Chapter 6
Flow Analysis Using Differential Methods
(Differential Analysis of Fluid Flow)
1
• In the previous chapter-• Focused on the use of finite control volume for the
solution of a variety of fluid mechanics problems.
• The approach is very practical and useful since it
doesn’t generally require a detailed knowledge of the
pressure and velocity variations within the control
volume.
• Typically, only conditions on the surface of the
control volume entered the problem.
• There are many situations that arise in which the
details of the flow are important and the finite
control volume approach will not yield the
desired information
2
• For example -• We may need to know how the velocity varies over the cross
section of a pipe, or how the pressure and shear stress vary
along the surface of an airplane wing.
•  we need to develop relationship that apply at a point,
or at least in a very small region ( infinitesimal volume)
within a given flow field.
•  involve infinitesimal control volume (instead of finite
control volume)
•  differential analysis (the governing equations are
differential equation)
3
•
In this chapter—
•
(1) We will provide an introduction to the differential
equation that describe (in detail) the motion of fluids.
(2) These equation are rather complicated, partial differential
equations, that cannot be solved exactly except in a few
cases.
(3) Although differential analysis has the potential for
supplying very detailed information about flow fields, the
information is not easily extracted.
(4) Nevertheless, this approach provides a fundamental basis
for the study of fluid mechanics.
(5) We do not want to be too discouraging at this point,
since there are some exact solutions for laminar flow that
can be obtained, and these have proved to very useful.
•
•
•
•
4
•
•
•
•
•
(6) By making some simplifying assumptions, many other
analytical solutions can be obtained.
for example , μ small 0 neglected
 inviscid flow.
(7) For certain types of flows, the flow field can be conceptually
divided into two regions—
(a) A very thin region near the boundaries of the system in
which viscous effects are important.
(b) A region away from the boundaries in which the flow is
essentially inviscid.
(8) By making certain assumptions about the behavior of the fluid
in the thin layer near the boundaries, and
using the assumption of inviscid flow outside this layer, a large
class of problems can be solved using differential analysis .
the boundary problem is discussed in chapter 9.
Computational fluid dynamics (CFD)  to solve differential eq.
5
One of the goals of differenti al analysis is to determine how these velocity
components specifical ly depend on x, y, z, t, for a particular problem.





 v
v dv
v
v

v w
u
a
z dt
y
x
t
u
u
u
u
w
v
u
ax 
z
y
x
t
ay  .    
az  .    
 (.)  (.)  
 (.)
 (.)
D(.)  (.)
 (v  )(.)

w
v
u

t
z
y
x
t
t
6
1 d () u v w  


 
   v  volumetric dilation rate
 dt
x y z
 
v  0
for an incompress ible fluid,
u v w
Note :
, ,
 linear deformation of the element
x y z
u v
,  angular deformation of the element
y x
7

W  the rotation vector



 w x i  w y j  wz k
1 w v  u w  v u 
 {(
 )i  ( 
) j  (  )k }
2 y z
z x
x y



i
j
k

1 


1  
1

 (  v )  curlV
2 x y z 2
2
u
v
w


  
Define   vorticity  2 w    v  curlV
From Eq. (6.12) such as wz 
1 v u
(  )
2 x y
 Rotationabout z - axis as an undeformed block (i.e. wOA   wOB ) only when
 Rotation around the z - axis is zero
u
v

y
x
v u

x y
 
or   v  0      Irrotational flow
8
§ 6.2.1 Differential Form of Continuity Equation
d  xyz  0



  d  ( d) 
xyz
cv
t
t
t
 

(
v
  n )dA  The rate of mass flow through the surfaces of the element
  ( vA) out   ( vA) in
9
T he flow in the x - direction
 ( u ) y 
x
x 2  T aylor series expansion- - neglectinghigh order
2
2
s
 ( u ) x  terms, such as (x ) , (x )   

u | y  u 
x
x 2 
2

u | x  u 
 Net rate of mass outflow in x - direction
 [ u 
u x
u x
 ( u )
]yz  [ u 
]yz 
xyz      (6.23)
x 2
x 2
x
similarly
 ( v)
xyz      (6.24)
y
 ( w)
 Net rate in z - direction 
xyz
z
 Net rate in y - direction 
10
 

d   ( v  n )dA  0
t cv
cs

u
v
w

xyz  [
xyz 
xyz 
xyz ]  0
t
x
y
z
u
v
w
Note :
xyz 
xyz 
xyz  Net rate of mass outflow
x
y
z
 u v w




 0      (6.27)
t
x
y
z
T he continuity equation in differential form.
Since


11
- - One of the fundamenta l equations of fluid mechanics
In vector form

 
   v  0      (6.28)
t
- - For steady flow of compressib le fluid


  ( v )  0
u v w
or


 0      (6.29)
x
y
z
- - For incompress ible flow

  const 
0
t
 
  v  0      (6.30)
u v w
or
 
 0      (6.31)
x y z
12
Example 6. 2
For an incompress ible flow
u  x2  y2  z2
v  xy  yz  z
w?
Determine : w , required to satisfy t he continuity equation.
Solution :
from the equation of continuity
u v w


0
x y z
 2

w
( x  y 2  z 2 )  ( xy  yz  z ) 
0
x
y
z
w

 2 x  ( x  z )  3 x  z
z
z2
Integratio n w  3 xz 
 c ( x, y )
2

13
§ 6.2.2 Cylindrical Polar Coordinates
14

 1  (rvr ) 1  ( v )  ( v z )



 0      (6.33)
t r r
r 
z
This is the differenti al form of the continuall y equation
in cylindrica l coordinate s
For steady, compressib le flow
1 
1 

(rvr ) 
( v )  ( v z )  0
r r
r 
z
For incompress ible flow ( steady or unsteady flow )
1  (rvr ) 1 v v z


0
r r
r  z
15
§ 6.2.3 The Stream Function
Continuity equation
 u v w



0
t
x
y
z
For the continuity equation of steady, incompressible, plane, & 2 - D flow

w
where
 0 (   cte)
 0 ( 2  D flow)
t
z
u v


 0      (6.36)
x y
Define a function  ( x, y )  the stream function,


where u 
; v
so that it satisfies the continuity eq.
y
x
u v  




 (
)  (
)0
x y x y
y
x
using stream
u
function
two
unknows

   one unknow

v

conservation of mass will be satisfied
16
In cylindrica l coordinate s , the continuall y equation for
Incompress ible , place, 2 - D flow.
1  (rvr ) 1 v
 1  (rvr ) 1  ( v )  ( v z )
0

0



r 
r r
z
r 
t r r
 vr 
1 
r 
v  

     (6.42)
r
Example 6.3.
17
Example 6.3 Stream Function
• The velocity component in a steady, incompressible, two
dimensional flow field are
u  2y
v  4x
Determine the corresponding stream function and show on a
sketch several streamlines. Indicate the direction of glow along
the streamlines.
18
Example 6.3 Solution
From the definition of the stream function

u
 2y
y

v
 4x
x
  y2  f1 (x)
  2x 2  f 2 (y)
  2x  y
2
2
Ψ=0
  2x  y  C
2
2
For simplicity, we set C=0
2
2
y
x

1
 /2
Ψ≠0
19
§ 6.3 Conservation of Linear Momentum
For the linear momentum

D ( mv ) sys D 
D

v d   P
dt
dt sys
dt

where P 




v d  v dm
sys
sys
From the Reynolds transport theoremfor the linear momentum

D ( mv ) sys
  
 

v d  
v  ( v  n )dA 
Fc.v.      (6.44)
dt
t cv
cs
For sm all C.V .




dv
Eq(6.44)  Fcv  m
or Fcv  ma
dt
T he Newtons 2nd law for a system


Fsys  ma



20
Figure 6.9 (p. 287)
Components of force acting on an arbitrary differential area.
21
Figure 6.10 (p. 287)
Double subscript notation for stresses.
22
Figure 6.11 (p. 288)
Surface forces in the x direction acting on a fluid element.
23
§ 6.3.2 Equation of Motion
Fx  ma x
Fx  ma x
Fz  ma z

Fbx  Fsx  ma x
Fby  Fsy  ma y
Fbz  Fsz  ma z
using m  xyz  d
 xx  yx  zx
u
u
u
u
 g x 


 (  u
v
 w )      (6.50a )
x
y
z
t
x
y
z
 xy  yy  zy
v
v
v
v
g y 


  (  u  v  w )      (6.50b)
x
y
z
t
x
y
z
 xz  yz  zz
w
w
w
w
g z 


 (
u
v
 w )      (6.50c)
x
y
z
t
x
y
z
General differenti al equation of motion for a fluid.
They are also applicable to any continuum (solid or fluid) in motion or at rest .
Unknowns - - - - - stresses Velocities
24
§ 6.4 Inviscid Flow
Some common fluid , such as air and water ,  air &  water  small
  air &  water  0  &   0
Flow field in which the shearing stresses are assumed to be negligible
are said to be inviscid , nonviscous , or frictionle ss .
 0 


For invisicid flow  
  &  0
 P   xx   yy   zz


compressiv e normal stress
25
§ 6.4.1 Euler’s Equations of Motion
From Eq (6.50a) (6.50b) & (6.50c) with   0 and   0
 xx   yy   zz   P
P
u
u
u
u
 (  u
v
 w )      (6.51a )
x
t
x
y
z
P
v
v
v
v
g y 
  (  u  v  w )      (6.51b)
y
t
x
y
z
P
w
w
w
w
g z 
 (
u
v
 w )      (6.51c)
z
t
x
y
z
 g x 
These equations are Commonly referred to as Eulers equations of motion

 
v
  
 g  P  [  (v  )v ]      (6.52)
t
Difficulty to solve .
Simplify (using )  to solve Eq (6.52)
26
§ 6.4.2 The Bernoulli Equation
In section 3. 2 Bernoulli equation  Newtons 2nd law
In this section Bernoulli equation  Eulers equation
Form Eulers equation

 
  
v
g  p  [  (v  )v ]      (6.52)
t

v
where
 0 ( steady state )
t
 
  
 g  p   (v  )v      (6.53)


Usi ng g   gz and vector identity
   1     
(v  )v  (v  v )  v  (  v )
2
Eq (6.53) becomes


  
   
 gz  p  (v  v )  v  (  v )
2


  
p 1  2

 v  gz  v  (  v )
 2
27

p

  
1 2
 v  gz  v  (  v )ds
 2
Let ds  a differenti al length along streamline



 dxi  dyj  dzk



   

p  1  2

 ds  (v )  ds  g (z )  ds  [v  (  v )]  ds

2

p p p
p  ds 
, ,
   dx, dy, dz 
x y z
p
p
p

dx 
dy 
dz  dp
( if steady state.)
x
y
z
Similarly
 2

(v )  ds  dv 2




z 
z  ds  ( k )  (dxi  dyj  dzk )  dz
z

 




[v  (  v )]  ds  0 because v  (  v )  in / out of papey.
  

 [v  (  v )]  ds
1 2
dp v 2

 dv  gdz  0  

 gz  const .
 2

2
dp
28
For Inviscid, incompress ible fluid
Inviscid flow


Steady flow
p v2

 
 gz  const 
 2
 Incompress ible flow
Flow along a streamline
29
§ 6.4.3 Irrotational Flow
Irrotation al flow
 
 
 1  
w  (  V )  0 or   V  0 or Vorticity  0 (  V  Vorticity)
2
v u 

x y  A general flow field could not

w v 
  satisfy t hese three equations.

y z 
However, a uniform flow does.
u w 

z x 

u  U (const .)

v0
 An example of an irrolation al flow

w0

30
§ 6.4.5 The Velocity Potential
 
 1  
- - For irrotation al flow w  (  v )  0 or   v  0
2
v u




i
j
k
x y
 



w v
 v 
0


x y z
y z
u
v
w
u w

z x



u
, v
, w
x
y
z
where  ( x, y, z ) is a scale function  velocity potential
 can be defined for a general 3 - D flow
 is restricted to 2 - D flow
 
 

 v  ui  vj  wk  
Note : The velocity potential
- - a consequenc e of the irrotation ally of the flow field .
The stream function
- - a consequenc e of conservati on of mass.
31
 
For an incompress ible fluid (  v  0)
 
and irrotation al flow (v   ) , it follows that
2
 2   2   2
   0 or
 2  2  0      (6.66)
2
x
y
z
Laplace equation
Inviscid , incompress ible ,
Irrotation al flow field .
This type of flow is commonly called a potential flow.
If  is known
from Eq.(6.66) with
boundary conditions
u

 v or v
w
can be determined
with Bernoulli equation

To calculate pressures

32
In Cylindrica l Coordinate s, r ,  , z

 (.)  1  (.)   (.) 
e z      (6.67)
e 
er 
(.) 
z
r 
r

  1    
e z      (6.68)
e 
er 
  
z
r 
r
where    (r ,  , z )




Since v r  v r er  v e  v z e z      (6.69)

1 

     (6.70)
; vz 
; v 
z
r 
r
2
1  2  2
1  
 2  0      (6.71)
(r )  2

2
r r r
z
r 
vr 
33
Example6.4
st ream funct ion
Given :   2 r 2 sin 2
m2 / s
& r m
Figure on right
Det ermine :
(a) velocit y pot ent ial
(b) pressure at point (2)
if P1  30kpa,   103 kg / m3, z1  z2
Solution:
( a ) stream function in cylindrical conditions
conservation of m ass


1  
1 
1
vr 
2r 2 sin 2  2r 2 (cos2 )  2  4r cos2

r  
r 
r
 


v  
v  
2r 2 sin 2  4r sin 2

r 
r
Irrotational flow
vr 


Velocity potential in cylindrical conditons
vr 

1 

; v 
; vz 
r
r 
z
   vr dr
 
 v dr   4r cos2dr  C ( )  2r cos2  C ( )    (1)
r
1
2
1
34
From Eq(1)
  2r 2 cos 2  C1 ( )      (1)
1 
Since v 
r 
1 
[2r 2 cos 2  C1 ( )]
r 

2
2
 4r sin 2  2r ( sin 2 )2 
C1 ( )



C1 ( )  0 or C1 ( )  C  const.      (2)

Eq (1) & (2)
  2r 2 cos 2  C      ( Ans)
 4r sin 2 
35
(b) For an irrotation al flow of a nonviscous , incompress ible fluid,
the Bernoulli equation can be written as
p
p1 1 2
1
( z1  z 2 )
 v1  gz1  2  v 22  gz 2
 2
 2
1
1
 p 2  p1  v12  v 22      (3)
2
2


sin ce v  v r er  v e  v 2  v r2  v2
 v 2  (4r cos 2 ) 2  (4r sin 2 ) 2  16r 2
At point (1), r  1m
v12  16r 2  16  v1  4m / s
At point (2), r  0.5m
v 22  16r 2  16  0.5 2  4  v 2  2m / s
From Eq (3)
p 2  30  10 3 pa 
1
 10 3 kg / m 3  (16  4)m 2 / s 2  36kpa      ( Ans)
2
36
§ 6.5 Some Basic, Plane Potential Flows
u v

y x
(Irrotatio nal flow) Using Eq (6.74)
From Eq (6.72) 
 


 2  2
(
)  (
) 2  2 0
y y
x
x
x
y
From Eq (6.66) for incompress ible , Irrotation al flow
u v
 
 

0 ( ) ( ) 0
x y
x x
y y
 2  2  2 
 2   2
x  y plane flow
 2  2  2  0     2  2  0
x
y
z
x
y
37
§ 6.8 Viscous Flow
To incorporat e viscous effects into the differenti al analysis of fluid motion,
we must return to the previously derived general equations of motion, Eq.6.50,
such as .

 xx  yx  zx
u
u
u
u

g





(

u

v

w
)      (6.50a )
 x
x
y
z
t
x
y
z

 xy  yy  zy
v
v
v
v


g





(

u

v

w
)      (6.50b)

y
x
y
z
t
x
y
z

 xz  yz  zz

w
w
w
w
 g z  x  y  z   ( t  u x  v y  w z )      (6.50c)

General differenti al equation of motion for a fluid .
There are more unknowns than equations.
It is necessary to establish a relationsh ip between t he stresses & velociti es.
38
§ 6.8.1 Stress - Deformation Relationships
For incompress ible , Newtonian fluids , it is known tha t the stresses
are linearly related to the rate of deformatio n.
u
u
 xx   P  2
     (6.125a )
x
x
v
v
 yy  P  2
  yy   P  2 
     (6.125b)
y
y
w
w
 zz  P  2
 zz   P  2
     (6.125c)
z
z
u v
 xy   yz   (  )      (6.125d )
y x
v w
 yz   zy   (  )      (6.125e)
z y
u w
 zx   xz   (  )      (6.125 f )
z x
 xx  P  2
39
In cylindrica l polar coordinate s
The stresses for Newtonian, incompress ible fluids
V r
         (6.126a )
r
1 V V r
  P  2 (
 )      (6.126b)
r 
r
V z
  P  2
         (6.126c)
z
 rr   P  2 
 
 zz
 V
1 V r
( )
]    (6.126d )
r r
r 
V 1 V z
 [

]      (6.126e)
z
r 
V r V z
 [

]        (6.126 f )
z
r
 r   r  [r
 z   z
 zr   rz
40
§ 6.8.2 The Navier–Stokes Equations
From Eq.(6.50a) ~ (6.50c) with (6.125a) ~ (6.125f)
and Eq. of continuity , Eq (6.31),
x - direction
P
 2u  2u  2u
u
u
u
u
g x 
 ( 2  2  2 )  (  u
v
 w )      (6.127 a)
x
t
x
y
z
x
y
z
y - direction
P
 2v  2v  2v
v
v
v
v
g y 
 ( 2  2  2 )  (  u
v
 w )      (6.127b)
y
t
x
y
z
x
y
z
z - direction
P
2w 2w 2w
w
w
w
w
g z 
 ( 2  2  2 )  (
u
v
 w )      (6.127c)
z
t
x
y
z
x
y
z
41