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4. (10) A Couette viscosimeter consists of two long co-axial
cylinders. The region betweenthe two cylinders is filled with the
fluid whose viscosity one wishes to determine. Whenthe cylinders
rotate relative to each other with known angular velocity, the
fluid viscositycan be determined by measuring the torque on one of
the two cylinders.
Derive the relation between the torque, the angular velocity and
the radius of the cylinders
assuming steady laminar flow. (A detailed analysis of both the
velocity and pressure fieldis required)
5. Stagnation point flow.
x, u
v, y
= 0
Figure 1: Stagnation point flow with a viscous layer close to
the wall.
The stagnation point flow is depicted in figure 1, and is a flow
where a uniform velocityapproaches a plate where it is slowed down
and turned to a flow parallel two the plates inboth the positive
and negative x-direction. At the origin we have a stagnation
point.
The inviscid stagnation point flow is given as follows
inviscid:
u= cxv = cy
z = = v
x
u
y = 0
Note that this flow has zero vorticity and that it does not
satisfy the no-slip condition(u = v = 0, y = 0). In order to
satisfy this condition we need to introduce viscosity, orequally,
vorticity.
a) (2) Write the equation for the two-dimensional vorticity in
the boundary layer close tothe wall.
b) (5) Assume u= xf(y)v= f(y)
Verify that this assumption satisfies the continuity equation
and derive an equation for f.Show that this form of the equation
for the vorticity can be integrated once yielding
f2 ff f =c2
Formulate the boundary conditions for this equation.
c) (5) Make the equation non dimensional with respect toand the
constant c introducedabove. Give the length and velocity scale for
the problem and write the nondimensional
form of the equation together with boundary conditions.
Good Luck!
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Solutions to exam in Fluid mechanics 5C1214, 2006-01-12
1. Potential flow and conformal mapping
a) Equation for the circlez=aei
Equation for the plate
=z+a2
z =aei +aei = 2a
ei +ei
2 = 2a cos ,
which is a real function taking the values from 2ato 2a, thus a
flat plate with length 4a.
b) Complex potential for a circular cylinder
F =Uzei +Ua2
zei +
i
2ln z
c) Complex velocity for the circular cylinder
W =dF
dz =Uei U
a2
z2ei +
i
2z
Complex velocity in the -plane
=dF/dz
d/dz =
Uei U
a2
z2ei +
i
2z
1
a2
z2
The flow field has singular points at the leading and trailing
edges of the flat plate. Resolvethe singularity at the trailing
edge by choosing the circulation so the numerator vanishes
at z=aUei Uei +
i
2a= 0
k = 4Uaei ei
2i = 4Ua sin .
d) Move the point z= a inside of the circle by putting the
center at z= . We stillwant a sharp trailing edge so the circle
crosses the real axis at z=a. The radius of sucha circle is a+ and
the circle is described by
z= + (a+)ei
2. See Kundu & Cohen p. 57.
3. See recitation number 3 and 12.
4. See Tritton page 108. M= 4r21r2221r22r2
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5. See lecture notes, example 3 on page 36.
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