SampleMidterm1 - P k u ∇ − = µ ~(where k is a constant...

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ChE 6200 Fall 2007 Sample Midterm Exam Duration: 1.5 hours 1. (5 points) Prove that ~ ~ ~ a f a f a f + = , where f is a scalar. 2. (10 points) Calculate ~ r , where ~ r is the position vector. 3. (10 points) Calculate the velocity gradient tensor ~ u for the flow field ~ ~ 2 ~ ~ ) 3 sin( k xz j e z i xy u at + + = , where a is a constant. 4. (5 points) Which of the following statements is true? - Stagnant simple fluids are not subject to shear stresses. - The continuum approximation cannot be used for compressible flow. - The no-slip boundary condition does not apply for non-Newtonian fluids. 5. (10 points) If continuity applies in a material volume, i.e. 0 ~ = + u Dt D ρ (where is the density and ~ u is the fluid velocity), show that Dt DV V u 1 ~ = , where V is volume. 7. (10 points) A relationship often used to model fluid flow in porous media is Darcy’s law:
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Unformatted text preview: P k u ∇ − = µ ~ (where k is a constant, ~ u is the fluid velocity and P is pressure). a ) Use Darcy’s law to obtain expressions for conservation of mass (continuity) for the porous flow. b ) Show that for an incompressible fluid, the differential equation from part ( a ) reduces to 2 = ∇ P 6. (10 points) You are setting up an experiment, where a balloon initially contains some water, and through a plunger (look at figure) you introduce additional amounts of water, expanding the balloon a bit. A friend of yours makes the observation that the balloon is defining essentially a material volume. Do you agree? Please justify your answer. If you don’t agree, try to imagine what the material volume really is, and briefly describe it (a sketch will do). plunger balloon...
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SampleMidterm1 - P k u ∇ − = µ ~(where k is a constant...

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