Due Monday Oct 14 or Wed Oct 16 at the latest

1. Suppose that f(x,t) is conserved along particle path in an incompressible flow (f could be the density for instance). If the fluid is also inviscid (i.e. nu=0), what is the rate of change of [w.grad f] along particle paths? where w is the vorticity, i.e. the curl of the velocity.
2. Acheson 2.5
3. Acheson 2.7
4. Acheson 5.19
5. Acheson 6.4
6. Consider the 2D boundary layer equations with F(x)=-A^2/x^3 (see class notes and Acheson Chapter 8) where A is a positive constant. Look for a self-similar solution of the form u=A f'(eta)/x with eta=y/g(x) and f'(eta)->1 as eta -> infinity. Find g(x) and f'(eta) explicitly (using a shooting numerical method) or show that f'(eta) does not exist. Try to discuss briefly what this corresponds to in physical terms.

Due TBA

1. Acheson 5.8
2. (a) Consider a spherical gas bubble of radius R(t) immersed in an infinite incompressible fluid. Assuming that R(t) is known, find the velocity induced in the fluid by the bubble oscillations and the pressure at the surface of the bubble on the liquid side in terms of R(t) and the pressure at infinity.
   (b) Assume that the gas is perfect and that the oscillations are adiabatic (i.e. no heat exchange, but temperature varies because of the compression-expansions)) in which case the gas pressure is related to the gas density as p = C rho^gamma where C is a constant and gamma is the ratio of specific heats. The surface tension in the gas-liquid interface is T. Assume that the pressure and density are uniform inside the gas bubble and neglect bulk viscosity. Derive the equation that governs the evolution of the bubble radius. [Assume also that there is no mass transfer at the bubble interface, i.e. no condensation or evaporation]. You must thoroughly justify other approximations that you make.
3. Acheson 3.8
4. A rigid sphere of radius R rests on a flat rigid surface, and a small amount of liquid surrounds the point of contact making a concave-planar lens whose diameter is small compared with R. The angle of contact of the liquid with each of the solid surfaces is zero, and the tension in the air-liquid interface is T. Show that there is an adhesive force of magnitude [4 pi RT] acting on the sphere. (The fact that this adhesive force is independent of the volume of liquid is noteworthy).
5. Consider a liquid (water) cylinder of radius a surrounded by a gas (air). Neglect gravity. Consider an axisymmetric perturbation of the cylinder radius: r=R+ e cos (kz) where z is the axial direction, k is a wavenumber, R is a constant to be determined and e is a constant with |e| << a. Show that the surface area will be reduced for constant volume provided |k a| < 1. This implies a surface tension instability for those wavenumbers but does not provide growth rate information.