ME 309 Fall 2008
Section 2 (Merkle)
Homework # 21
Due Mon 20 Oct 2008
Problem 211
.
The text develops the velocity profile for the flow between parallel plates
from an analysis of a control volume.
Repeat this analysis by starting from the Navier
Stokes equations.
Specifically, given that the flow is fully developed in the
x
direction, is
planar with no variations and zero velocity in
z
, is incompressible without body forces
and is at steady state, find the velocity profile between the plates.
The continuity
equation is given in Eq. 5.20 and the NavierStokes equations in Eq. 5.27.
You may use
only the
x
and
y
components of the NavierStokes and ignore the
z
component.
Solution:
Given:
NavierStokes equations plus the continuity equation;
Continuity
(conservation of mass):
0
=
∂
∂
+
∂
∂
+
∂
∂
z
w
y
v
x
u
NavierStokes
x
direction
∂
∂
+
∂
∂
+
∂
∂
+
=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
2
2
2
2
2
2
z
u
y
u
x
u
g
x
p
z
u
w
y
u
v
x
u
u
t
u
x
μ
ρ
ρ
ρ
ρ
ρ
y
direction
∂
∂
+
∂
∂
+
∂
∂
+
=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
2
2
2
2
2
2
z
v
y
v
x
v
g
y
p
z
v
w
y
v
v
x
v
u
t
v
y
μ
ρ
ρ
ρ
ρ
ρ
Assumptions:
incompressible, no body forces, twodimensional, fully developed
Analysis:
Starting from the continuity equation, the first term is dropped because of the fully
developed assumption; the third term is dropped because of the twodimensional
assumption. This leaves the result:
0
=
∂
∂
y
v
, or, upon integrating,
v
= const.
Evaluating this ‘velocity profile’ for the
v
component by evaluating the constant of
integration at the wall where
v
= 0, we obtain:
v
= 0
Apply to the
y
component of the momentum equation:
∂
∂
+
∂
∂
+
∂
∂
+
=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
2
2
2
2
2
2
z
v
y
v
x
v
g
y
p
z
v
w
y
v
v
x
v
u
t
v
y
μ
ρ
ρ
ρ
ρ
ρ
The first term is zero because of steady state condition
The second term is zero because of fully developed flow and because
v
= 0
The third term is zero because
v
= 0
The fourth term is zero because
v
= 0 and because there is no variation in
z
The first term on the RHS is zero because there is no gravity. The viscous terms (in
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 Spring '08
 MERKLE
 Fluid Dynamics, Thermodynamics, Force, ∂x, Navier–Stokes equations

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