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Tentamen Str omningsmekanik a.k. 5C1203
Final exam in course 5C1203 1/6 2001 9-13 in M36Examiner: Prof.
Dan HenningsonThe point value of each question is given in
parenthesis and you need more than 20 points topass the course
including the points obtained from the homework problems.Copies of
pages 348-355 in Acheson can be used for the exam as well as a book
of basic mathformulas and a calculator.
1. a)(5) Derive the complex potential of a doublet by taking the
limit of the potentialobtained when a source and sink move
together. Assume the strength of the source
and of the sink is equal and that the strength increases
inversely proportional to thedistance between them. Recall that the
complex potential of a source is
F (z ) = m/ 2 ln z.
b)(5) Show that the streamlines of a doublet are circles through
the origin.
2. (5) Consider the unsteady ow,
u = u0 , v = kt, w = 0,
where u0 and k are positive constants. Show that the streamlines
are straight lines,and sketch them at two different times. Also
show that any uid particle follows aparabolic path as time
proceeds.
3. (10) Incompressible ow of a viscous uid between two porous,
innite, parallel sur-faces is driven by a pressure gradient p = F
ex , where F is constant and ex isparallel to the surfaces. At the
same time uid ows through the porous surfaces atthe constant
velocity V out from one surface and in to the other. The surfaces
arelocated at y = 0 and y = h. Compute the velocity eld.
Investigate what happens if is small or really large, and when V
becomes very small.
V
V
p = F e x
y
y = h
x
z
The exam continues on the next page.
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4. (10) A Couette viscosity meter consists of two long coaxial
cylinders with lengthh. The region between these cylinder is lled
with the uid that viscosity is to be
measured for. As the cylinders rotate relative to each other
with known angularvelocities the viscosity can be determined by
measuring the torque on one of thecylinders. Derive the relation
between torque and and angular velocities for stationarylaminar
ow.
1
2
R 1
R 2
5. (5) For surface waves on water with nite depth h the phase
speed is given by therelation c2 = gk tanh( kh). Compare the group
velocity to the phase speed in the limitof long and short waves.
Make a rough 2D sketch of a wave train resulting from an
impulse perturbation and explain what is happening.6. A uid is
owing in a 2D jet out from a slit in a wall and into a large space
lled with
the same uid that is practically motionless. Let x be the
coordinate in the directionof the jet and y parallel to the
wall.a)(2)Assume that the ow is governed by the boundary layer
equations and motivatethe usage of the equation,
uux
+ vuy
= 2 uy 2
()
b)(3) Show by integrating ( ) that the momentum ux, M , in the
direction of the jetis constant in x. Hint: Integrate by parts and
use incompressibility.
M =
u2 dy ()
c)(5) Assume that the stream-function is ( x, ) = um (x)g(x)f ()
where the similar-ity variable is = y/g (x). Show, using (), that
the maximum velocity of the jet atx is um (x) = C/ g(x), where C is
a given constant. Show that the thickness of the jet grows
proportional to x2 / 3 , and that f () satises,
f + k
2f f + kf
2= 0 ,
where k is a constant.
Good Luck!
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Solutions to exam in Str omningsmekanik a.k. 5C1203,
2001-06-01
1. a)lim
ln(z + ) ln(z )2
= ddz
ln(z ) = 1z
b)
= yx2 + y2
= k x2 + y 12k
2
= 14k2
2. See Acheson problem 1.8.
3.
u =F h
V
y
h
1 eV y/
1
eV h/ , V, 0
4.u =
1R 22 R 21
(2 R 22 1 R21 )r (2 1 )
R 22 R 21r
M = h 2
0 r r
2 d = 4h (2 1 )R 22 R 21
R 22 R 21
5. Long waves k 0 gives c2g = gh. Short waves k gives c2g = g/
4k. Longer waves
travel faster.
0 0.5 1 1.5 2 2.5 32
1.5
1
0.5
0
0.5
1
1.5
2
6. See Acheson problem 8.4.
