Homework 6 Solution Manual

# Homework 6 Solution Manual - HOMEWORK 6 SOLUTION MANUAL...

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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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the 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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Homework 6 Solution Manual - HOMEWORK 6 SOLUTION MANUAL...

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