Hydrodynamics and instabilities of a falling liquid film with an insoluble surfactant

In this study, we investigate the linear and weakly nonlinear stability of a liquid film flowing down an inclined plane with an insoluble surfactant. First, the nonlinear evolution equations of a liquid film thickness and surfactant concentration are derived using the long-wave expansion method at a moderate Reynolds number (0 < Re <= 20). The linear stability of the flow is examined using the normal-mode method, and the linear stability criterion and critical Reynolds number Re-c are obtained. The results reveal the destabilizing nature with increasing Reynolds number Re and the stabilizing nature with increasing Marangoni number M. Second, the nonlinear equations described by the complex Ginzburg-Landau equation are obtained using the multiple-scale method to investigate the weakly nonlinear stability of the system. The results show that a new linear instability region appears above the neutral stability curve caused by the solute-Marangoni effect, which develops into a supercritical stable zone under the influence of nonlinear factors. Increasing M generally improves the stability of the flow but continuing to increase M under the condition of M > M-c (critical Marangoni number) improves the nonlinear instability in the region and transforms part of the unconditional stability zone into a subcritical instability zone. The increase in Re extends an explosive unstable zone and reduces the threshold amplitude in the subcritical unstable zone. In contrast, the unconditional stable zone decreases and disappears after increasing Re to a specific value, which reflects the destabilizing effect of Re on the nonlinear zone of the flow.