Problem 1 - Fluid between two parallel plates is subjected to shear flow. The upper plate is held at T₀, and the lower plate is also at T = T₀ for x < 0.  Far upstream, the fluid is at T = T₀.

For x ≥ 0, the temperature of the lower plate is a linearly increasing temperature, T = Ax + T₀. It is assumed that the velocity field is unchanged by the heating of the fluid.

a) If the Péclet number is very large, qualitatively sketch where you expect to see a thermal boundary layer on the above sketch.

b) Using dimensionless quantities, state under what conditions you would assume a thermal boundary layer to exist.

c) Write down the differential equation and boundary conditions governing the temperature distribution in the thermal boundary layer.

d) Give the profile of the boundary layer.  Write down the resulting ODE and boundary conditions for the temperature.  If appropriate, writing the solution in integral form is acceptable.

e) How does the heat flux at the bottom wall depend on x?

Problem 2 -

A plate of density ρ_m, thickness W and length L is hinged at one edge in contact with a stationary substrate and liquid separates the plate and the substrate by a very, very small angle α.  Assume the plate extends to ∞ in the direction orthogonal to the page. The plate is released at t = 0 at α = α₀ and it settles under the influence of gravity toward the substrate.

Develop an equation that describes how α depends on t.