Stability of thin film flowing down the outer surface of a rotating non-uniformly heated vertical cylinder

Abstract

We investigated an incompressible viscous liquid film flow over a rotating vertical cylinder of radius R and of infinite length rotating with a uniform angular velocity $$\varvec{\Omega }$$ about its axis. The surface of the vertical cylinder is non-uniformly heated where the temperature varies linearly in the downstream direction. The flow is assumed to be axisymmetric, and the component of the velocity along the azimuthal direction is assumed to be constant. The surface tension of the liquid is assumed to vary linearly with temperature such that as the temperature increases, the surface tension decreases. This gives rise to Marangoni stress over the free surface of the thin film. Using the long-wave approximation method, we derived a free surface evolution equation. For linear stability analysis, we used a normal mode approach and found that the Marangoni number plays a double role. There exists a critical Marangoni number $$\left( \mathrm{Mn}^{*}\right) $$ such that for $$\mathrm{Mn} \mathrm{Mn}^{*}$$ it plays a destabilizing role. We also found that as the rotation number $$\mathrm{Ro}$$ increases, the destabilizing zone increases but it decreases with the increment of the radius R of the cylinder. We further performed a weakly nonlinear analysis of the flow using the method of multiple scales. The study reveals that the Marangoni number Mn, the radius R and the rotation number $$\mathrm{Ro}$$ have substantial effects on different stability zones. The study also reveals that in the supercritical stable (subcritical unstable) zone, the threshold amplitude of the nonlinear disturbance increases (decreases) with the increment of Mn and $$\mathrm{Ro}$$ but decreases (increases) with the increment of R. The nonlinear wave speed in the supercritical stable zone decreases with the increment of Mn and $$\mathrm{Ro}$$, whereas it increases with the increment of R. We also examined the effect of thermocapillarity and rotation on the profile of the steady travelling wave solutions of the leading order part of the evolution equation.