Effect of Viscosity and Membrane on the Oscillations of Superposed Fluids

摘要

Two-dimensional small-amplitude viscous, as well as nonviscous, wave motion is considered in a medium consisting of two superposed incompressible fluids of finite depths, separated at the interface by a membrane. The upper one has a free surface and the lower one is bounded below by a rigid bottom. This is rather a crude model of a general problem in biology. Here one is interested in studying similar motions in a medium consisting of several layers of fluid separated by membranes across which there is migration of particles. It is found that the membrane affects the dispersion relation but does not lead to the damping of the waves. The frequency in the viscous case decreases with viscosity. The modulus of decay of the amplitude is 0{(ν1ν2)1/2/[ρ1(ν1)1/2+ρ2(ν2)1/2]}, where ν1, ν2 are the viscosities and ρ1, ρ2 are the densities of the upper and the lower fluid, respectively.