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CHAPTER 2
DIFFERENTIAL FORMULATION
OF THE BASIC LAWS
2.1 Introduction

Differential formulation of basic laws:

Conservation of mass

Conservation of momentum

Conservation of energy
2.2 Flow Generation
(i) Forced convection. Motion is driven by mechanical means.
(ii) Free (natural) convection. Motion is driven by natural forces.
2.3 Laminar vs. Turbulent Flow

Laminar flow: no fluctuations in velocity, pressure, temperature, …

Turbulent flow: random fluctuations in velocity, pressure, temperature, …

Transition from laminar to turbulent flow: Determined by the Reynolds number:

Flow over a flat plate: Ret  V xt /ν  500,000

Flow through tubes: Ret = u D v  2300
u
u
turbulent
laminar
t
t
Fig. 2.1
2.4 Conservation of Mass: The Continuity Equation
2.4.1 Cartesian Coordinates
2

The principle of conservation of mass is applied to an element dxdydz
Rate of mass added to element - Rate of mass removed from element =
Rate of mass change within element
m y 
y
dy
dx
m x
x
 (m y )
y
m x 
dy
 (m x )
dy
y
m y
(a )
(b )
Fig. 2.2
(2.1)

Expressing each term in terms of velocity components gives continuity equation
 


 u   v   w  0
t x
y
z

(2.2a)
This equation can be expressed in different forms:
 u v w 




u
v
w
  

0
t
x
y
z
 x y z 
(2.2b)
or
or


D
   V  0
Dt
(2.2c)


  V  0
t
(2.2d)
For constant density (incompressible fluid):

 V  0
2.4.2 Cylindrical Coordinates
(2.3)
3
 1 
 r v r   1   v     v z   0

t r r
r 
z
(2.4)
2.4.3 Spherical Coordinates



1 
1

 v sin    1   v  0
 2
 r 2 vr 
t r r
r sin  
r sin  


(2 .5)
2.5 Conservation of Momentum: The Navier-Stokes Equations of Motion
2.6 2.5.1 Cartesian Coordinates

Application of Newton’s law of motion to the
element shown in Fig. 2.5, gives



dy
(a)
dx
Application of (a) in the x-direction, gives
 F
x


 F  ( m)a
dz
y
x
 ( m)a x
(b)
z
Fig. 2.5
Each term in (b) is expressed in terms of flow
field variables: density, pressure, and velocity components:
Mass of the element:
 m   dxdydz
(c)
Acceleration of the element a x :
ax 
du Du
u
u
u u

u
v
w

dt Dt
x
y
z t
Substituting (c) and (d) into (b)

 Fx  
Du
dxdydz
Dt
(e)
Forces:
(i) Body force

 Fx body  g x dxdydz
(ii) Surface force
(g)
(d)
4
Summing up all the x-component forces shown in Fig. 2.6 gives
  xx  yx  zx 

δF

 x surface  x  y  z dxdydz


(h)
Combining the above equations

 xx  yx  zx
Du
  gx 


Dt
x
y
z
(2.6a)
y-direction:

 xy  yy  zy
Dv
 gy 


Dt
x
y
z
(2.6b)
z-direction:

 yz  zz

Dw
  g z  xz 

Dt
x
y
z
(2.6c)
x-direction:
By analogy:
IMPORTANT

THE NORMAL AND SHEARING STRESSES ARE EXPRESSED IN TERMS
OF VELOSICTY AND PRESSURE. THIS IS VALID FOR NEWTOINAN
FLUIDS. (See equations 2.7a-2.7f).

THE RESULTING EQUATIONS ARE KNOWN AS THE NAVIER-STOKES
EQUAITONS OF MOTION

SPECIAL SIMPLIFIED CASES:
(i) Constant viscosity




 1
DV

 g  p     V   2V
Dt
3



(2.9)
(2.9) is valid for: (1) continuum, (2) Newtonian fluid, and (3) constant viscosity
(ii) Constant viscosity and density




DV

 g  p   2V
Dt
(2.10)

(2.10) is valid for: (1) continuum, (2) Newtonian fluid, (3) constant viscosity and
(4) constant density.

The three component of (2.10) are
x:
  2u  2u  2u 
 u
p
u
u
u 
u
v
 w   g x 
  2  2  2 
 x
x
y
z 
x
y
z 
 t

 
(2.10x)
5
y-
 
  2v  2v  2v 
 v
p
v
v
v 
u
v
 w   g y 
  2  2  2 
 x
x
y
z 
y
y
z 
 t

(2.10y)
 2w 2w 2w 
 w
p
w
w
w 
  g z 
u
v
w
  2  2  2 
 x
x
y
z 
z
y
z 
 t

(2.10z)
 
z-
2.5.2 Cylindrical Coordinates
The three equations corresponding to (2.10) in cylindrical coordinates are (2.11r), (2.11  ),
and (2.11z).
2.5.3 Spherical Coordinates
The three equations corresponding to (2.10) in spherical coordinates are (2.11r), (2.11  ),
and (2.11  ).
2.6 Conservation of Energy: The Energy Equation
dz
y
2.6.1 Formulation
dy

dx
The principle of conservation of energy is
applied to an element dxdydz
x
A
Rate of change of
internal and kinetic
energy of element

B
z
Net rate of internal and kinetic
energy transport by convection
C
Net rate of heat
addition by conduction

Fig. 2.7
(2.14)
_
D
Net rate of work done by
element on surroundings

The variables u, v, w, p, T, and  are used to express each term in (2.14).

Assumptions: (1) continuum, (2) Newtonian fluid, and (3) negligible nuclear,
electromagnetic and radiation energy transfer.

Detailed formulation of the terms A, B, C and D is given in Appendix A

The following is the resulting equation
 cp
DT
Dp
   kT   T
 
Dt
Dt

(2.15) is referred to as the energy equation

 is the coefficient of thermal expansion, defined as
(2.15)
6
 

1   
  T  p
(2.16)
The dissipation function  is associated with energy dissipation due to friction. It is
important in high speed flow and for very viscous fluids. In Cartesian coordinates 
is given by
2
2

2  2
2 
2
 v w 
v
 u 
 w    u v 
 w u  

 
  2               

 
 x 
y 
z y 
 z    y x 
 x z  



 

2
2  u v w 




3  x y z 
(2.17)
2.6.2 Simplified Form of the Energy Equation

Cartesian Coordinates
(i) Incompressible fluid. Equation (2.15) becomes
c p
DT
   kT  
Dt
(2.18)
(ii) Incompressible constant conductivity fluid. Equation (2.18) is simplified further if
the conductivity k is assumed constant
DT
 k 2T  
Dt
(2.19a)
  2T  2T  2T 
 T
T
T
T 
  k 
u
v
w
 2  2   
2


t

x

y

z
y
z 


 x
(2.19b)
c p
or
 c p 
(iii) Ideal gas. (2.15) becomes
 cp
DT
Dp
   kT 
 
Dt
Dt
(2.22)
 cv

DT
   kT  p  V  
Dt
(2.23)
or

Cylindrical Coordinates. The corresponding energy equation in cylindrical
coordinate is given in (2.24)

Spherical Coordinates. The corresponding energy equation in cylindrical
coordinate is given in (2.26)
7
2.7 Solutions to the Temperature Distribution

The flow field (velocity distribution) is needed for the determination of the
temperature distribution.
IMPORTANT:

Table 2.1 shows that for constant density and viscosity, continuity and
momentum (four equations) give the solution to u, v, w, and p. Thus for this
condition the flow field and temperature fields are uncoupled (smallest
rectangle).

For compressible fluid the density is an added variable. Energy equation and
the equation of state provide the fifth and sixth required equations. For this
case the velocity and temperature fields are coupled and thus must be solved
simultaneously (largest rectangle in Table 2.1).
TABLE 2.1
Basic law
No. of
Equations
Unknowns
p
  k
u
v
w
1
u
v
w
Momentum
3
u
v
w
Equation of State
1
T
p
 

Viscosity relation
   ( p, T )
1
T
p

1
T
p
Energy
1
Continuity
Conductivity relation
k  k ( p, T )
TT

p
k
2.8 The Boussinesq Approximation

Fluid motion in free convection is driven by buoyancy forces.

Gravity and density change due to temperature change give rise to buoyancy.

According to Table 2.1, continuity, momentum, energy and equation of state must be
solved simultaneously for the 6 unknowns: u. v, w, p, T and 

Starting with the definition of coefficient of thermal expansion  , defined as
 
1   
  T  p
(2.16)
8
or
 
This result gives

1   
  T  T
(f)
      (T  T )
(2.28)
Based on the above approximation, the momentum equation becomes



DV
1
   g T  T  
 p  p    v  2V
Dt

(2.29)
2.9 Boundary Conditions
(1) No-slip condition. At the wall, y  0

V ( x,0, z , t )  0
(2.30a)
or
u ( x,0, z, t )  v( x,0, z, t )  w( x,0, z, t )  0
(2.30b)
(2) Free stream condition. Far away from an object ( y   )
u( x, , z, t )  V
(2.31)
Similarly, uniform temperature far away from an object is expressed as
T ( x, , z, t )  T
(2.32)
(3) Surface thermal conditions. Two common surface thermal conditions are used in the
analysis of convection problems. They are:
(i) Specified temperature. At the wall:
T ( x,0, z, t )  Ts
(2.33)
(ii) Specified heat flux. Heated or cooled surface:
k
T ( x,0, z, t )
  qo
y
(2.34)
2.10 Non-dimensional Form of the Governing Equations: Dynamic and Thermal
Similarity Parameters


Express the governing equations in dimensionless form to:

(1) identify the governing parameters

(2) plan experiments

(3) guide in the presentation of experimental results and theoretical solutions
Dimensional form:

Independent variables: x, y, z and t
9

Unknown variables are: u, v, w, p and T . These variables depend on the four
independent variables. In addition various quantities affect the solutions. They
are

p , T , V , Ts , L , g , p , and

Fluid properties c p , k, , , and 

Geometry
2.10.1 Dimensionless Variables

Dependent and independent variables are made dimensionless as follows:

* V
,
V 
V
p 
*
( p  p )
 V 2
,
(T  T )
,
T 
(Ts  T )
*

* g
g  ,
g
(2.35)
V
t
L
Using (2.35) the governing equations are rewritten in dimensionless form.
x* 

x
,
L
y* 
y
,
L
z* 
z
,
L
t* 
2.10.2 Dimensionless Form of Continuity
D
Dt

*

   V *  0
No parameters appear in (2.37)
2.10.3 Dimensionless Form of the Navier-Stokes Equations of Motion

DV *
Gr *  *
1 *2 
* *


T
g


P

 V*
Re
Dt *
Re 2

(2.38)
Constant (characteristic) quantities combine into two governing parameters:
Re 
Gr 
V L V L

, Reynolds number (viscous effect)

v
g Ts  T L3
v2
, Grashof number (free convection effect)
2.10.4 Dimensionless Form of the Energy Equation

(2.37)
Consider two cases:
(i) Incompressible, constant conductivity
(2.39)
(2.40)
10
DT *
Dt

*

1
Εc *
 *2T * 

RePr
Re
(2.41a)
Constant (characteristic) quantities combine into two additional governing
parameters:
Pr 
cp
k

/
v
 , Prandtl number (heat transfer effect)
k / c p 
V 2
,
Εc 
c p (Ts  T )
Eckert number (dissipation effect – high speed, large viscosity)
(2.42)
(2.43)
(ii) Ideal gas, constant conductivity and viscosity
DT *
Dt 


Dp * Εc *
1
 *2T *  Εc * 

RePr
Re
Dt
(2.41b)
No new parameters appear.
2.10.5 Significance of the Governing Parameters


Dimensionless temperature solution:
T *  f ( x* , y * , z * , t * ; Re, Pr, Gr, Ec)
(2.45)
NOTE:

Six quantities: p , T , Ts , V , L , g and five properties c p , k, , , and  , are
replaced by four dimensionless parameters: Re, Pr, Gr and Ec.

Special case: negligible free convection and dissipation: Two governing
parameters:
T *  f ( x* , y * , z * , t * ; Re, Pr)
(2.46)

Geometrically similar bodies have the same solution when the parameters are the
same.

Experiments and correlation of data are expressed in terms of parameters rather
than dimensional quantities.

Numerical solutions are expressed in terms of parameters rather than dimensional
quantities.
2.10.6 Heat Transfer Coefficient: The Nusselt Number

Local Nusselt number:
Nu x = f ( x * ; Re, Pr, Gr, Ec)

Special case: negligible buoyancy and dissipation:
(2.51)
11
Nu x = f ( x * ; Re, Pr )

Free convection, negligible dissipation
Nu x = f ( x * ; Gr, Pr )

2.9
(2.52)
(2. 53)
For the average Nusselt number, x * is eliminated in the above.
Scale Analysis

A procedure used to obtain order of magnitude estimates without solving governing
equations.