Cascade of structures in long-wavelength Marangoni instability

Abstract

The long-wavelength instability for thermocapillary-driven convection in two dimensions is studied numerically. The system under consideration consists of a horizontal periodical liquid layer bounded from below by a rigid wall and from above by a deformable free surface. The liquid is heated from the bottom wall and cooled from above. The problem can be approximated by the Stokes equation and has two dimensionless parameters. One parameter is the dynamic Bond number which is the ratio between gravity and thermocapillary force. The other is the static Bond number, which describes the ratio between gravity and the surface tension. Using the boundary integral method we present full-scale direct numerical simulations of the long-wavelength Marangoni instability in two dimensions. The time evolution of the free surface leads to the formation of drained regions (so-called “dry spots”). The simulations demonstrate a remarkable complexity of the touchdown process, involving a deep cascade from large to increasingly small structures. In the behavior of the minimum height of the interface hmin at large time simple scaling dependence on time was not observed. Extrapolation of hmin exhibits infinite-time singularity. The dependence of the size of drained region on the parameters is discussed.