A NOTE ON SURFACE-FILMS AND SURFACE-WAVES

The boundary condition for the continuity of the shear stress between a surface film and an underlying liquid is cast in the form u + l(u(y) + nabla-v) = 0, where: u and upsilon are the horizontal and vertical components of the velocity at the interface between the film and the liquid (y < 0); nabla is the gradient operator in the horizontal plane; l is a complex function of the frequency sigma that has the dimensions of length and may be expressed in terms of the surface-film parameters or, more conveniently, regarded as a phenomenological parameter to be determined by direct measurement. The complex dispersion relation for a surface wave of frequency sigma and wavenumber k is derived in the form sigma = sigma-0f(epsilon, lambda), where sigma = sigma-0(k) is the relation for an inviscid fluid, epsilon = kl-nu, lambda = l/l-nu, l-nu = (2-nu/sigma)1/2, and nu is the kinematic viscosity.