1.1

The convective derivative. Convection carries a contaminant concentration with the fluid velocity. Let the velocity be uniform in the x direction and equal to U. If the concentration is initially constant inside a circle, x ² + y ² = 1, and zero elsewhere, what will the concentration be after some time t?

A circular patch of dye is released into a large channel. The velocity profile is not uniform, but it can be approximated locally by U + Sy, where U and S are constants. Describe how the dye patch will evolve. Ignore molecular diffusion; focus on pure convection. Show that after some time t the dye will be contained in an ellipse.

1.2

Reynolds number scaling. Derive the nondimensional forms (1.20) and (1.21) of the Navier Stokes equations. Write out the x and y components of these equations for two-dimensional flow.

1.3

One-dimensional flow analysis. Flow enters a duct from the left, with velocity u, pressure p, and constant density. It exits at right through two channels. The exit areas are A ₂ = 1/2 A ₁ and A ₃ = 1/3 A ₁. This is inviscid flow, and Bernoulli s equation is applicable.

Derive a formula for the force on the divider. Explain how you chose to split the flow between the two downstream channels.

1.4

Ejector pump. A jet of air enters...