,

q = z+ (w+ Ndx)dt;

AR = x+ (u+ ldy)dt,

Rr = y+ dy+ (v+ mdy)dt,

r = z+ (w+ ndy)dt;


20/65

AS = x + (u+ dz)dt,

Ss = y+ (v+ dz)dt,

s = z+ dz+ (w+ dz)dt.

, , , o

, , , dt

= 16

dxdydz

pqr pqr

pqr

pqr

u = 0

pqr

z


21/65

pqs

prs

qrs

1

6dxdydz= pqrs+prs + qrspqr.

pqs = 1

3pqs(p+ q+ s),

prs = 1

3prs(p+ r + s),

qrs = 1

3qrs(q + r + s),

pqr = 1

3pqr(p+ q + r).

pqr = pqs+ prs+ qrs

1

6dxdydz= 1

3p.qrs1

3q.prs1

3r.pqs +

1

3s.pqr;


22/65

dxdydz= 2pqr.s 2pqs.r 2prs.q 2qrs.p.

AQ= AP+ Q; Qq= P p + q; q= p+ ,

AR= AP+ R; Rr= P p + r; r= p+ ,

AS=AP+ S; Ss = P p + s; s= p+

p

dxdydz= 2pqr. 2pqs. 2prs.

pqr PpqQ

PprqQ PprR+ RrqQ

pqr =1

2P R(P p + Rr) +

1

2RQ(Rr+ Qq)1

2P Q(P p + Qq);


23/65

P R= R, RQ= Q R, P Q= Q,

pqr = 12

R(P pQq) +12

Q(Rr P p) = 12

Qr 12

Rq.

pqs = 12

P S(P p + Ss) +1

2SQ(Ss + Qq)1

2P Q(P p + Qq),

pqs =

1

2

S(P p + Ss) +1

2

(Q

S)(Ss + Qq)

1

2

Q(P p + Qq)

pqs= 12

S(P pQq) + Q(Ss P p) =12

Qs12

Sq.

prs = 12

P R(P p + Rr) +1

2RS(Rr+ Ss)1

2P S(P p + Ss),

prs = 12

R(P p + Rr) +12

(SR)(Rr+ Ss)12

S(P p + Ss)

prs= 12

R(P p Ss) +12

S(Rr P p) =12

Sr 12

Rs.

dxdydz= (Qr Rq)+ (SqQs)+ (Rs Sr),


24/65

1

6(Qr Rq)+1

6(SqQs) +1

6(Rs Sr).

Q= dx + Ldxdt q = Mdxdt = Ndxdt

R= ldydt r= dt + mdydt = ndydt

S=dzdt s= dzdt = dz+ dzdt.

Qr Rq = dxdy(1 + Ldt + mdt + LMdt2 MLdt2)

SqQs = dxdz(dt Ldt2 + Mdt2)

Rs Sr = dydz(dtmdt2 + ldt2),

1

6dxdydz

1 +

+Ldt +Lmdt2 +Lmdt3

+mdt Mldt2 Mldt3

+dt +Ldt2 Lndt3

+mdt2 +Mndt3

ndt2 Nmdt3

Ndt2 +Nldt3


25/65

o= 16

dxdydz

dt

0 = L+ m + +dt(Lm + L+ mMl N n)

+dt2(Lm+ Mn+ Nl LnMlNl).

L + m +

= 0

u, v, w L= du/dx m= dv/dy =

dw/dz,

x, y, z u, v, w

du

dx+

dv

dy+

dw

dz = 0.

du/dx + dv/dy+dw/dz= 0


26/65

AL = x, Ll = y

l =

du=Ldx+ ldy+ dz+ Ldt,

dv=Mdx+ mdy+ dz+ Mdt,

dw=Ndx+ ndy+ dz+Ndt.

L+ m + = 0,

dt = 0.

A, L, l,


27/65

du

dx+

dv

dy+

dw

dz = 0.

dt


28/65

AL = x, Ll = y;

AL AB

du=Ldx+ ldy+ Ldt,

dv=Mdx+ mdy+ Mdt;

L + m= 0.

= udt AL

= vdt AB;

dt, dx udt dy vdt,

du=Ludt + lvdt +

Ldt,

dv=Mudt + mvdt+Mdt,

L = ut, M =v/t.


29/65

alongAL: 2(Lu+ lv+ L)

alongAB : 2(Mu+ mv+ M)


30/65

dp= Rdx + rdy+Rdt, lm = no = dx

ln= mo= dy ; =dxdy.


31/65

= p, = p+Rdx,

=p + rdy =p + Rdx + rdy.

=dx(p + 12

Rdx).

=dx(p + 12

Rdx + rdy).

=rdxdy. dy(p + 1

2rdy) dy(p + Rdx + 1

2rdy)

= Rdxdy.

=R, =r.

R.

R= 2Lu+ 2lv+ 2L


32/65

R= 2Lu 2lv 2L,

r= 2Mu+ 2mv+ 2M

r= 2Mu 2mv 2M

dp= dx 2(Lu + lv+ L)dx 2(Mu+ mv+ M)dy+Rdt.

dx

Rdt,

d(Lu+ lv+ L)dy

=d(Mu+ mv+ M)

dx

du/dx= L, du/dy= l, dv/dx= M, dv/dy= m,

Ll+ udLdy

+ lm+ vdldy

+dLdy

=M L+ udMdx

+ mM+vdmdx

+dMdx


33/65

(L+ m)(l M) + u

dL

dy dM

dx

+ v

dl

dydm

dx

+

dLdy dM

dx = 0.

Ldx+ldy+Ldt Mdx+mdy+Mdt

dL

dy

= dl

dx

; dm

dx

=dM

dy

; dL

dy

=dl

dt

and dM

dx

=dM

dt

,

(L + m)(l M) + u

dl dMdx

+ v

dl dM

dy

+

dl dMdt

= 0

l = M : du/dy = dv/dx.

udx+vdy

udx + vdy

=du dv


34/65

udx + vdy

p,

M=l,

dp= dx

2u(Ldx + ldy)

2v(ldx + mdy)

2

Ldx

2

Mdy+

Rdt.

Ldx + ldy = du Ldt ldx+ mdy= dv Mdt,

dp= dx 2udu 2vdv + 2Ludt + 2Mvdt 2Ldx 2Mdy+ Rdt.

dp= dx 2udu 2vdv 2Ldx 2Mdy

L =du/dt M =dv/dt,

dp= dx

2udu

2vdv

2

du

dt

dx

2

dv

dt

dy.

M =l.


35/65

udx + vdy

t S

dS=udx+vdy dt= 0.

dS= udx + vdy + Udt.

du/dt= dU/dx dv/dt= dU/dy. U=dS/dt.

du

dt dx +dv

dt dy=dU

dx dx +dU

dydy

= U.

dU=K dx + kdy, dU/dx= K dU/dy= k.

dU

dxdx+

dU

dydy = K dx + kdy = dU.

U=dS/dt,

dp= dx 2udu 2vdv 2dU


36/65

p= Const.+ x uu vv 2dSdt

,

S dt= 0, udx + vdy.

=V = (uu + vv) p = Const.+ x uu vv 2dS/dt. S dS S=

(udx + vdy),

ds. =p/f.

dS/dt= U,

dp= dx pf

ds 2V dV 2dU.

p= e

s/f e

s/f

(dx 2V dV 2dU)

dV

dU.


37/65

p= x V V 2U1f

es/f

es/f(x V V 2U)ds.

udx+ vdy

du/dx+ dv/dy = 0, udy +vdx

udx+ vdy

udy+ vdx

,

AL, AB, AC,

du = Ldx+ ldy+ dz+ Ldtdv = Mdx+ mdy+ dz+ Mdtdw = Ndx+ ndy+ dz+Ndt;

L+ m+ = 0

du

dx+

dv

dy+

dw

dz = 0.


38/65

dt, .

=udt AL, =vdt

AB, = wdt AC.

alongAL = u + Ludt + lvdt + wdt +

Ldt,

alongAB = v+ Mudt+ mvdt + wdt +Mdt,alongAC = w+ Nudt + nvdt + wdt +Ndt.

alongAL = 2(Lu+ lv+ w+ L),alongAB = 2(Mu+ mv+ w+

M),

alongAC = 2(Nu+ nv+ w +N).

AC

AL AB AC,

= 1 p

dp= Rdx+ rdy+ dz,


39/65

AL: R, AB: r, AC: .

R= 2(Lu+ lv+ w+ L),r= 2(Mu+ mv+ w+ M),= 1 2(Nu+ nv+ w +N).

dp = Rdx+ rdy +dz

dR/dy= dr/dx; dR/dz= d/dx; dr/dz= d/dy.

I

uLy+ vly+ wy+ Ly +Ll+ lm+ n =uMx+ vmx+ wx+ Mx +ML+ mM+ N,


40/65

II

uLz+ vlz+ wz+ Lz +L + l+ =uNx+ vnx+ wx+Nx +NL+ nM+ N ,

III

uMz+ vmz+ wz+ Mz +M+ m+ =uNy+ vny+ wy+Ny +Nl+ nm+ n.

Ly =lx; mx=My; y =lz; x= Mz ; Ly =lt; Mx= Mt,Lz =x; lz =y; nx = Ny; x = Nz; Lz =t; Nx= Nt,Mz =x; Ny =nx; mz =y; y =nz; Mz =t; Ny =nt.

dl dM

dt

+udl

dM

dx

+vdl

dM

dy

+wdl

dM

dz

+(l M) (L+ m)+nN=

d dN

dt

+u

d dN

dx

+v

d dN

dy

+w

d dN

dz

+(N) (L+ )+lnM=

d dn

dt

+u

d dn

dx

+v

d dn

dy

+w

d dn

dz

+( n) (m + )+MNl= 0.


41/65

l= M; = N; = N;

dudy =

dvdx;

dudz =

dwdx ;

dvdz =

dwdy .

udx+vdy+

wdz

u, v, w x, y, z

udx + vdy + wdz

dt= 0

du=Ldx + Mdy+ Ndz

dv=Mdx+ mdy+ ndz

dw=Ndx+ ndy+ dz


42/65

R, r,

R= 2(Lu+ Mv+ Nw+ L),r= 2(Mu+ mv+ nw+ M),= 1 2(Nu+ nv+ w +N).

dp= dz 2u(Ldx+ Mdy+ Ndz)2v(Mdx+ mdy+ ndz)2w(Ndx + ndy+ dz)2Ldx 2Mdy 2Ndz

=dz 2udu 2vdv 2wdw 2Ldx 2Mdy 2Ndz.

L =du/dt; M =dv/dt; N =dw/dt,

p= C z uu vv ww 2

du

dtdx +

dv

dtdy+

dw

dtdz

.

udx+vdy + wdz

=S.

t

dS= udx + vdy + wdz+ Udt,


43/65

du/dt= dU/dx; dv/dt= dU/dy; dw/dt= dU/dz.

dU

dxdx+

dU

dydy+

dU

dzdz= dU

p= C

z

uu

vv

ww

2U,

p= C z uu vv ww 2dSdt

.

uu+ vv + ww V

,

p= C z V V 2dSdt

.

S udx+vdy +wdz.

t dt,

dS/dt,

U =S/t.

dS/ds.


44/65

udx + vdy + wdz

du/dx+dv/dy+dw/dz= 0.

udx+ vdy + wdz+ Udt

p= C z uu vv ww 2U.

udx+vdy +wdz

du/dx + dv/dy + dw/dz= 0.


45/65

p = Cz uu vv ww 2U.

udx +

vdy + wdz

du/dx+dv/dy+ dw/dz = 0. u = dS/dx, v = dS/dy,

w= dS/dz,

ddS

dx2 +

ddS

dy2 +

ddS

dz2 = 0.


46/65

S= (Ax+ By + Cz)n.

dS/dx= nA(Ax+ By + Cz)n1 ddS/dx2 =n(n1)AA(Ax+ By +Cz)n2, ddS/dy2 ddS/dz2.

n(n 1)(Ax+ By + Cz)n2(AA + BB + CC) = 0.

n= 0 n= 1.

S=Ax + By +

Cz,

AA + BB +

CC= 0.

S= (Ax+ By + Cz)n


47/65

S=+ x + y + z+ (Ax+ By + Cz)n

+ (Ax+ By+ Cz)n

+ (Ax+ By+ Cz)n

+ (Ax+ By+ Cz)n

etc.

AA + BB + CC= 0;

AA + BB + CC = 0;

AA + BB + CC = 0 etc.

I. S= A,

II. S= Ax+ By + Cx,

III. S= Axx+ Byy+ Czz+ 2Dxy+ 2Exz+ 2F yz

(A + B+ C= 0),

IV. S= Ax3 + By3 +Cz3 + 3Dxxy+ 3Fxxz+ 3Hyyz+ 6Kxyz

A+ E+ G= 0; B+ D+ I= 0; C+ F+ H= 0;


48/65

+Ax4 + 6Dxxyy+ 4Gx3y+ 4Hxy3 + 12Nxxyz

V. S= +By4

+ 6Exxzz+ 4Ix3

z+ 4Kxz3

+ 12Oxyyz+By4 + 6Exxzz+ 4Ix3z+ 4Kxz3 + 12Oxyyz

where A + D+ E= 0 G + H+ P = 0,

B+ D+ F = 0 I+ K+ O= 0,

C+ E+ F = 0 L+ M+ N= 0.

x+y+z

LZxx + MZyy +

NZzz,

L + M+ N

+Ax5 + 5Dx4y+ 5Dx4z+ 10Gx3yy + 10Gx3zz+ 20Kx3yz+ 30NxyyzzS= +By5 + 5Exy4 + 5Ey4z+ 10Hx2y3 + 10Hy3zz+ 20Lxy3z+ 30Oxxyzz

+Cz5 + 5F xz4 + 5Fyz4 + 10Ixxz3 + 10Iyyz3 + 20Mxyz3 + 30Pxxyyz,


49/65

A + G + G = 0; D+ H+ O= 0; D + I+ P = 0B+ H+ H = 0; E+ G + N= 0; E+I+ P = 0; K+ L + M= 0C+ I+I= 0; F+ G + N= 0; F+ H+ O= 0.

S=A,

p= C2dAdtz.

S=Ax + By + Cz,

u= A; v= B &w = C.


50/65

,

p= CzAABBCC2xdAdt 2y dBdt 2zdCdt.

S=Axx+Byy +Czz+2Dxy +2Exz+2F yz A+ B + C = 0 :

u= 3Ax + 2Dy+ 2Ez;v = 2By + 2Dx + 2F z;w = 2Cz+ 2Ex + 2F y,

w= 2Ex + 2F y 2(A+ B)z.

p = C z AACC 2xdAdt 2y dB

dt z dC

dt.


51/65

o;

== dx =

QQ + qq+

= = dy =

RR+ rr+

= o= dz =

SS+ ss +

= = dx2 + dy2 = (QR)2 + (q

r)2 + (

)2

= o =

dx2 + dz2 =

(Q S)2 + (q s)2 + ( )2

= ro =

dy2 + dz2 =

(R S)2 + (r s)2 + ( )2.

QR+ qr+ = 0;QS+ qs + = 0;

RS+ rs + = 0.

, ,


52/65

1 = 1 + 2Ldt; l+ M= 0

1 = 1 + 2mdt; + N= 0

1 = 1 + 2dt; + n= 0

L = 0, m = 0, = 0, M =l,N= n= .

du= + ldy+ dz

dv= ldx + dz

dw= dx dy.

l=M, = N

n = .


53/65

=dxdydz,

AL: = 2dxdydz(Lu + lv+ w+L

) = 2dxdydz(uux+ vuy+ wuz+ ut)

AB: = 2dxdydz(Mu+ mv+ w+M) = 2dxdydz(uvx+ vvy+ wvz+ vt)AC: = 2dxdydz(Nu+ nv+ w +N) = 2dxdydz(uwx+ vwy+ wwz+ wt)

udx+ vdy+ wdz

= S.

dS = udx+ vdy+ wdz+ Udt. du/dy = dv/dx;

du/dz= dw/dx;du/dt= dU/dx

AL:= 2dxdydzudu + vdv+ wdw+ dU

dx ,

ux du

dx .


54/65

AB:= 2dxdydzudu + vdv + wdw+ dUdy

,

AC := 2dxdydzudu + vdv+ wdw+ dU

dz .

uu+vv +ww +2U=T ,

dT =K dx+ kdy + dz

dxdydz

AL: =Kdxdydz

AB: =kdxdydz

AC: =dxdydz.

,


55/65

T =uu + vv + ww + 2U

p= CzT,

,

p= C+ (Qdx + qdy+ dz) T.

Qdx + qdy+ dz

.


56/65

u= 0, v= 0

w= 0. T = 2U,

p= C+

(Qdx+ qdy+ dz) 2U.

U t,

U C,

p= C+

(Qdx+ qdy+ dz) ,

,

(Qdx+ qdy+ dz)

Qdx+qdy + dz =P,

p= C+ P.

p= C

z.


57/65

p= C+ P

P =

P =E, C=

E;

p= PE. p= Cz, z= C,


58/65

=rr,

=f f, =, dt +d. t, d/dt. V = f f/rr. dy = dx dz = dx;

u=f frr

11 + +

;

v=f frr

1 + +

;

w=f frr

1 + +

;

. f f


59/65

uu+vv + ww = V V = f4/r4. rr

udx+vdy +wdz

=S,

dS=f f

rr

dx (1 + + )1 + +

=f f

rr dx

1 + + .

dx

1 + +

= ds, dS = ffdsrr . t, .

S= ffdsrr .

U=dS/dt. ffdsrr

dS/dt = U = ddt ffdsrr ;

T =f4

r4 +

2ddt

ffds

rr .


60/65

p= C+

(Qdx + qdy+ dz)f

4r4 2d

dt

ffds

rr .

Qdx+qdy + dz


61/65


62/65

12

0 0 1

dx + Ldxdt Mdxdt 1

ldydt dy+ mdydt 1

= 12

dxdy+ 12

(L+ m)dxdydt + 12

(Lm lM)dxdydtdt 0 =L + m= u/x + v/y.

udx+vdy

=d, ,


63/65

L+ m = 0.

l M

d

dt

u

y v

x

=

d

dt= 0,

= 0

l M, N, n = U,

DDt U + [ U]= 0,

D/Dt


64/65

U, S=

1

2

[U] + [U]T

,

U= S.

D

Dt = S,

lM= N= n= 0

A+B+C = 0,

AA, BB, C C


65/65

p= C+

(Qdx + qdy+ dz) T

Qdx+ qdy+ dz

Documents

De Motu Fluidorum from L. Euler