Chapter 9 Viscous flow

26 CHAPTER 9. VISCOUS FLOW
At this point, the radial pressure gradient is not yet specified, and we have
not used the mass conservation equation so far. So the mass conservation
equation can be used to obtain the radial pressure gradient. It is convenient
to integrate the mass conservation equation over the width of the channel to
get
∫ h(r ∗ )
0
dz ∗ 1 r ∗ ∂
∂r ∗r∗∂u∗ r
This can be simplified to provide
1
r ∗
1
∂
r ∗ ∂r ∗r∗
∂
∂r ∗r∗
∂ (
∂r ∗
∫ h(r ∗
∂r + )
∗ 0
∂ ∫ h(r ∗ )
∂r ∗ 0
− h(r∗ ) 3
12
dz ∗∂u∗ z
∂z ∗ = 0 (9.103)
dz ∗ u ∗ r − 1 = 0
∂p ∗ )
− 1 = 0 (9.104)
∂r ∗
This equation can be integrated to provide the radial pressure gradient,
∂p ∗
∂r ∗ = − 6r∗
h(r ∗ ) 3 − C 1
r ∗ h(r ∗ ) 3 (9.105)
The condition that the pressure is finite throughout the channel requires
that the constant C 1 is zero. A second integration with respect to the radial
coordinate provides the pressure.
p ∗ =
3
(1 + (r ∗2 /2)) 2 + C 2 (9.106)
The pressure C 2 is obtained from the condition that the dimensional pressure
scales as (µU/R) in the limit r ∗ → 0, and therefore the scaled pressure is
proportional to ǫ 3 in this limit.
p ∗ =
3
(1 + (r ∗2 /2)) 2 (9.107)
The scaled force in the z ∗ direction is obtained by integrating the axial
component of the product of the stress tensor and the unit normal,
∫
F z = 2π
r dr(T zz n z + T zr n r ) (9.108)