HOMEWORK 6 SOLUTION MANUAL
Marat K
Problem 1 (5 points):
The Bernoulli equation is:
Eq1
Since the velocity at the stagnation point is zero and maximum pressure is found at locations
with minimum velocity we should rewrite the Bernoulli equation as:
Eq2
Problem 2 (5 points):
Problem 2 Schematic
Start with the Bernoulli equation (Eq1) again and rewrite it as:
Eq3
And realize that in an incompressible flow
= Q
Eq4
Combining Eq3 with Eq4 will yield Eq5
Eq5
Solving for Q in Eq5 will yield the following:
Eq6
Eq6 is a little messy so keep track of all the units and make sure they simplify to units of
volumetric flow rate. This is the easiest way to verify that your final solution makes sense. An
alternative way to obtain Q is presented below. Combine Eq3 and 4 to yield Eq7:
Eq7
Then realize that:
Eq8
Problem 3 (10 points):
Please read pages 227 to 237 in the textbook since they discuss this problem in detail. Also, refer
to the diagram in Figure 3.22 for this problem. You can solve this problem using potential flow
or stream function but we will use the latter in this derivation. First derive the stream function for
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View Full Documentthe entire flow by summing the stream function for uniform flow and a source. If do that
correctly you should end up with Eqn 3.75, which is Eq10 here.
Eq9
Eq10
Remember that stream lines are always parallel to the velocity vectors so fluid particles can
never pass through the stream lines. Because of this we can think of a region that is completely
enclosed by a single stream function as a solid body. To find the enclosing stream function you
should first compute velocities from the global stream function as:
Eq11
Eq12
At the stagnation points Vr and Vθ are zero. Solving Eq11 and Eq12 simultaneously will yield:
Eq13
Eq14
Eq 13 is the answer for part c. Next, plug Eq13 and 14 into Eq 10 to find the stream function that
passes through the stagnation points:
Eq15
Since the stream function that passes through the stagnation points is also the stream function
that lies on the body you can use Eq10 and 15 to find r as a function of θ on the body .
Eq16
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 Fall '09
 Fluid Dynamics, Velocity, stream function

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