A derivation of a phase field model with fluid properties

We develop a phase-field model for the solidification of a pure material that includes convection in the liquid phase. The model permits the interface to have an anisotropic surface energy, and allows a quasi-incompressible thermodynamic description in which the densities in the solid and liquid phases may each be uniform. The solid phase is modeled as an extremely viscous liquid, and the formalism of irreversible thermodynamics is employed to derive the governing equations. We investigate the behavior of our model in two important simple situations corresponding to the solidification of a planar interface at constant velocity: density change flow and a shear flow. In the former case we obtain a non-equilibrium form of the Clausius-Clapeyron equation and investigate its behavior by both a direct numerical integration of the governing equations, and an asymptotic analysis corresponding to a small density difference between the two phases. In the case of a parallel shear flow we are able to obtain an exact solution which allows us to investigate its behavior in the sharp interface limit, and for large values of the viscosity ratio.

show abstract

Section: Introductionmentioning

confidence: 99%

A phase-field model of solidification with convection

Anderson

McFadden

Wheeler

2000

Physica D: Nonlinear Phenomena

176

137

View full text Add to dashboard Cite

show abstract

“…There have been a number of other phase-field type descriptions of solidification that include convection in the melt [17][18][19][20][21]. The AMW model (see also Ref.…”

Section: Introductionmentioning

confidence: 99%

A phase-field model with convection: sharp-interface asymptotics

Anderson

McFadden

Wheeler

2001

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We have previously developed a phase-field model of solidification that includes convection in the melt [Physica D 135 (2000) 175]. This model represents the two phases as viscous liquids, where the putative solid phase has a viscosity much larger than the liquid phase. The object of this paper is to examine in detail a simplified version of the governing equations for this phase-field model in the sharp-interface limit to derive the interfacial conditions of the associated free-boundary problem. The importance of this analysis is that it reveals the underlying physical mechanisms built into the phase-field model in the context of a free-boundary problem and, in turn, provides a further validation of the model. In equilibrium, we recover the standard interfacial conditions including the Young-Laplace and Clausius-Clapeyron equations that relate the temperature to the pressures in the two bulk phases, the interface curvature and material parameters. In nonequilibrium, we identify boundary conditions associated with classical hydrodynamics, such as the normal mass flux condition, the no-slip condition and stress balances. We also identify the heat flux balance condition which is modified to account for the flow, interface curvature and density difference between the bulk phases. The interface temperature satisfies a nonequilibrium version of the Clausius-Clapeyron relation which includes the effects of curvature, attachment kinetics and viscous dissipation.

show abstract

“…The density profile is given by Eqn. (48), and the resulting vertical component of velocity is small, with w = O(S).…”

Section: Equilibrium Equationsmentioning

confidence: 99%

“…We therefore choose the pressure and temperature as independent variables, and work with a (48) where the solid and liquid densities ps and pL of the bulk phases are constants, and r(</>) is a smooth monotonic function that has r(0) = 0 and r(l) = 1; we will take r(</>) = <j > 2 { 3 -2 </>).…”

Section: Equilibrium Equationsmentioning

confidence: 99%

“…A variety of situations have been studied ranging from spinodal decomposition to thermocapillary flow; the review by Anderson et al [47] and references therein provide further details. An early attempt to include fluid motion within a phase-field model is due to Caginalp and Jones [48,49] Tonhardt and Amberg have also performed two-dimensional numerical studies using adaptive finite elements to study the effects of a shear flow on dendritic growth morphology [51,52] [42]. Unlike previous diffuse interface models, which incorporate fluid motion coupled to a conserved order parameter description [47], we adopt a nonconserved order parameter, </>, in line with our aim of directly extending conventional phase field models of solidification to account for convection.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation