Effect of gravity on the stability of thermocapillary convection in a horizontal fluid layer

Abstract

Smith & Davis (J. Fluid Mech., vol. 132, 1983, pp. 119–144) considered the stability of thermocapillary convection in a horizontal fluid layer with an upper free surface generated by a horizontal temperature gradient. They showed that for a return-flow velocity profile, the convection will become unstable in the hydrothermal mode with waves propagating upstream obliquely. Their findings provided a theoretical explanation for the defects often found in crystals grown by the floating-zone technique and in thin-film coating processes. Their predictions were verified experimentally by Riley & Neitzel (J. Fluid Mech., vol. 359, 1998, pp. 143–164) in an experiment with 0.75 mm thick layer of silicone oil. Their results with 1 and 1.25 mm thick layers show that as the thickness of the layer is increased, the angle of propagation, the frequency of oscillation and the phase speed of the hydrothermal wave instability decrease, while the wavelength stays nearly constant. We have extended the linear stability analysis of the problem with the effect of gravity included. It is found that when the Grashof number Gr is increased from zero, the angle of propagation first increases slightly, reaches a maximum and then decreases steadily to zero at Gr = 18. The phase speed, the frequency of oscillation and the wavelength of the instability waves all decrease with increasing Grashof number. For Gr larger than 18, there is the onset of the instability into travelling transverse waves. We have also carried out energy analysis at the time of the instability onset. It is found that the major contribution to the energy of the disturbances is from the surface-tension effect. As the gravitational effect is increased, there is a reduction in the kinetic energy supply to sustain the motion of the disturbances. We also found that it requires more kinetic energy to sustain the hydrothermal mode of instability than that required for the travelling transverse mode of instability. As a result, with increasing Grashof number, the kinetic energy available for the disturbances decreases, causing the angle of propagation to gradually decrease until finally reaching zero at Gr = 18.