University of Nottingham School of Mathematical Sciences


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This coursework counts for 5% of the total module mark and will be marked out of 20. The maximum marks available for each section are indicated in brackets. Credit will be given for relevant work associated with solving the problems as well as the answers.

Solutions should be submitted firmly attached to a completed Coursework Submission Sheet and in accordance with the instructions given on that sheet.

 Incompressible Newtonian fluid of density ρ and viscosity μ occupies the space (x, y)(- ∞, ∞) x [0, h], between two boundaries at y = 0 and y = h. The boundary at y = 0 is stationary, but the boundary at y = h is accelerated such that its velocity at time t is at , for some constant a R.

 1. Show that a solution of the form u = [u0 (y) + tu1 (y)]satisfies the incompressibility condition · u = 0 and that (u · ) u = 0.

2. By considering components of the Navier-Stokes equations, show that if there is no pressure gradient in the x-direction then the pressure must be a function of time, t, alone.

3. Assuming that the pressure is uniform, i.e., p = p0 a constant, and that the velocity has the form given in Q1, find the fluid velocity.

4. For what times do we have ‘flow reversal’, where one region of fluid is moving in the opposite direction to the of the fluid? Sketch the velocity profile for: t = — ρh2/3μ; t = ρh2/24μ; t = ρh2/6μ.

5. Calculate the viscous stresses on the lower and upper boundary and show that they point in the same direction when

— ρh2/3μ < t < ρh2/6μ

6. Verify that

\displaystyle\frac{d}{\displaystyle{dt}}\biggr \int_0^{h} ρu (y, t) dy = μ \biggl[\displaystyle\frac{\partial u}{\displaystyle{\partial y}}\biggr]_0^{h},

where u (y, t) = u0 (y) + tu1 (y), and interpret the result physically.

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