Liquid film flow along a substrate with an asymmetric topography sustained by the thermocapillary effect

Abstract

We investigate flow in a thin liquid film over a “thick” asymmetric corrugated surface in a gas-liquid bi-layer system. Using long-wave approximation, we derive a nonlinear evolution equation for the spatiotemporal dynamics of the liquid-gas interface over the corrugated topography. A closed-form expression indicating a non-zero value for a liquid flow rate is derived in a steady state of the system. Through numerical investigations we study the nonlinear dynamics of the liquid-gas interface with respect to topographical variations of the solid surface, different thermal properties of the liquid and the solid, and different values of the Marangoni number. We find the existence of a critical value for the Marangoni number Mc, so that for M > Mc, the liquid film ruptures, whereas for M < Mc, the interface will remain continuous. In a broad variety of parameters, the interface attains a deformed steady state with a nonzero average flow rate through the system, thus the described mechanism may be used as a means of transport in microfluidic devices. We carry out the Floquet stability analysis of periodic steady states with respect to spatial replication and show that in the framework of the time-independent evolution equation, the system is unstable to long wave perturbations. We demonstrate that in a finite periodic setting, the system may evolve within a certain parameter range into a metastable state which may be manipulated by varying the Marangoni number M in time in order to increase, control, and sustain the average flow rate through the system. We also show that in the case of a solid substrate with the thermal conductivity lower than that of the liquid, the flow rate through the system may be significantly increased with respect to the opposite case.