Behavior of a free liquid sheet subject to a temperature difference between both surfaces

Behavior of a free viscous liquid sheet subject to a temperature difference between both surfaces is analytically examined by considering the thermoca pillary effect. Under the long wave approximation, nonlinear evolution equa tions of the sheet thickness and velocity are derived for sufficiently smal l Prandtl number, while a nonlinear relation between the position of sheet centerline and the sheet thickness is obtained. It is shown that the surfac e profiles deviate from the symmetric (dilational) to the asymmetric with f lat troughs and large crests as the temperature difference increases. It is also shown that there exist certain critical temperature difference Delta Theta (c) above which the sheet becomes linearly unstable. In the neighborh ood of Delta Theta (c), a weakly nonlinear equation of the sheet thickness is obtained and numerically solved for the instability. However, since Delt a Theta (c) is estimated to be so large in both linear and nonlinear analys es, the sheet may be substantially stable as far as the small Prandtl numbe r is considered.