Solutions of a comprehensive dispersion relation for waves at the elastic interface of two viscous fluids

Abstract

Wave propagation in a system of two arbitrary fluids separated by a contaminated interface in the form of an insoluble monolayer is investigated. The effects of four parameters (density ratio, R; kinematic viscosity ratio, V; Marangoni number, P; and squared wave-Reynolds number, σ) on the solutions of a comprehensive dispersion relation for these waves are analyzed. In the first part of the manuscript, several special cases are considered wherein the comprehensive dispersion relation is modified to a simple polynomial equation and its numerical solutions are obtained for σ∈[0,1]. A bifurcation is observed at σ=σb in two of the solutions in each case; and the system is shown to be overdamped for σ<σb, and underdamped for σ>σb. The relative influences of {R,V,P} on σb are also analyzed, and it is found that in general, σb is a strong function of {R,V}, but a weak function of P. In the second part of the manuscript, numerical solutions of the comprehensive dispersion relation are obtained for waves in a decane-water system, in the limit of {P≫1,σ≫1}; and these solutions are classified to correspond to a transverse wave or to a longitudinal wave. The presence of a maximum/minimum in the frequency of oscillation and in the damping rate is observed for the transverse/longitudinal waves. For both the transverse and longitudinal wave modes, the relative deviation of the numerical solution (obtained from the comprehensive dispersion relation) from the respective approximate solution (obtained from a simple dispersion relation) is calculated. The relative deviation, though small, cannot be neglected because it accounts for the effects of elasticity and of capillarity/gravity; and disregarding it in the analyses will fail to yield the inherent maximum/minimum in the frequency of oscillation and in the damping rate for the transverse/longitudinal waves.