Stability of a thin viscous fluid film flowing down a rotating non-uniformly heated inclined plane

Abstract

A thin viscous liquid film flowing down a rotating non-uniformly heated inclined plane is investi- gated. It is assumed that the rotation is small and the region of investigation is very far from the axis of rotation, so that the centrifugal force has a dominant role in the instability of the investigated region. Therefore, we have derived a Benney-like free surface evolution equation on the basis of a long-wave (small wave number) approximation and not included the Coriolis effects in the expansion of the dependent variables. Further, a linear and a weakly nonlinear study have been carried out. The linear study reveals that the growth of linear perturbation is independent of the Rossby number Ro, that is invariant with the Coriolis effect, while the linear phase speed cr depends on Ro as well as on the Taylor number Ta. It also reveals that as Ta increases the stable zone decreases, and the influence of Ta is stronger for greater inclination of the rotating inclined plane. Again, it is found that the Marangoni number Mn has similar qualitative (destabilizing) behavior as Ta, but the destabilizing behavior of Ta is more at high Mn than at low Mn. The relation between the parameters Ta and Ro gives a unified parameter Tarot, which reflects the effect of rotation, and we found that the linear phase speed cr first decreases with Tarot up to a critical value and then increases with Tarot. This is due to the fact that the Coriolis force is dominant at very small rotation, while for relatively large rotation the centrifugal force dominates the flow field. The weakly nonlinear study reveals that the effect of rotation appears in the form of both Coriolis and centrifugal force into the growth of finite amplitude perturbations in contrast to the growth of the infinitesimal perturbations where the rotation arrives in the form of centrifugal force only. Also, it plays significant role in the different stability zones, amplitudes of sub/super critical disturbances and nonlinear phase speed.