Physics 326 Homework #3
due in course homework box by Fri 1 pm
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written so
Physics 326 Homework #4
due Friday, 1 pm HARD DEADLINE
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions
Currently the ODE and boundary conditions are in dimensional form. To make the solution to the ODE
general, lets re-write it in terms of dimensionless parameters:
f y
2A
F where y
f
f
d 2 f d
d
8.
Very Low-Reynolds Number (aka Creeping, aka Stokes) Flows (Re < 1)
Consider the governing equations for an incompressible fluid, neglecting body forces, in dimensional form:
u 0
(7.29)
u
u u p 2
2.
Note that for this flow the non-linear convective terms in the Navier-Stokes equations, (u)u, did not
drop out as they have in the previous exact solutions.
3.
In flows around objects with surface
Two additional useful relations can be found if we take the curl of both sides of the momentum equation:
p 2 u
2 0
and if we take the divergence of both sides of the momentum equation (and using co
6.
Note that if u is a Stokes flow solution, then u = -u is also a solution since:
u 0 and p 2u p p
In addition, 2 0 and 2 p 0 where u u .
Hence, Stokes flows are kinematically reversible and flow ar
9.
Stokes Flow Around a Sphere
Now lets examine the creeping flow around a sphere of radius, R, in a uniform stream of velocity, U. For
axi-symmetric creeping flows it is convenient to use a stream fu
2
d2
2 n
n4
2 2 r n 2 n 3 2 n n 1 2 r 0
r
dr
n 1,1, 2,3
A
f (r ) Br Cr 2 Dr 3
r
The corresponding stream function and velocities are:
A
(r , ) Br Cr 2 Dr 3 sin 2
r
A B
ur 2 3 C Dr cos
r
r
A
Now lets examine the Navier-Stokes equation in the x-direction:
2u 2u
u
u
u
p
x u x x u y x 2x 2x f x
x
y
x
y
t
x
After simplifying:
2
u x
u x
p
u x
ux 2ux
ux
uy
fx
x x 2 y 2
t
x
0
6.
We can also use the solution approach presented here to determine the drag on a spherical droplet of a
fluid (with dynamic viscosity i) in a different fluid (of dynamic viscosity o). The general di
The drag force acting on the sphere surface (r=R) is found by integrating the pressure and viscous forces in
the horizontal direction over the entire spheres surface:
r r R sin
R
dr R cos d
F r r R s
10. Lubrication Flow
One very important application of creeping flows is in the study of lubrication problems. Lets consider the
example of a simple, stationary, planar slipper pad bearing as shown in
Review Questions
1. Describe several common assumptions used to simplify the Navier-Stokes equations.
2. Describe several common boundary conditions used when solving the Navier-Stokes equations.
3. A
Thus, this flow could be considered a creeping flow.
Using the simplifications just discussed, the Navier-Stokes equation in the x-direction reduces to:
p
2u
2
x
y
Note that we expect the magnitude
UL
1.6 MPa 160 atm!
h02
A more accurate analysis of the flow would also include variations in the fluid viscosity due to the
large pressure variations.
2.
Note that a truly symmetric bearing and flo
We know that since were using a stream function the continuity equation is automatically satisfied. To
make sure we satisfy the momentum equations we substitute the velocity components into the Navier
6.
Oscillating Flat Plate (aka Stokes Second Problem, aka the Rayleigh Problem)
Consider the incompressible, constant viscosity, Newtonian fluid flow resulting from the sinusoidal
oscillation of an in
3.
Poiseuille Flow
Consider the steady flow of an incompressible, constant viscosity, Newtonian fluid within an infinitely
long, circular pipe of radius, R.
r
R
z
Well make the following assumptions:
Now lets apply boundary conditions to determine the unknown constants c1 and c2. First, note that the
fluid velocity in a pipe must remain finite as r0 so that the constant c1 must be zero (this is a
4.
Laminar flow in an elliptical cross-section pipe can be determined by considering the simplified
Navier-Stokes equation in the z-direction but using Cartesian coordinates (assuming ux=uy=0):
2 u z
4.
Starting Flow Between Two Parallel Plates
Consider a flow starting from rest between two parallel flat plates. The bottom plate is fixed while the top
plate moves impulsively at t > 0 with constant
Note that as t , the flow profile should approach the Couette flow profile derived previously, i.e.:
y
u x y, t U
(7.7)
h
Hence, lets investigate a solution of the form:
y
ux ux U
(7.8)
h
Substituting
Since Eqn. (7.9) and the boundary and initial conditions (7.10) - (7.12) are linear, we can add the together
the solutions in Eqn. (7.20) so that they satisfy the given initial condition (Eqn. (7.10).
6.
Starting Flow in a Circular Pipe
Consider the unsteady flow of an incompressible, constant viscosity, Newtonian fluid within an infinitely
long, circular pipe of radius, R.
r
R
z
Well make the foll
We know that as t the flow should approach the Poiseuille flow solution found in the previous section,
i.e.:
uz r , t
R 2 dp
r2
1 2
4 dz R
Hence, lets investigate a solution of the following for
0 for t 0
u * y 0, t
1 for t 0
*
u y , t remains finite
Note that since the velocity is dimensionless, it must depend only on dimensionless quantities. The only
dimensional quantities in the PDE are
3.
If we hold both boundaries stationary and move the fluid using only a pressure gradient (note that flow
in the positive x-direction occurs for dp/dx < 0), the velocity profile becomes:
h 2 p y
y
u
5.
Impulsively Started Flat Plate (aka Stokes First Problem, aka the Rayleigh Problem)
Now lets consider the incompressible, constant viscosity, Newtonian fluid flow resulting from the sudden
movement