Use the complex method for obtaining the quasi steady

 Use the complex method for obtaining the quasi-steady solution.

u
)

Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei ( 李雨 )
Quasi-steady internal flow (3)
,
cosh (1 )
2
iK
A
i a
ω
ω
ν
=
 
+
 
 
Apply B.C.’s 
0 = B
Then




















+






+
− =
a i
y i
K
i y w
ν
ω
ν
ω
ω
2
) 1 ( cosh
2
) 1 ( cosh
1 ) (
( )
1
cosh
2
( , ) Re ( ) Re 1
1
cosh
2
i t i t
i y
N
iK
a
u y t w y e e
i
N
ω ω
ω
 
+  
 
 
 
 
 
 
= =  − 
 
+
 
 
 
 
 
 
 
 
 
and
ν ω / 2 a N ≡
with
a dimensionless frequency
↓

Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei ( 李雨 )
Quasi-steady internal flow (4)
For small N ,
......
2
1
1 ......
2
1
2
1
1
2
1
cosh
2
2
2
+






+ ≈ +





 +
+ ≈
+
a
y
iN
a
y
N
i
a
y
N
i
( )








− − ≈
2
2 2
1 cos
2
1
) , (
a
y
t K
a
t y u ω
ν
Thus
 the quasi velocity has a parabolic profile.
 same as the steady solution if the steady pressure gradient
is replaced by the periodic pressure gradient under this low
frequency limit (i.e., for N << 1).

Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei ( 李雨 )
Quasi-steady internal flow (5)
For large N ,
a
y
N
i
a
y
N
i
a
y
N
i
e e e
a
y
N
i
2
1
2
1
2
1
2
1
2
1
2
1
cosh
+ +
−
+
≈








+ =






+
1
1
2
( , ) Re 1
i y
N
i t
a
iK
u y t e e
ω
ω
+
 
−
 
 
 
 
≈ −
 
 
 
 
Thus
sin
( , ) Re
i t
iK K t
u y t e
ω
ω
ω ω
−
 
≈ =
 
 
For region not close to the wall (called the core region ),
it further reduces to (note that y / a < 1)
∞ → N
as
which is also the solution with the viscous term set to zero.
Thus the flow in the core region is essentially inviscid !

Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei ( 李雨 )
Quasi-steady internal flow (6)
For region next to the wall (called the boundary layer region ),
we may re-scale the problem by setting
,
a y
N
a
η
−
=
1
2 2
sin 1
( , ) Re 1 sin
2
i
i t
iK K t K
u y t e e e t
η
η
ω
ω
ω η
ω ω ω
+
− −
 
 
 
= − ≈ − + −
 
   
 
   
) 1 ( O = η
such that within the boundary layer.
We then have
↑
same as the core flow
↑
viscous effect

Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei ( 李雨 )
Other examples :
 the pulsating flow in a circular straight pipe driven
by an imposed periodic pressure gradient,
 the oscillating plane Couette flow driven by the
harmonic oscillation of the upper and/or lower
plate(s),
 the pulsating flow in a circular straight pipe driven
by the axial oscillation of the pipe wall.
Quasi-steady internal flow (7)

Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei ( 李雨 )
Flow with circular streamlines (1)
 We have found that the nonlinear convection term
in the Navier-Stokes equation is identically zero for
cases with parallel streamlines.