Effect of Interfacial Tension on Internal Waves Based on Boussinesq Equations in Two-Layer Fluids

An analytical model associated with internal waves in the presence of interfacial tension bounded by a rigid floating plate and rigid bottom in a two-layer fluid is presented based on Boussinesq equations. The general governing equations and boundary conditions of internal wave motion under interfacial tension are described. The detail of the Boussinesq equations associated with interfacial tension with two depth parameters that indicate the specific elevations in upper- and lower-layer fluids are obtained based on an expansion of velocity potentials as a power series in the dispersive effect. For simplicity, the one-dimensional linearized Boussinesq equations and dispersion relation with interfacial tension are obtained. The first-and second-order internal wave amplitudes with interfacial tension based on a perturbation technique are derived, and the super-and subharmonic interactions of second-order internal waves are obtained. The accuracy of the analytical result of internal wave displacement is verified by comparing it with computational fluid dynamics model simulations without introducing interfacial tension, and it is observed that the wave profiles and their peak amplitudes agree well. Furthermore, the linearized dispersion relation is compared with one recently obtained. The influences of interfacial tension on internal wave properties are studied by analyzing dispersion relation, fluid particle velocities, first-and second-order harmonic waves on super-and subharmonic transfer functions, and shoaling gradients. It is found that smaller values of interfacial tension and wavenumber ratio result in stronger second-order effect on the internal wave profile. Finally, an application of interfacial tension is demonstrated in the presence of surfactants for partitioning oil-water interface.