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 or incorrect reasoning. Answers given without explanati
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 solutions or incorrect reasoning. Answers given without e
2.
Planar Couette-Poiseuille Flow
Consider the steady flow of an incompressible, constant viscosity Newtonian fluid between two infinitely
long, parallel plates separated by a distance, h.
g
h
y
fluid
x
Well make the following assumptions:
1. The flow is
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 function in spherical polar coordinates, (r, ,
), to desc
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 around symmetric objects will produce
symmetric streamlin
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 continuity):
p 2 u
(7.35)
2 p 0
Both the vorticity and
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 curvature (e.g. a torpedo-shaped object), this solution
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 u
t
Recall that when the Reynolds number is very small
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 2 dy
2
2A
2A
F
F
2A
df
df d
dy d dy
2A
2A
F
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 B
u 3 2C 3Dr sin
r
r
Applying the boundary conditions
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 differential
equation is the same but the boundary condit
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 sin dA p r R cos dA where dA 2 R 2 sin d
dr
dA 2 r
2
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 the figure below:
L
stationary slipper pad bearing
p
f
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. At what (rule of thumb) Reynolds number does transition
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 of the pressure gradient in the x-direction to be of th
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 flow would result in zero lift on the bearing since the pr
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 NavierStokes equations (simplified using our assumptions):
ux
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 infinitely long flat plate. The geometry of the problem i
7.
(Planar) Stagnation Point Flow (aka Hiemenz Flow)
Consider the flow in the vicinity of a stagnation point:
y
x
Well make the following assumptions in the analysis of this flow:
0 and uz constant
z
u y
u
x
0
t
t
fx f y fz 0
1.
The flow is planar.
2.
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 y
0 (#4)
(#5)
0 (#3)
0 (#2)
0 (#3)
2
u x 1 p
x
y
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:
1.
The flow is axi-symmetric and there is no swirl velo
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 type of
kinematic boundary condition). Also, the pipe w
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 2 u z 1 dp
(Poissons equation!)
2
x 2
y
dz
where 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 velocity, U. There are no pressure gradients in the fl
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 back into Eqn. (7.3) and the boundary and initial cond
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). Note that we can
add together the constants for the ne
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 following assumptions:
1.
The flow is axi-symmetric and the
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 form:
R 2 dp
r2
uz uz
1 2
4 dz R
THE REMAINDER OF TH
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 t, y, and . We can form only one dimensionless variable
Physics 326: Fall 2016
Lecture 11
1
Differential equations, phase portraits, and chaos.
Most of our interest in dynamics is not for discrete maps like the logistic or tent or Henon, with
discrete generation index j, but for differential equations with a c
Physics 326: Fall 2016
Lecture 18
1
Hamiltons Canonical equations of motion
Well now move onto the next level in the formalism of classical mechanics, due initially to Hamilton
around 1830. While we wont use Hamiltons approach to solve any further it is u
Physics 326: Fall 2016
Lecture 15
1
Torque free Euler equations
Recall from last lecture, we were discussing the torque-free Euler equations, which describe the free
rotation of a body in space. In body axes cfw_e1 , e2 , e3 chosen so that I is diagonal,
Physics 326: Fall 2016
Lecture 20
1
Continuum Mechanics
Today we begin continuum mechanics, specifically waves on a string.
Consider a string of tension T (T is possibly a function of x) and mass per unit length (x),
with transverse (i.e sideways) displac
Physics 326: Fall 2016
Lecture 13
1
Planar rigid body motion
1.1
Planar rigid body kinetics, Moments of inertia
The bodys motion is to be characterized by the center of mass motion and the angular velocity .
The velocity of any other point on the body can