Nonlinear Wave Behavior of the Sinuous Mode on a Free Liquid Sheet with Variable Surface Tension and Viscosity

Abstract

We investigate nonlinear wave behavior of a free liquid sheet subject to a temperature difference between both surfaces. Considering the temperature dependence of the surface tension and viscosity, we analytically obtain a nonlinear evolution equation for the sinuous (antisymmetric) mode under the membrane approximation. For infinitesimal disturbances or without temperature difference, the equation is reduced to the linearized K–dV equation in which the mode is neutrally stable. When weak nonlinearity is considered in the equation, however, the mode is modulationally unstable for the temperature difference above a critical value. Numerical analysis on the equation shows that the sheet becomes unstable with steepening and distortion above the critical temperature difference. In particular, for large temperature difference, the bulge (symmetric) mode is additionally induced on the sinuous mode and this induced mode is expected to cause the breakup of the sheet. On the other hand, the temperature dependence of the viscosity affects on the growth rate of the mode due to the instability.