Effect of Permeable Boundary in the Benard-Marangoni Convection in a Relatively Hotter or Cooler Liquid Layer

We utilize the modified linear stability analysis to study the problem of the onset of convection in a horizontal liquid layer heated from below with permeable lower boundary on which boundary condition of the type specified by Beavers and Joseph [2] is applicable, while the driving mechanism is surface tension at the upper free surface. A Fourier series method is used to obtain the characteristic value equation for the Marangoni number M. It is established numerically that the principle of exchange of stabilities is valid for the problem under consideration. The numerical results are obtained for both thermally conducting and thermally insulating cases of the lower boundary, for various values of the boundary parameter K characterizing the permeable nature of the lower boundary. It is shown that there is a smooth increase in the value of the critical Marangoni number with increase in value of K irrespective of whether the liquid layer is relatively hotter or cooler. Further, it is observed that the limiting cases of the parameter K, give rise to the particular cases namely, when the lower boundary surface is either dynamically free ( K →0 ) or rigid ( K→ ∞).