For the Poisseuille flow of the previous question, derive the
principal strain rates and the principal strain directions.
If you put a cup of coffee at the center of a
rotating turn table and wait, eventually, the coffee will be
executing a “solid body rotation” in which the
velocity field is, in cylindrical coordinates:
where is the angular velocity of the turn table. Draw
samples of the streamlines of this flow. Find the vorticity and the
strain rate tensor for this flow, using the expressions in
appendices B and C. Show that indeed the coffee moves as a solid
body; i.e. the fluid particles do not deform, and that for a solid
body motion like this, indeed the vorticity is twice the angular
velocity.
An “ideal vortex flow” is described
in cylindrical coordinates by
where is some constant. Draw samples of the streamlines of this
flow. In cylindrical coordinates, nabla is given by
Evaluate
for this flow. Compare the answer with the vorticity
, for which you can find the correct
expressions in appendix B. Explain why the determinant does not
give the correct result for the vorticity.
The “circulation “ along a
closed contour is defined as
For both the solid body rotation of question 2, and the
vortex flow of question 3, find the circulation along the
unit circle in the -plane. Next, only for the vortex
flow, find the circulation along the closed curve consisting of
the following segments:
The part of the curve , from
to ;
vertically downward to the curve
,
at ;
following the curve
, from
to the point , hence , ;
in a straight line along the direction to the point ,
, ;
in a straight line from , , to
the starting point , , .
Note that in cylindrical coordinates
According to the Stokes theorem of Calculus III, you should have
where the second integral is over the inside of the contour. So
instead of integrating the circulation as you did in
question 4, you could have integrated the component of
vorticity normal to the circle over the inside of the circle. Show
that if you do that integral using the vorticity that you found for
solid body rotation in question 2, you do indeed get the
same answer as you got in question 4. Fine. But now show
that if you do the integral of the vorticity over the inside of the
circle for the vortex flow of question 3, you do not
get the same answer for the circulation as in question 4.
Explain which value is correct. And why the other value is wrong.
Following Friday‘s lecture, you would of course love to do
some integrals of the form
and
. Here is your chance. Do
question 5.1(b) and 5.1(d) and explain their physical meaning. Take
the surfaces , , , and to be one
unit length in the -direction. (To figure out the correct
direction of the normal vector at a given surface point,
note that the control volume in this case is the right half of the
region in between two cylinders of radii and and of unit
length in the -direction. The vector is a unit normal
vector sticking out of this control volume.)