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Computational Fluid Dynamics (CFD) Mechanical Engineering Department, USU
Page 1 of 11
Created by AMBARITA Himsar
Persaman Pembentuk Aliran
(Governing Equations)
The fundamental governing equations for fluid flow and heat transfer are developed by three conservation
laws of physics. They are the law conservation of mass, conservation of momentum, and conservation of energy.
These laws will be discusses in the Cartesian coordinate.
2.1.1 The Law conservation of mass
Consider a small element of fluid in two-dimensional case with dimension xδ and yδ as shown in
figure 2.1. The main concept here is that the rate of increase mass in the control volume is equal to the net of mass
flow through the inlet and the outlet ports.
∑∑ −=∂
∂
outin
mmt
M&& (2.1)
where M is the mass instantaneously trapped inside the fluid element and m& is the mass flow rate through the
faces of the element.
Fig 2.1 A fluid element for conservation of mass in two-dimensional case
Using the symbols in the figure, equation can be extended to
( ) xyy
vvyx
x
uuxvyuyx
tδδ
ρρδδ
ρρδρδρδρδ
∂
∂+−
∂
∂+−+=
∂
∂ (2.2)
Solving this equation and dividing the remains by the element size of yxδδ yields,
( ) ( )0=
∂
∂+
∂
∂+
∂
∂
y
v
x
u
t
ρρρ (2.3)
In order to develop the similar equation for three-dimensional flow, the same element of fluid is shown in
figure 2.2. In the figure the velocity in the z-direction is named as w. By using the concepts depicted in the figure,
equation (2.1) gives
( )
yxzz
wwzxy
y
vv
zyxx
uuyxwzxvzyuzyx
t
δδδρ
ρδδδρ
ρ
δδδρ
ρδδρδδρδδρδδρδ
∂
∂+−
∂
∂+−
∂
∂+−++=
∂
∂
(2.4)
Solving this equation and dividing the remains by the element size of zyx δδδ yields
( ) ( ) ( )0=
∂
∂+
∂
∂+
∂
∂+
∂
∂
z
w
y
v
x
u
t
ρρρρ (2.5)
Using divergence operator, equation (2.5) can be written as
( ) 0=⋅∇+∂
∂Vρ
ρ
t (2.6)
Computational Fluid Dynamics (CFD) Mechanical Engineering Department, USU
Page 2 of 11
Created by AMBARITA Himsar
Fig 2.2 A fluid element for conservation of mass in three-dimensional case
The conservation mass equation shown in equation (2.5) can be written as
0=
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂
z
w
y
v
x
u
zw
yv
xu
tρ
ρρρρ (2.7)
By introducing material derive, it defines as
( ) ( ) ( ) ( ) ( )z
wy
vx
utDt
D
∂
∂+
∂
∂+
∂
∂+
∂
∂= (2.8)
And also divergence operator,
z
w
y
v
x
u
∂
∂+
∂
∂+
∂
∂=⋅∇ v (2.9)
Equation (2.7) can be written in a simple form as
0=⋅∇+ vρρ
Dt
D (2.10)
The above equation is a general form of the law of conservation of mass or also known as continuity equation. In the
case of incompressible flow, which means temporal and spatial variations in density are negligible, this equation can
be simplified by dropping DtDρ from the equation. In the tensor notation, the continuity equation can be
written as
( ) 0=∂
∂+
∂
∂i
i
uxt
ρρ
(2.11)
, where ix , 3,2,1=i referred to zyx ,, axes, respectively.
2.1.2 The Law conservation of momentum
This law is also known as Newton’s second law of motion. It says resultant forces that act upon an object
equals to acceleration multiplied by the mass of the object. A small element of fluid in two-dimensional case with
dimension xδ and yδ is shown in Fig 2.3. In the two-dimensional case, forces in the x-direction and y-direction
are only considered. In the figure only the forces in x-direction are presented. Forces act upon the element can be
segregated into surface forces and body forces. The surface forces are generated by pressure, normal stress, and
Computational Fluid Dynamics (CFD) Mechanical Engineering Department, USU
Page 3 of 11
Created by AMBARITA Himsar
shear stress distributions, respectively. The body force, denoted as f, is defined as force per unit mass acting on the
centre of fluid element. In a real problem this force can be gravitational, electric, and magnetic forces.
xδ
yδyp δ
yxxδσ
yxx
pp δδ
∂
∂+
yxx
xxxx δδ
σσ
∂
∂+
xyx δτ
xyy
yx
yx δδτ
τ
∂
∂+
xf
Fig 2.3 A fluid element for conservation of momentum in two-dimensional case
The Newton’s second law in x-direction can be written as
∑ = xx maF (2.12)
,where xF and xa are the resultant forces and acceleration in x-direction, respectively. By substituting all forces
depicted in the figure and using the definition of acceleration DtDuax = , equation (2.11) can be expanded as
Dt
Dumyxfy
yyx
xyx
x
ppp xyx
yx
yxx
x
x =+
−
∂
∂++
−
∂
∂++
∂
∂+− δρδτδ
ττδσδ
σσδδ (2.13)
Solving this equation and substituting mass yxm δρδ= yields
Dt
Duyxyxfyx
yyx
xyx
x
px
yxx δρδδρδδδτ
δδσ
δδ =+∂
∂+
∂
∂+
∂
∂− (2.14)
Divide this equation by yxδδ , we get a more compact equation as follows:
x
yxx fyxx
p
Dt
Duρ
τσρ +
∂
∂+
∂
∂+
∂
∂−= (2.15)
In order to provide more complete momentum equation a small element of fluid for three-dimensional
case is shown in Fig 2.4. In the figure only forces in x-direction are only shown. As a note for the three-dimensional
case, there are six normal and shear stresses act on the surfaces. These forces, two forces sourced by pressure
distribution and force sourced by body force are depicted in the figure.
Substituting these forces into the definition of Newton’s second law in equation (2.11) yields
Dt
Duzyxzyxfyxz
z
zxyy
zyxx
zyxx
ppp
xzx
zx
zx
yx
yx
yx
xx
xx
δδρδδδρδδδτδτ
τ
δδτδτ
τδδδσ
σδδδ
=+
−
∂
∂+
+
−
∂
∂++
∂
∂++
∂
∂+−
(2.16)
Solving this equation and divide by zyx δδδ , results in a more compact equation as follows.
x
zxyxxx fzyxx
p
Dt
Duρ
ττσρ +
∂
∂+
∂
∂+
∂
∂+
∂
∂−= (2.17a)
Computational Fluid Dynamics (CFD) Mechanical Engineering Department, USU
Page 4 of 11
Created by AMBARITA Himsar
xδzδ
yδzyp δδ zyx
x
pp δδδ
∂
∂+
zyxx δδσ zyxx
xxxx δδδ
σσ
∂
∂+
zxyx δδτ
zxyy
yx
yx δδδτ
τ
∂
∂+
yxzx δδτ
yxzz
zxzx δδδ
ττ
∂
∂+
xf
Fig 2.4 A fluid element for conservation of momentum in three-dimensional case
Using the similar way, the equations in y- and z-directions are
y
zyyyxyf
zyxy
p
Dt
Dvρ
τστρ +
∂
∂+
∂
∂+
∂
∂+
∂
∂−= (2.17b)
, and
z
zzyzxz fzyxy
p
Dt
Dwρ
σττρ +
∂
∂+
∂
∂+
∂
∂+
∂
∂−= (2.17c)
, respectively. The above equation was generated by element of the fluid is moving with the flow or known as
non-conservation form. Thus the terms of substantial derivative must be converted into conservation form. For
instance, the conversion process of DtDu is shown in the following.
ut
u
Dt
Du∇⋅+
∂
∂= Vρρρ (2.18)
Expanding the following derivates and recalling the vector identify for divergence of the product scalar times a
vector give
( )t
ut
u
t
u
∂
∂+
∂
∂=
∂
∂ ρρ
ρ (2.19)
And
( ) ( ) ( ) uuu ∇⋅+⋅∇=⋅∇ VVV ρρρ (2.20)
Substituting equation (2.19) and equation (2.20) into equation (2.18) yields
( ) ( ) ( )VV ρρρρ
ρ ⋅∇−⋅∇+∂
∂−
∂
∂= uu
tu
t
u
Dt
Du (2.21)
which it can be arranged into
( ) ( ) ( )
⋅∇+
∂
∂−⋅∇+
∂
∂= VV ρ
ρρ
ρρ
tuu
t
u
Dt
Du (2.22)
The last term of this equation is equal to zero as shown in equation (2.6). Thus equation (2.22) can written as
( ) ( )Vut
u
Dt
Duρ
ρρ ⋅∇+
∂
∂= (2.23)
Substituting equation (2.23) into equation (2.17) results in momentum equation in x-direction in conservation form.
( ) ( )x
zxyxxx fzyxx
pu
t
uρ
ττσρ
ρ+
∂
∂+
∂
∂+
∂
∂+
∂
∂−=⋅∇+
∂
∂V (2.24a)
Similarly, the equations in y- and z-directions, respectively, are
( ) ( )y
zyyyxyf
zyxy
pv
t
vρ
τστρ
ρ+
∂
∂+
∂
∂+
∂
∂+
∂
∂−=⋅∇+
∂
∂V (2.24b)
( ) ( )z
zzyzxz fzyxz
pw
t
wρ
σττρ
ρ+
∂
∂+
∂
∂+
∂
∂+
∂
∂−=⋅∇+
∂
∂V (2.24c)
Computational Fluid Dynamics (CFD) Mechanical Engineering Department, USU
Page 5 of 11
Created by AMBARITA Himsar
Equations (2.24) are also known as Navier-Stokes equation in conservation form.
If the stress versus strain rate curve of fluids are plotted, there are two phenomena can be drawn. They are
fluid with linier curve and one with non-linier curve. The fluids with linier curve are known as Newtonian fluids, as
an example is water. The fluids with non-linier curve are known as non-Newtonian fluids, as an example is blood. In
the present dissertation we only consider the Newtonian fluids. For these fluids, the normal stress can be formulated
as follows.
( )x
uxx
∂
∂+⋅∇′= µµσ 2V (2.25a)
( )y
vyy
∂
∂+⋅∇′= µµσ 2V (2.25b)
( )z
wzz
∂
∂+⋅∇′= µµσ 2V (2.25c)
And shear stress
∂
∂+
∂
∂==
y
u
x
vyxxy µττ (2.26a)
∂
∂+
∂
∂==
x
w
z
uzxxz µττ (2.26b)
∂
∂+
∂
∂==
z
v
y
wzyyz µττ (2.26c)
,where µ is the gradient of the stress versus strain rate curve or known as the molecular viscosity (very popular as
dynamic viscosity) and µ ′ is the second viscosity. These two viscosities are related to the bulk viscosity ( )κ by
expression
µµκ ′+=32 (2.27)
In general, it is believed that the bulk viscosity is negligible except in the study of structure of shock waves and in
the absorption and attenuation of acoustic waves. In other words, for almost all fluids bulk viscosity is equal to zero
or 0=κ . Thus the second viscosity becomes
µµ32=′ (2.28)
As a note this hypothesis was introduced by Stokes in 1845. Although the hypothesis has still not been definitely
confirmed, however, it is frequently used to the present day. The present work is included.
Substituting the hypothesis and the normal and shear stresses equations into equation (2.24) we obtain the
complete Navier-Stokes equations.
( ) ( ) ( ) ( )
xfz
u
x
w
zx
v
y
u
y
z
w
y
v
x
u
xx
p
y
uw
y
uv
x
uu
t
u
ρµµ
µρρρρ
+
∂
∂+
∂
∂
∂
∂+
∂
∂+
∂
∂
∂
∂+
∂
∂−
∂
∂−
∂
∂
∂
∂+
∂
∂−=
∂
∂+
∂
∂+
∂
∂+
∂
∂2
3
2
(2.29a)
( ) ( ) ( ) ( )
yfy
w
z
v
zy
u
x
v
x
z
w
x
u
y
v
yy
p
y
vw
y
vv
x
uv
t
v
ρµµ
µρρρρ
+
∂
∂+
∂
∂
∂
∂+
∂
∂+
∂
∂
∂
∂+
∂
∂−
∂
∂−
∂
∂
∂
∂+
∂
∂−=
∂
∂+
∂
∂+
∂
∂+
∂
∂2
3
2
(2.29b)
( ) ( ) ( ) ( )
zfy
w
z
v
yz
u
x
w
x
y
v
x
u
z
w
zz
p
y
ww
y
vw
x
uw
t
w
ρµµ
µρρρρ
+
∂
∂+
∂
∂
∂
∂+
∂
∂+
∂
∂
∂
∂+
∂
∂−
∂
∂−
∂
∂
∂
∂+
∂
∂−=
∂
∂+
∂
∂+
∂
∂+
∂
∂2
3
2
(2.29c)
These equations can be written with more compact by using tensor equation as
Computational Fluid Dynamics (CFD) Mechanical Engineering Department, USU
Page 6 of 11
Created by AMBARITA Himsar
( ) ( )i
k
k
ij
i
j
j
i
jij
jii fx
u
x
u
x
u
xx
p
x
uu
t
uρµδµ
ρρ+
∂
∂−
∂
∂+
∂
∂
∂
∂+
∂
∂−=
∂
∂+
∂
∂
3
2 (2.30)
Where 3,2,1,, =kji referred to zyx ,, axes, respectively.
2.1.3 The Law conservation of energy
In this section, the third physical principle that is energy is conserved is applied. It says the rate change of
energy inside ( )E& an element is equal to sum of the net heat flux ( )Q& into the element and rate of work done
W& on element by body and surface forces. This law can be written as
WQE &&& += (2.31)
The rate of work done on element by body and surface forces will firstly evaluated. Consider a small element of
fluid as shown in Fig 2.5. The considered forces here are forces due to pressure field, due to normal and shear
stresses, and due to body force. As a note the definition of the rate of work done on element is the force multiple by
velocity. Thus, all forces must be considered here. However, it will be very if all forces are drawn in the same
element. In order to make it simple, only the forces in x-direction are shown in the figure. These forces will be firstly
evaluated and the similar way will be employed to evaluate work by forces in y- and z-direction then.
xδzδ
yδzyup δδ zyx
x
upup δδδ
∂
∂+
)(
zyu xx δδσ zyxx
uu xx
xx δδδσ
σ
∂
∂+
)(
zxu yx δδτ
zxyy
uu
yx
yx δδδτ
τ
∂
∂+
)(
yxu zx δδτ
yxzz
uu zx
zx δδδτ
τ
∂
∂+
)(
xuf
Fig 2.5 Work done on element by forces in x-direction
Using the definition, the rate of work by forces in x-direction is calculated by the following equation.
∑= xx uFW& (2.32)
Substituting all the forces shown in the above figure gives
zyxfuyxuzz
uuzxuy
y
uu
zyuxx
uuzyx
x
upupupW
xzx
zx
zxyx
yx
yx
xx
xx
xxx
δδδρδδτδτ
τδδτδτ
τ
δδσδσ
σδδδ
+
−
∂
∂++
−
∂
∂++
−
∂
∂++
∂
∂+−=
)()(
)()(&
(2.33)
Solving this equation and defining zyxV δδδδ = yields
Vfuz
u
y
u
x
u
x
upW x
zxyxxx
x δρττσ
+
∂
∂+
∂
∂+
∂
∂+
∂
∂−=
)()()()(& (2.34a)
Similar way gives the work rate by forces in y- and z-directions, respectively, as
Computational Fluid Dynamics (CFD) Mechanical Engineering Department, USU
Page 7 of 11
Created by AMBARITA Himsar
Vfvz
v
y
v
x
v
y
vpW y
zyyyxy
y δρτστ
+
∂
∂+
∂
∂+
∂
∂+
∂
∂−=
)()()()(& (2.34b)
Vfwz
w
y
w
x
w
z
wpW z
zzyzxz
z δρσττ
+
∂
∂+
∂
∂+
∂
∂+
∂
∂−=
)()()()(& (2.34c)
In total, the net rate of work done on the fluid element is the sum of these terms. Thus the net rate of work is
( ) ( ) ( )
( ) Vwvuz
Vwvuy
wvux
pW
zzzyzx
yzyyyxxzxyxx
δρσττ
δτστττσ
⋅+++
∂
∂+
++
∂
∂+++
∂
∂+⋅∇−=
Vf
V&
(2.35)
The next term is the net rate of heat flux into the fluid element. There are two sources of this heat flux. The
first is due to heat generation inside the element, such as heat adsorption, chemical reaction, or radiation. The second
is heat transfer to the element across the surfaces due to temperature difference. Define the volumetric heat
generated inside the element as q& and heat transfer rates across the surface in x-, y-, and z-directions are xq& , yq& ,
and zq& , respectively. All of theses sources are shown in Figure 2. 6. Using all of those sources shown in the figure,
thus the net rate of heat flux into the element can be calculated as
zyxqyxzz
qqq
zxyy
qqqzyx
x
qqqQ
z
zz
y
yy
x
xx
δδδρδδδ
δδδδδδ
&&
&&
&&&
&&&&
+
∂
∂+−+
∂
∂+−+
∂
∂+−=
(2.36)
Solving this equation yields
zyxz
q
y
q
x
qqQ zyx δδδρ
∂
∂+
∂
∂+
∂
∂−=
&&&&& (2.37)
xδzδ
yδzyq x δδ& zyx
x
qq x
x δδδ
∂
∂+
&&
yx
q z
δδ&
yx
z
zq
q
z
z
δδ
δ
∂∂
+
&
&
zx
qy
δδ
&z
xy
yqq
y
yδ
δδ
∂∂+
&&
zyxq δδδ&
Fig 2.6 Heat flux across the surfaces of fluid element
The heat flux in the above equation can be calculated by using Fourier’s law, is proportional to the local temperature
Computational Fluid Dynamics (CFD) Mechanical Engineering Department, USU
Page 8 of 11
Created by AMBARITA Himsar
gradient. They arex
Tkqx
∂
∂−=& ,
y
Tkq y
∂
∂−=& , and
z
Tkq z
∂
∂−=& , the heat fluxes in x-, y-, and z-directions
respectively. Here, k is the thermal conductivity. Thus, the equation (2.37) can be written as
Vz
Tk
zy
Tk
yx
Tk
xqQ δρ
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂+= && (2.38)
Finally we will calculate the rate change of energy inside the fluid element in the equation (2.31). Here,
the energy is the total energy inside the fluid element. It is the sum of internal energy and kinetic energy due to
velocity of the element. On one hand, according to the classical thermodynamics, the internal energy is related to the
sum of the translational, rotational, and electronics of its molecules. In this dissertation we will not explore into the
molecules energy calculation. We only define that all of these energies are defined as internal energy per mass of
fluid element, which it is denoted as i . On the other hand, the kinetic energy of the fluid element can be calculated
by considering all of the components of the velocity. Here the kinetic energy per mass is 22V ,
where2222
wvuV ++= . Using these explanations, the rate change of energy inside the fluid element can be
calculated using the following equation:
zyxV
iDt
DE δδδρ
+=
2
2
& (2.39)
Substituting the above developed equations into equation (2.31) we get the energy equation in general form.
( )
( ) ( ) ( ) Vf
V
⋅+++∂
∂+++
∂
∂+++
∂
∂
+⋅∇−
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂+=
+
ρστττστττσ
ρρ
zzzyzxyzyyyxxzxyxx wvuz
wvuy
wvux
pz
Tk
zy
Tk
yx
Tk
xq
Vi
Dt
D&
2
2
(2.40)
This equation is known as energy equation in non-conservation form and it contains energy in terms of total energy,
the internal and kinetic energy. As a note the above equation is just one of many different forms of energy equation.
Furthermore, it does not clearly show the relation of all parameters. For instance, if we want to use this equation to
calculate the temperature field, it seems to be veiled in the left hand side of the equation. Since so, this equation
need to be converted into a more specific form.
In order to convert the energy equation into a more specific form, we call again the momentum equation in
equation (2.17). Consider the momentum equation in x-direction and multiple by component of velocity gives
x
zxyxxx ufz
uy
ux
ux
pu
Dt
Duu ρ
ττσρ +
∂
∂+
∂
∂+
∂
∂+
∂
∂−= (2.41)
By using the definition that ( ) xABxBAxAB ∂∂+∂∂=∂∂ , the above equation can be written as
( ) ( ) ( ) ( ) ( )
xzxyx
xx
zxyxxx
ufz
u
y
u
x
u
x
up
z
u
y
u
x
u
x
up
Dt
uD
ρττ
σττσ
ρ
+∂
∂−
∂
∂−
∂
∂−
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂−=
22
(2.42a)
Using the similar way for momentum in y- and z-direction yields
( ) ( ) ( ) ( ) ( )
yzyyy
xy
zyyyxy
vfz
v
y
v
x
v
y
vp
z
v
y
v
x
v
y
vp
Dt
vD
ρτσ
ττστ
ρ
+∂
∂−
∂
∂−
∂
∂−
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂−=
22
(2.42b)
( ) ( ) ( ) ( ) ( )
zzzyz
xz
zzyzxz
wfz
w
y
w
x
w
z
wp
z
w
y
w
x
w
z
wp
Dt
wD
ρστ
τσττ
ρ
+∂
∂−
∂
∂−
∂
∂−
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂−=
22
(2.42c)
Adding the all equations (2.42) and using the definition 2222
wvuV ++= results in an equation. Subtracting the
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Created by AMBARITA Himsar
equation resulted from the energy equation in general form of equation (2.40), we obtain
z
w
y
w
x
w
z
v
y
v
x
v
z
u
y
u
x
u
z
w
y
v
x
up
z
Tk
zy
Tk
yx
Tk
xq
Dt
Di
zzyzxzzyyyxyzxyx
xx
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂−
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂+=
στττστττ
σρρ &
(2.43)
The above energy equation is the equation in the non-conservation form and the left hand it contains the internal
energy only. In other words, the kinetic and body force terms have dropped out. The normal and shear stresses do
appear in the equation. It is very convenient to convert these terms into the velocity components. To do so, calling
the relationships in the equation (2.25) to (2.26) for the Newtonian’s fluid. Thus, the equation (2.43) is converted
into
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂−
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂+=
y
w
z
v
x
w
z
u
x
v
y
u
z
w
y
v
x
u
z
w
y
v
x
up
z
Tk
zy
Tk
yx
Tk
xq
Dt
Di
zyzxyxzzyy
xx
τττσσ
σρρ &
(2.44)
Substituting the normal and shear stresses relationships, yield
( ) ( )
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂
+⋅∇′+⋅∇−
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂+=
222222
2
222y
w
z
v
x
w
z
u
x
v
y
u
x
u
y
v
x
u
pz
Tk
zy
Tk
yx
Tk
xq
Dt
Di
µ
µρρ VV&
(2.45)
In order to make this equation more easy look, all of the viscous effects are grouped into a factor. The factor is
known as dissipation function Φ , which can be rewritten from the above equation as
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂+
∂
∂
+
∂
∂+
∂
∂+
∂
∂′=Φ
222222
2
222y
w
z
v
x
w
z
u
x
v
y
u
x
u
y
v
x
u
z
w
y
v
x
u
µ
µ
(2.46)
Using this function, the energy equation developed so far can be written as
( ) Φ+⋅∇−+
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂= Vpq
z
Tk
zy
Tk
yx
Tk
xDt
Di&ρρ (2.47)
The material derivate term in the left hand side of this equation shows that it is still in the non-conservation form. In
the conservation form it can be written as
( ) ( ) ( ) ( )
( ) Φ+⋅∇−+
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂=
∂
∂+
∂
∂+
∂
∂+
∂
∂
Vpq
z
Tk
zy
Tk
yx
Tk
xz
wi
y
vi
x
ui
t
i
&ρ
ρρρρ
(2.48)
In order to convert this equation so it contains the temperature in the left hand side, the equation of state which
shows the relationship between internal energy and temperature can be used. For instance, we uses a simple
relationship of internal energy cTi = , where c is the heat capacity of the fluid. Substituting this relationship, we
get the equation
( ) ( ) ( ) ( )
( ) Φ+⋅∇−+
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂
=∂
∂+
∂
∂+
∂
∂+
∂
∂
Vpqz
Tk
zy
Tk
yx
Tk
x
z
cwT
y
cvT
x
cuT
t
cT
&ρ
ρρρρ
(2.49)
As a note, the objective of solving the energy equation is to obtain the temperature distribution in the flow field.
Since so, it needs to be presented in the terms of temperature terms. It is now clearly shown that the energy equation
in the term of temperature only.
The energy equation shown in equation (2.49) can be written with more compact by using tensor equation
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Created by AMBARITA Himsar
as
( ) ( )Φ++
∂
∂−
∂
∂
∂
∂=
∂
∂+
∂
∂q
x
up
x
Tk
xx
cT
t
cT
i
i
iii
&ρρρ
(2.50)
Where 3,2,1,, =kji referred to zyx ,, axes, respectively. If some assumptions are proposed, some of terms in
the energy equation (2.50) are vanished. For instance, if density is constant or incompressible fluid the term
ii xup ∂∂ will be equal to zero. In addition, if viscous dissipation is negligible, the term Φ will be dropped from
the equation. And also, if the internal heat generated inside the element is zero, it will be dropped as well.
2.1.4 Summarize of the governing equations
Although the equations resulted seems to be very complicated, however they come from three very simple
conservation laws, mass is conserved, momentum is conserved, and energy is conserved. In the case of three
dimensional, these laws generate five differential equations. They are a coupled system of nonlinear partial
differential equations. Thus, they are very difficult to solve analytically. There is no general solution to these
equations. Some optimistic people may say not yet found and solution has not been reported. In other words, this
does not mean that no general solution exists but the scientists just have not been able to fine one. These equations
are an open problem without analytical solution for almost 200 years. Clay mathematics institute, a private
non-profit foundation based in Cambridge, Massachusetts has called the Navier-Stokes equations as one of the seven
most important open problems in mathematics. The foundation has been offering one billion US dollars for a
solution or a counter-example. To date, no body has been awarded this money. The analytical solution is still open.
The other method to solve those equations is numerical method. This the main concern of this chapter. In this
method, the equations will be solved iteratively to find a solution as close as possible to the exact solution. How this
method works will be discussed in the next section.
We will now summarize all of the governing equations. There are several forms that can be used to present
the governing equations. Some forms have been used in the previous section. The other possible form will be used to
summarize these equations. In transient three-dimensional of compressible Newtonian fluids, these forms are as
follows.
The continuity equation
( ) 0=+∂
∂Vρ
ρdiv
t (2.51)
The momentum equations
x-momentum: ( ) ( ) ( ) uS
x
pugraddivudiv
t
u+
∂
∂−=+
∂
∂µρ
ρV (2.52a)
y-momentum: ( ) ( ) ( )
vSy
pvgraddivvdiv
t
v+
∂
∂−=+
∂
∂µρ
ρV (2.52b)
x-momentum: ( ) ( ) ( ) wS
z
pwgraddivwdiv
t
w+
∂
∂−=+
∂
∂µρ
ρV (2.52c)
The energy equation
( ) ( ) ( ) TSTgradkdivcdivt
cT+=+
∂
∂Vρ
ρ (2.53)
Where uS , vS , wS , and TS are the sources terms related to u, v, w, and T, respectively. These sources can be
calculated by comparing these equations with the previous forms.
The main objective of casting these equations into forms as shown is that to bring out their commonality.
Observing equation (2.51) to (2.53) clearly shows it. If we introduce a general variable φ the conservative form of
all governing equations can be written in the following form
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( ) ( ) ( ) φφρφρφ
Sgraddivdivt
+Γ=+∂
∂V (2.54)
In words sum of rate of increase of φ of fluid element and net rate of flow of φ out of fluid element is equal to
sum of rate of increase of φ due to diffusion and rate of increase of φ due to source. The equation (2.54) is
known as transport equation for property φ . It is clearly shown the equation can be divided into four terms. They
are the transient rate of change, the convective term, diffusive term ( Γ is diffusion coefficient), and source term. So
we closing this section by saying solving equation (2.54) numerically can used to solve all of the governing
equations. The method to solve this equation will be discussed in the next section.