Linear instability analysis of thermocapillary convection in a bilayer system

The linear instability analysis of thermocapillary convection in a bilayer system consisting of silicon oil 10cS and fluorinert FC70 liquids was discussed in this paper. The bilayer system was bound below by a rigid plate and above by a free surface with a passive gas. The two immiscible liquids were separated by an interface. A constant horizontal temperature gradient was imposed along the interfaces. Two typical cases were studied: (i) streamwise homogeneous disturbances (the streamwise disturbance wave number α = 0); (ii) spanwise homogeneous disturbances (the spanwise disturbance wave number β = 0). When α = 0, it was found that convection in the two layers may occur in the form of stationary mode or oscillatory mode. The oscillatory mode takes the form of traveling wave, which propagates in either spanwise direction. When β = 0, convection in the two layers occurs in the form of oscillatory mode, which takes the form of traveling wave propagating in the same direction as base flow. The three dimensional analysis suggested that disturbance in the spanwise direction was the major cause of destabilization in the system. The influences of Biot number Bi and the depth ratio h on the unstable modes in the bilayer system were discussed.