Cascade of structures in long-wavelength Marangoni instability

The long-wavelength instability for thermocapillary-driven convection in tw o dimensions is studied numerically. The system under consideration consist s of a horizontal periodical liquid layer bounded from below by a rigid wal l and from above by a deformable free surface. The liquid is heated from th e 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 betw een gravity and the surface tension. Using the boundary integral method we present full-scale direct numerical simulations of the long-wavelength Mara ngoni instability in two dimensions. The time evolution of the free surface leads to the formation of drained regions (so-called "dry spots"). The sim ulations demonstrate a remarkable complexity of the touchdown process, invo lving a deep cascade from large to increasingly small structures. In the be havior of the minimum height of the interface h(min), at large time simple scaling dependence on time was not observed. Extrapolation of h(min) exhibi ts infinite-time singularity. The dependence of the size of drained region on the parameters is discussed, (C) 1999 American Institute of Physics. [S1 070-6631(99)01806-1].