The Stability of Viscous Fluid Flow under Pressure between Parallel Planes

Extract

If we consider, by the method of small oscillations, the stability of a viscous fluid flow in which the undisturbed velocity is parallel to the axis of x and its magnitude U is a function of y only (x, y, z being rectangular Cartesian co-ordinates), and if we assume that any possible disturbance may be analysed into a number (usually infinite) of principal disturbances, each of which involves the time only through a single exponential factor, then it has been proved by Squire, by supposing the disturbance analysed also into constituents which are simple harmonic functions of x and z, and considering only a single constituent, that if instability occurs at all, it will occur for the lowest Reynolds number for a disturbance which is two-dimensional, in the x, y plane. Hence only two-dimensional disturbances need be considered. The velocity components in the disturbed motion will be denoted by (U + u, v). Since only infinitesimal disturbances are considered, all terms in the equations of motion which are quadratic in u and v are neglected. When u and v are taken to be functions of y multiplied by e^i(αx−βi), the equation of continuity becomes

and the result of eliminating the pressure in the equations of motion then gives the following equation for v, where ν is the kinematic viscosity of the fluid: