ENU 4134 Homework SOLUTIONS #0, Fall 2011 – D. Schubring
1
ENU 4134 – Homework SOLUTIONS #0 – Fall 2011
Task 1. The NavierStokes Equation
ρ
∂~v
∂t
+
~v
·
(
∇
~v
)
=
∇
p
+
μ
∇
2
~v
+
~
f
(1)
The terms on the left side represent unsteady acceleration (explicit time dependence) and con
vective acceleration (change in velocity with position due to acceleration of fluid particles). The
terms on the left are the pressure gradient, viscous effects, and the body force. The version of the
equation above uses
~
f
, a body force per unit volume. Your book prefers
ρ
~
f
, an accleration or body
force per unit mass.
The latter is more convenient when the body force is gravity, which is the
usual case.
Task 2. Tube Flow
a. The appropriate equation is the zdirection NS equation in cylindrical coordinates with
~
f
=
0.
ρ
∂v
z
∂t
+
ρv
r
∂v
z
∂r
+
ρv
θ
r
∂v
z
∂θ
+
v
z
∂v
z
∂z
=

∂p
∂z
+
μ
"
1
r
∂
∂r
r
∂v
z
∂r
+
1
r
2
∂
2
v
z
∂θ
2
+
∂
2
v
z
∂z
2
#
(2)
All LHS terms are zero (due to, in order, steadystate assumption,
v
r
= 0,
v
θ
= 0, and
v
z
is
not a function of
z
). On the RHS, the final two terms are eliminated since
v
z
is not a function
of
θ
or
z
. The equation simplifies (greatly) to:
dp
dz
=
μ
r
d
dr
r
dv
z
dr
(3)
This is a second order ODE, which requires two BC’s. The first is that
dv
z
/dr
= 0 (or
v
z
(
r
)
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 Fall '11
 Schubring
 Fluid Dynamics, Thermodynamics, Heat Transfer

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