Unified inviscid dispersion relation and Ohnesorge number effect on the rivulet instability

Abstract

Effects of the Ohnesorge (Oh) number and contact angle on the instability of a sessile rivulet are theoretically studied in this paper for both the pinned contact line (PCL) and moving contact line with invariant contact angle (MCL) conditions. Inviscid dispersion relations of unstable modes for rivulets with different static contact angles are shown to be transformed analytically into a unified form for both PCL and MCL conditions, reflecting the same intrinsic instability mechanism. The growth rate becomes lower at higher Oh due to the stabilization effect of viscosity. It is found for both PCL and MCL conditions that the deviation of the maximum growth rate from its inviscid counterpart varies as Oh1/2 when Oh is so small that the viscous effect mainly exists in a thin layer near the wall. For the MCL condition, the maximum growth rate of the unstable mode decreases and increases monotonically with the augment of the static contact angle at low and high Oh, respectively, while at moderate Ohnesorge numbers, increasing the static contact angle will first enlarge and then decrease the maximum growth rate for cases with hydrophilic bottom walls.