region does not grow with x because the approaching flow confines the
vorticity generated associated with the noslip condition next to the wall.
Similarly, we can solve the axial symmetric problem numerically, and
find
109
75
.
2
/
=
ν
δ
B
for the axial symmetric stagnation flow.
(386b)
The shear stress at the wall is calculated as
ν
µ
ν
µ
µ
τ
B
F
n
Bx
B
F
n
Bx
x
v
y
u
y
w
)
0
(
'
'
1
0
)
0
(
'
'
1
0
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
=
=
,
(387)
which increases linearly with x. There is no contribution from the term
0

/
=
∂
∂
y
y
v
µ
. For the planar case, we have
2
=
n
, and
23
.
1
)
0
(
'
'
=
F
according to the numerical result. The shear stress at wall is usually
written in dimensionless form as
R
1
)
0
(
'
'
2
2
1
2
−
=
≡
n
F
u
C
I
w
f
ρ
τ
,
(387a)
where
f
C
is called the skin friction coefficient, and
ν
/
x
u
I
=
R
is the
Reynolds number. Finally, with the velocity field obtained, we can
evaluate the pressure field by integrating (381a).
(iii) Other nonlinear exact solutions
There are also some other nonlinear problems having exact solutions.
For examples, the flow induced by a rotating disk (the “von Karman’s
viscous pump”, Problem 4 of the Homework), the steady jet from a point
source of momentum (called the Squire or Landau’s jet, see Batchelor’s
book), and some other interesting problem in Yih’s book (pp.332334).
(IV) Concluding Remarks
For steady incompressible flow with constant viscosity, the Reynolds
number is the only dimensionless parameter governing the flow.
Mathematically, exact solution is valid for all Reynolds number.
Unfortunately, this is not true! As claimed by L.D. Landau and E. M.
Lifshitz (1959), “Yet not every solution of the equation of motion, even if
it is exact, can actually occur in nature. The flows that occur in nature
must not only obey the equations of fluid dynamics, but also be stable.”
For example, we may observe parabolic velocity profile in a circular pipe
under the steady, fully developed condition only when the Reynolds
number (based on the mean velocity and pipe diameter) is less than a
110
critical value, say, less than 2100. When the Reynolds number exceeds
the critical value, the flow becomes unstable, and we observe turbulent
flow when the Reynolds number is sufficiently large. The shape of the
mean turbulent profile is more flat (may be approximated as a
1/7power profile) in comparing with the parabolic profile. The book by
Drazin & Reid is a good introduction to the study of the stability of fluid
motion (P. G. Drazin and W. H. Reid, “Hydrodynamic stability,”
Cambridge University Press, 1981).
111
Figure 315: The Fortran program for solving the planar stagnation flow
using shooting method (with fourth order RungeKutta
method).
112
Figure 316: Numerical results for the planar stagnation flow.
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 Fluid Dynamics, Fluid Mechanics, mechanics, dt