544 CHEG Transport Phenomena I First Hour Exam Closed Books and Notes 1). (10 points) Consider a particle freely suspended (no net force or torque on the particle) at zero Reynolds number in the pure straining flow uj = Ejk xk where Ejk is a symmetric, second order, physical tensor. We wish to examine the resulting angular velocity of the particle i due to Ejk. a). What is the most general tensorial relationship for an arbitrarily shaped particle between i and Ejk? What can you say in general about this tensor? b). Show that if the particle is a sphere, the angular velocity is zero. c). (the nasty part) Show that if the particle is a body of revolution with fore-and-aft symmetry whose orientation is specified by the director pi, then the tensor for part (a) may be reduced to a single term (e.g., glop multiplied by one unknown scalar constant). 2). (40 points) Consider the system depicted below. A film of depth d, density and viscosity is on top of a plate which is oscillated back and forth in the x direction with amplitude u = U0 sin t. The fluid above the film is air, so the boundary condition at y = d is just the zero shear stress condition. You may take the flow to be unidirectional. We are interested the in asymptotic behavior at large times (e.g., after initial transients have died away). =0 y u = U0 sin t a). Render the governing equation and boundary conditions dimensionless. What is the dimensionless group that appears in the problem? b). Solve for the velocity distribution for all values of this dimensionless parameter, leaving the problem in complex form. c). Asymptotic limit 1: low frequencies. Explicitly solve for the velocity distribution in the limit of low dimensionless oscillation frequencies. What is the amplitude of the shear stress at the lower wall, and what is the amplitude of the velocity at the upper surface in this limit? (Hint: this is most easily solved via a regular perturbation expansion rathe...