Diffuse interface tracking of immiscible fluids: Improving phase continuity through free energy density selection

Abstract

Diffuse interface (DI) tracking methods frequently adopt the double-well energy density function to describe the free energy variation across an interface, leading to phase interpenetration and spontaneous drop shrinkage when applied to immiscible two-phase systems. While the observed continuity losses can be limited by constraints placed on the interfacial width and mobility parameter, the associated increase in computational cost and mesh requirements has limited DI methods to 2D planar and axi-symmetric flow.Using the proposed temperature-variant simplified energy density function (TVSED), the effect of the metastable thermodynamic region on phase continuity is examined using a reduced temperature parameter, T R . The solutions obtained by the proposed DI implementation and a volume-of-fluid (VOF) based solver are discussed for two common benchmark simulations (collapse of a column of water, and droplet deformation/relaxation in simple shear). While comparable solutions are obtained from the T R = 0 and VOF simulations; the large metastable region of the more commonly used double-well function (T R = 1) promoted inter-mixing in the bulk phases, diluting the inertia transferred between impacting fluids and under-predicting deformation in shear. The fundamental mechanisms responsible for spontaneous drop shrinkage are eliminated at T R = 0, allowing for constraints on the interfacial width and mobility to be relaxed. The comparable performance of T R = 0 and VOF simulations is a promising indication of the potential for application to complex 3D flows at reduced mesh resolution relative to existing DI methods.Crown

Comparison of energy stable simulation of moving contact line problems using a thermodynamically consistent Cahn–Hilliard Navier–Stokes model

“…For the choice of the relaxation parameter s in W s , see Remark 2, several points must be considered. To reduce the intermixing between the phases and increase the rate at which the equilibrium profile of ϕ is reestablished after a deformation, it is desirable to exhibit a large spinodal region and subsequently a small metastable region [53]. The metastable region of the bulk energy potential W s is located between 1 > |ϕ| > ξ −1 = 1 − 1 s , while the metastable region for W poly is located between 1 > |ϕ| > √ 3 −1 ≈ 0.577.…”

A viscoelastic two-phase solver using a phase-field approach

“…1. The double well function is the simplest non-singular function of the mass concentration [32]. Arguably it is one of the most frequently considered expressions in PF models, for modeling immiscible fluids [7], as it reduces the numerical difficulties associated with common approaches that exhibit a singular behaviour [33,34].…”

“…For all numerical simulations with the PF method, δ is related to the desired number of cells, k, used for discretising the interface thickness. For a numerical (uniform) grid with characteristic cell size h, the interface width is defined as δ = hk and thus, the interface thickness is correlated to the numerical mesh as = hk/(4 √ 2 tanh −1 (0.9)) [15,32,35]. Furthermore, the interface thickness is usually related to the problem of interest by the Cahn number, defined as Cn = /L, where L is the characteristic length of the problem and provides a measure for approaching the sharp-interface limit (Cn → 0) [15,36,37].…”

“…where ρ is the density, p the pressure, τ the extra stress tensor, g the gravitational acceleration vector and the last term, f st , is added to account for the forces due to interfacial tension. The modeling of the surface forces in the Navier-Stokes equation is key to a successful numerical code, and various models have been proposed for that reason [7,8,22,25,32]. Here, the work of Badalassi et al [24] and Ding et al [27] is adopted and the applied forces are evaluated based on the gradients of the concentration across the interface:…”

Recoil of Drops During Coalescence on Superhydrophobic Surfaces

“…The solver uses the Cahn-Hilliard contact line model [Eqs. (3), (4)] coupled with the Navier-Stokes equations (Yue et al, 2010;Donaldson et al, 2011) to advance the evolving drop shapes.…”