Bab II Persamaan Pembentuk Aliran

Computational Fluid Dynamics (CFD) Mechanical Engineering Department, USU

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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)


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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


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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)


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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)


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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


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( ) ( )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


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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


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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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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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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.

Bab II Persamaan Pembentuk Aliran

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