Daniel Anderson

The spreading of volatile liquid droplets on heated surfaces

A two‐dimensional volatile liquid droplet on a uniformly heated horizontal surface is considered. Lubrication theory is used to describe the effects of capillarity, thermocapillarity, vapor recoil, viscous spreading, contact‐angle hysteresis, and mass loss on the behavior of the droplet. A new contact‐line condition based on mass balance is formulated and used, which represents a leading‐order superposition of spreading and evaporative effects. Evolution equations for steady and unsteady droplet profiles are found and solved for small and large capillary numbers. In the steady evaporation case, the steady contact angle, which represents a balance between viscous spreading effects and evaporative effects, is larger than the advancing contact angle. This new angle is also observed over much of the droplet lifetime during unsteady evaporation. Further, in the unsteady case, effects which tend to decrease (increase) the contact angle promote (delay) evaporation. In the ‘‘large’’ capillary number limit, matche...

A phase-field model of solidification with convection

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.

Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys

We consider the solidification of a binary alloy in a mushy layer and analyse the system near the onset of buoyancy-driven convection in the layer. We employ a neareutectic approximation and consider the limit of large far-field temperature. These asymptotic limits allow us to examine the rich dynamics of the mushy layer in the form of small deviations from the classical case of convection in a horizontal porous layer of uniform permeability. Of particular interest are the effects of the asymmetries in the basic state and the non-uniform permeability in the mushy layer, which lead to transcritically bifurcating convection with hexagonal planform. We obtain a set of three coupled amplitude equations describing the evolution of small-amplitude convecting states in the mushy layer. These equations are analysed to determine the stability of and competition between two-dimensional roll and hexagonal convection patterns. We find that either rolls or hexagons can be stable. Furthermore, hexagons with either upflow or downflow at the centres can be stable, depending on the relative strengths of different physical mechanisms. We determine how to adjust the control parameters to minimize the degree of subcriticality of the bifurcation and hence render the system globally more stable. Finally, the amplitude equations reveal the presence of a new oscillatory instability.

Thin interface asymptotics for an energy/entropy approach to phase-field models with unequal conductivities

Karma and Rappel [Phys. Rev. E 57 (1998) 4342] recently developed a new sharp interface asymptotic analysis of the phase-field equations that is especially appropriate for modeling dendritic growth at low undercoolings. Their approach relieves a stringent restriction on the interface thickness that applies in the conventional asymptotic analysis, and has the added advantage that interfacial kinetic effects can also be eliminated. However, their analysis focussed on the case of equal thermal conductivities in the solid and liquid phases; when applied to a standard phase-field model with unequal conductivities, anomalous terms arise in the limiting forms of the boundary conditions for the interfacial temperature that are not present in conventional sharp interface solidification models, as discussed further by Almgren [SIAM J. Appl. Math. 59 (1999) 2086]. In this paper we apply their asymptotic methodology to a generalized phase-field model which is derived using a thermodynamically consistent approach that is based on independent entropy and internal energy gradient functionals that include double wells in both the entropy and internal energy densities. The additional degrees of freedom associated with the generalized phase-field equations can be used to eliminate the anomalous terms that arise for unequal conductivities.

The case for a dynamic contact angle in containerless solidification

Containerless solidification, in which the melt is confined by its own surface tension, is an important technique by which very pure materials can be produced. The form of the solidified product is sensitive to conditions at the tri-junction between the solid, the melt and the surrounding vapor. An understanding of the dynamics of tri-junctions is therefore crucial to the modelling and prediction of containerless solidification systems. We consider experimentally and analytically the simple system of a liquid droplet solidifying on a cold plate. Our experimental results provide a simple test of tri-junction conditions which can be used in theoretical analyses of more complicated systems. A new dynamical boundary condition at the tri-junction is introduced here and explains the surprising features of solidified water droplets on a cold surface.

A new oscillatory instability in a mushy layer during the solidification of binary alloys

We consider the solidification of a binary alloy in a mushy layer and analyse the linear stability of a quiescent state with specific interest in identifying an oscillatory convective instability. We employ a near-eutectic approximation and consider the limit of large far-field temperature. These asymptotic limits allow us to examine the dynamics of the mushy layer in the form of small deviations from the classical case of convection in a horizontal porous layer of uniform permeability. We consider also the limit of large Stefan number, which incorporates a key balance necessary for the existence of the oscillatory instability. The model we consider here contains no double-diffusive effects and no region in which a statically stable density gradient exists. The mechanism underlying the oscillatory instability we discover is instead associated with a complex interaction between heat transfer, convection and solidification.

A diffuse-interface description of internal waves in a near-critical fluid

We present a diffuse-interface treatment of the internal gravity waves which have been observed experimentally by Berg et al. in xenon near its thermodynamic critical point. The results are compared with theoretical predictions by Berg et al. that were obtained using separate models above and below the critical temperature Tc. The diffuse-interface model applies both above and below Tc, and is formulated by using the density as an order parameter. The diffuse interface is represented as a transition zone of rapid but smooth density variation in the model, and density gradients appear in a capillary tensor, or Korteweg stress term, in the momentum equation. We obtain static density profiles, compute internal wave frequencies and compare with the experimental data and theoretical results of Berg et al. both above and below the critical temperature. The results reveal a singularity in the diffuse-interface model in the limit of incompressible perturbations.

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

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.

Capillary rise of a liquid into a deformable porous material

We examine the effects of capillarity and gravity in a model of one-dimensional imbibition of an incompressible liquid into a deformable porous material. We focus primarily on a capillary rise problem but also discuss a capillary/gravitational drainage configuration in which capillary and gravity forces act in the same direction. Models in both cases can be formulated as nonlinear free-boundary problems. In the capillary rise problem, we identify time-dependent solutions numerically and compare them in the long time limit to analytically obtain equilibrium or steady state solutions. A basic feature of the capillary rise model is that, after an early time regime governed by zero gravity dynamics, the liquid rises to a finite, equilibrium height and the porous material deforms into an equilibrium configuration. We explore the details of these solutions and their dependence on system parameters such as the capillary pressure and the solid to liquid density ratio. We quantify both net, or global, deformation ...

A model for wetting and evaporation of a post-blink precorneal tear film

We examine a fluid dynamic model for the evolution of a precorneal tear film that includes evaporation of the aqueous layer and a wetting corneal surface. Our model extends previous work on the break-up time for a post-blink tear film to include a more realistic model for evaporation. The evaporation model includes the effects of conjoining pressure and predicts the existence of an equilibrium adsorbed fluid layer that serves as a model for a wetting corneal surface/mucin layer. The model allows the prediction of dewetting rates that are compared with experimental measurements. By choosing an expected thickness where evaporation and conjoining pressure balance, we obtain qualitative agreement for the opening rate with in vivo observations.