Drop formation during coating of vertical fibres

Abstract

When the coating film around a vertical fibre exceeds a critical thickness hc, the interfacial disturbances triggered by Rayleigh instability can undergo accelerated growth such that localized drops much larger in dimension than the film thickness appear. We associate the initial period of this strongly nonlinear drop formation phenomenon with a self-similar intermediate asymptotic blow-up solution to the long-wave evolution equation which describes how static capillary forces drain fluid into the drop. Below hc, we show that strongly nonlinear coupling between the mean flow and axial curvature produces a finite-amplitude solitary wave solution which prevents local finite-time blow up and hence disallows further growth into drops. We thus estimate hc by determining the existence of solitary wave solutions. This is accomplished by a matched asymptotic analysis which joins the capillary outer region of a large solitary wave to the thin-film inner region. Our estimate of hc = 1.68R3H–2, where R is the fibre radius and H is the capillary length H = (σ/ρg)½, is favourably compared to experimental data.

Thermocapillary effect on the dynamics of viscous beads on vertical fiber

“…Kalliadasis and Chang [5] solved the evolution equation derived by Frenkel [4] and showed that the mean flow can arrest the drop formation process when the thickness of the film is less than the critical value, h c , observed by Quéré [3] in which h c ∼ a 3 H −2 , where H = (σ/ρg) 1/2 . The nonlinear dynamics of Frenkel's equation has also been investigated by Kerchman and Frenkel [6] and Chang and Demekhin [7].…”

Stability of viscous film flow coating the interior of a vertical tube with a porous wall

“…Frenkel derived a simple Benney-like equation for the evolution of the film thickness using a thin-film approximation [8]. The nonlinear dynamics of Frenkel's equation has also been investigated by Kalliadasis and Chang [9], Kerchman and Frenkel [10], and Chang and Demekhin [11].…”

Liquid film dynamics in horizontal and tilted tubes: Dry spots and sliding drops

“…Increasing the precursor film thickness to η = 0.01 has marginal effect; increasing by a further order of magnitude (Figure 11(c)) increases drop speeds but again causes drops in vertical tubes to fall increasingly slowly compared to tilted tubes. We infer that the drop dynamics are controlled by a balance between dissipation at the advancing contact line and release of potential energy (as in [5,10]), so that wider drops are subject to greater dissipation and therefore travel slower. (In the most extreme case, α = π/2 in figure 11(d), the front of the drop wraps entirely around the interior of the tube to become a collar for t 8.)…”

“…[13] (exploiting the fact that, to leading order, substrate curvature in a curved tube has the same effect as gravity in a straight tube) and [16] (who included higher-order terms); the limit α = π/2, ∂ θ = 0 was considered in, for example, refs. [5,6,20]. Full details of the derivation of this class of equation may be found in these earlier works.…”

“…Weak gravitational forcing causes a collar to move down a vertical tube, accumulating fluid at its leading edge and depositing it at its rear. A very slowly moving collar deposits a very thin trailing film, allowing the collar's volume to increase over time to the extent that solutions of (2.1) can exhibit finite-time blow-up; in practice, the film thickens so much that the thinfilm assumption underlying lubrication theory is violated [5]. Multiple collars exhibit coarsening, with large collars traveling faster than smaller ones and subsequently merging with them; this also promotes blow-up [6].…”