Reference no: EM133215445

Membrane filtration modules are often configured as bundles of hollow fibers to cause the filtrate to pass outward through the fiber wall. Let us model the flow through one fiber by considering a cylindrical tube of radius R and length L. The mean axial velocity at the tube inlet is Uo. The radial fluid velocity at the tube wall is Vw. To make the problem more interesting, assume that the tube is also rotating with an angular speed Ω. The objective is to determine the velocities and pressure gradient in the tube. Ignore entrance/exit effects, and assume R << L. Neglect gravity.

1. Use order of magnitude analysis explicitly to derive the simplified form of the three equations of motion. Assume inertial terms to be negligible as your starting point. However, show all other steps.

2. Obtain an expression for Vθ using appropriate boundary conditions.

3. Obtain an expression for Vz. Your answer may have a pressure gradient term. d) Derive an expression for the pressure gradient in terms of known parameters.

Two similar rigid, plane rectangular plates are joined along the edge of length b by a smooth, oil-tight hinge. The plates make a small angle α = α(t) with one another. The space between the plates is filled with a Newtonian oil, which is squeezed out as the plates are pushed together by a force F perpendicular to each plate, applied at the outer edge. Assume α << 1, and a << b so Vx is negligible. Assume creeping flow in your analysis and neglect gravity.

1. Use order of magnitude analysis and any relevant information to simplify the equations of motion.

2. Show that P =? f(z). Derive an explicit expression for Vy (in terms of known parameters and/or pressure gradient term(s)).

3. Calculate ∂P/∂y at the outer free edge, i.e. y = a, in terms of α ≡ dα/dt and known parameters.