Boussinesq model for two-fluid system with surface- and interfacial tension

A Boussinesq model associated with surface and internal waves with surface tension and interfacial tension in a two-layer fluid is presented in the application range of weakly dispersive nonlinear. The nonlinear form of model definitions in terms of surface and interfacial tensions along with velocity potentials are provided. Based on the perturbation technique, the Boussinesq equations for both layers are derived. The present Boussinesq result is validated against existing published results and it has a good level of agreement. Further, the present model results of crests of surface and interface displacements are compared with existing published numerical OpenFOAM simulations. The dispersion relations for fast- and slow modes are derived in the quadratic form of wave frequencies in the presence of surface and interfacial tensions. The second-order surface and internal wave displacements along with velocity potentials for upper- and lower-layer fluids are presented. The effects of surface and interfacial tensions on dispersion characteristics, harmonic transfer functions, and shoaling gradients on different physical parameters associated with the present model are studied by analyzing several numerical results. In the end, a real physical problem is demonstrated in the application of surface tension with solvents in the gas-liquid interface.