STABILITY OF A DROPLET PINNED IN A CHANNEL UNDER GRAVITY

Abstract

The stability of a two-dimensional, incompressible water droplet, with two cylindrical-caps that is pinned in a channel, is investigated through the development of an analytical model based on Yong-Laplace relationship. The center of mass of the droplet is derived analytically by assuming a perfectly 2-D circular shape of the droplet cap. The derived analytical expressions are validated through the use of CFD. In the simulations, FLUENT with a 2-D pressure based solver is utilized, and Gambit with 2-D rectangular mesh is used to generate the grid. The pinned droplet states are measured by the location of the center of mass. The stability of the droplet states without gravity is evaluated by the growth rate σH of the Hamiltonian of the system computed by CFD for various drop sizes.

When a droplet is suspended on the straight channel and under no gravity conditions, it is proven analytically and through the use of CFD that there is a critical droplet volume, Vcr, where asymmetric droplet states appear in addition to the basic symmetric states when the drop volume V >Vcr. It is demonstrated that when V<Vcr the symmetric droplet states become stable and the growth rate of the disturbance decays. However, when V>Vcr and the growth rate &sigma;H is positive, the symmetric states become unstable and the asymmetric states become stable. The bifurcation of asymmetric states at Vcr has a pitchfork nature, and the growth rate &sigma;H increases with the volume size.

When the channel holding the droplet is contracted, the pitchfork bifurcation diagram of the droplet system changes into two separate branches of equilibrium states. The analytical expression of those stability branches has been developed for various contraction ratios. The primary branch describes a gradual and stable change of the droplet from a nearly symmetric to asymmetric state as the droplet volume, V, is increased. The secondary branch appears at a modified critical volume, Vmcr, and describes two additional asymmetric states for V>Vmcr. It is demonstrated that the large-amplitude states along the secondary branch are stable whereas the small-amplitude states are unstable.

When the capillary length and the channel width have the same order of magnitude, the effect of gravity is not negligible. An analytical expression is developed to find the effect on gravity, in terms of &beta;, on the behavior of a vertically suspended droplet in straight channel. When gravity is considered, the droplet stability behaves similar to that of the contracted channel apart from one conditions; unlike the contracted channel, there exists a maximum volume on each of the primary and secondary branch where the droplet no longer sustains its weight. The maximum volume on the primary branch, Vp_max, is smaller than the maximum volume on the secondary branch, Vs_max. A critical &beta; value, &beta;cr, is also found. That critical value describes the maximum condition at which the droplet will have only one range of solutions at the primary branch, and no longer sustain stability on the secondary branch. All analytical solutions are validated with CFD.