Behavior of a Free Liquid Sheet Subject to a Temperature Difference between Both Surfaces

Abstract

Behavior of a free viscous liquid sheet subject to a temperature difference between both surfaces is analytically examined by considering the thermocapillary effect. Under the long wave approximation, nonlinear evolution equations of the sheet thickness and velocity are derived for sufficiently small Prandtl number, while a nonlinear relation between the position of sheet centerline and the sheet thickness is obtained. It is shown that the surface profiles deviate from the symmetric (dilational) to the asymmetric with flat troughs and large crests as the temperature difference increases. It is also shown that there exist certain critical temperature difference ΔΘ_c above which the sheet becomes linearly unstable. In the neighborhood of ΔΘ_c, a weakly nonlinear equation of the sheet thickness is obtained and numerically solved for the instability. However, since ΔΘ_c is estimated to be so large in both linear and nonlinear analyses, the sheet may be substantially stable as far as the small Prandtl number is considered.