Study of interface coupling in three-layer viscous fluid systems

Linear stability analysis is performed to study the effect of viscosity and surface tension on a system composed of three superimposed immiscible incompressible Newtonian fluids under the gravity field. The eigenvalue problem is formulated in a general way to highlight an analytical solution and other solutions to be determined numerically for a given set of physical parameters. The behaviour of these solutions is analysed using a coupling parameter equal to the product between the thickness of the middle layer and the wave number of the disturbance. When this parameter is large enough, the study of this three-layer fluid system is reduced to the study of two-layer fluid subsystems. The solutions were determined for a three-layer fluid system of interest with a gravitationally unstable interface at the bottom and a gravitationally stable interface at the top to highlight respectively the Rayleigh–Taylor instability and a gravity wave as well as the coupling between these two phenomena. The temporal evolution of the physical quantities is obtained by solving the initial value problem. For this purpose, single-mode disturbances at the two interfaces are imposed as initial conditions. Direct numerical simulations performed with an in-house code initialized by the interface positions and the fluid velocities allow us to compare their temporal evolutions. Results show an excellent agreement with the theory until the amplitude of one interface becomes too large in the same way as the nonlinear effects of the flow. Three main regimes are observed depending on which solutions initially influence the flow the most.