Critical conditions for the buoyancy-driven detachment of a wall-bound pendant drop

Abstract

We investigate numerically the critical conditions for detachment of an isolated, wall-bound emulsion droplet acted upon by surface tension and wall-normal buoyancy forces alone. To that end, we present a simple extension of a diffuse-interface model for partially miscible binary mixtures that was previously employed for simulating several two-phase flow phenomena far and near the critical point [A. G. Lamorgese et al. “Phase-field approach to multiphase flow modeling,” Milan J. Math. 79(2), 597–642 (2011)] to allow for static contact angles other than 90◦. We use the same formulation of the Cahn boundary condition as first proposed by Jacqmin [“Contact line dynamics of a diffuse fluid interface,” J. Fluid Mech. 402, 57–88 (2000)], which accommodates a cubic (Hermite) interpolation of surface tensions between the wall and each phase at equilibrium.We show that this model can be successfully employed for simulating three-phase contact line problems in stable emulsions with nearly immiscible components. We also show a numerical determination of critical Bond numbers as a function of static contact angle by phase-field simulation

Phase-field modeling of interfacial dynamics in emulsion flows: Nonequilibrium surface tension

“…Subsequently, as the drop shape changes and a neck is formed, the nonequilibrium surface tension becomes nonuniform with peak values of about twice the initial equilibrium value located at the advancing tip and in the necking region near the minimum neck radius. This result is in favor of our previous argument to explain a discrepancy ( Lamorgese and Mauri, 2016 ) between our numerically determined static contact angle dependence of the critical Bond number for detachment of a wall-bound pendant drop and its sharp-interface counterpart based on a static stability analysis after numerical integration of the Young-Laplace equation. In fact, a sharp-interface analysis together with its assumption of constant surface tension are unable to account for the reduced tendency to detachment due to a sharpening of concentration gradients in the necking region which leads to an effective increase in the nonequilibrium surface tension or an increase in the critical Bond number.…”

“…Also, such extensions for binary fluids have been discussed in Sibley et al (2013c ) within a diffuse-interface description of liquid-vapor flows. At first, 3D simulations of buoyancy-driven detachment were run ( Lamorgese and Mauri, 2016 ) in a computational domain of size L x = L z = π 2 N a √ , L y = N a √ ( N = 64 ), with a pendant droplet (having a radius of 12 a ) of the minority phase deposited on the upper wall with a 90 °contact angle (at t = 0 ) and embedded in a continuous phase (with both phases at equilibrium), for calculating critical Bond numbers corresponding to a pinchoff event as a function of static contact angle. We also looked at the nonequilibrium surface tension [ Eq.…”

Modeling soft interface dominated systems: A comparison of phase field and Gibbs dividing surface models

“…From an order-of-magnitude estimate for γ based on Eq. ( 208) in the vicinity of the critical point, we obtain K ∼ (∆φ) 2 eq √ Ψ − 2, which was found to overestimate the actual value of K found in simulation [370]. As a practical matter, any uncertainty in the relation employed for evaluating γ (equivalent to an approximate choice of K) translates into an uncertain specification of equilibrium contact angle and this leads to a misrepresentation of the static contact angle as compared to its actual (prescribed) value [370].…”

“…This is the constitutive equation that has been used in all of the works on binary mixtures by Mauri and coworkers (see [359][360][361][362][363][364][365]351,[366][367][368]355,296,[369][370][371]). Note that, apart from the transport coefficients, namely, viscosity η, thermal conductivity k and diffusivity D, this model contains two parameters, Ψ and a.…”

“…Here we discuss this problem in the context of a diffuse interface model for regular binary mixtures under isothermal conditions, consisting of the Stokes/Cahn-Hilliard system together with a Cahn boundary condition based on a Hermite interpolation formulation as first proposed by Jacqmin [348]. The Cahn boundary condition which follows after minimization of the augmented free energy functional can be rewritten as [370,371] n…”