We consider the problem of nonlinear thermal-solutal convection in the mushy zone accompanying unstable directional solidification of binary systems. Attention is focused on possible nonlinear mechanisms of chimney formation leading to the occurrence of freckles in solid castings, and in particular the coupling between the convection and the resulting porosity of the mush. We make analytical progress by considering the case of small growth Péclet number, δ, small departures from the eutectic point, and infinite Lewis number. Our linear stability results indicate a small O(δ) shift in the critical Darcy-Rayleigh number, in accord with previous analyses. We find that nonlinear two-dimensional rolls may be either sub- or supercritical, depending upon a single parameter combining the magnitude of the dependence of mush permeability on solids fraction and the variations in solids fraction owing to melting or freezing. A critical value of this combined parameter is given for the transition from supercritical to subcritical rolls. Three-dimensional hexagons are found to be transcritical, with branches corresponding to upflow and lower porosity in either the centres or boundaries of the cells. These general results are discussed in relation to experimental observations and are found to be in general qualitative agreement with them.

I. DETERMINING MASTER CURVES FOR THE INERTIAL AND VISCOUS REGIMEFig. S 1: Time history of drop shapes as a function of film viscosity (µo), indicated to the left. At the point of detachment from the needle the drop shape is independent of the film viscosity.An inertial (τ ρ ) and a viscous (τ µ ) time scale are found to describe the drop deformation after contact with the oil film ( Fig. 5), where Fig. S1 shows the time history of the drop shapes. A similar scaling law is also expected for the apparent radius (r). By scaling r with the initial drop size (R) and time (t) with τ ρ or τ µ two distinct spreading regimes appear (Fig. S2). The inertial scaling ( Fig. S2a) shows that the data for Oh < 0.2 follows nearly the same curve, where as Oh > 0.2 the data remains scattered. If r is instead scaled with τ µ the data for Oh < 0.2 is reduced to nearly a single curve (Fig. S2b). The data in the transitional regime (0.2 < Oh < 2) follows neither of the two scaling laws.A log-log plot of the non-dimensional apparent radius (r/R) in Figs. S2a,b shows that it follows a power-law like behavior (Figs. S3, S4). We fit a master curve through each data set to determine the range of the exponents and the pre-factors for the two regimes. Since the experimental data show that these two regimes are bounded within a certain time span we only use the data in these regions for the fitting, which is illustrated by dashed square in Figs. S3a, S4a. Each data set inside the dashed area is fitted onto a curve of the form r/R = a(t/τ ρ ) b or r/R = a(t/τ µ ) b , where a and b are constants determined by the best fit. For clarity, the data in the transitional regime (0.2 < Oh < 2) has been omitted below.The inertial regime is defined before drop detachment from the needle (t/τ ρ < 2). Each data set is fitted by a single curve (Fig. S3), which is represented by the dashed line. We find the the inertial regime to be represented by a curve r/R ≈ 1.1( t τρ ) 0.45 , where the pre-factor varies between 1.09 -1.13 and the exponent is ≈ 0.45.

The stability of the flow in a half-zone configuration is analysed with the aid of direct numerical simulation. The work is concentrated on the small Prandtl numbers relevant for typical semiconductor melts. The axisymmetric thermocapillary flow is found to be unstable to a steady non-axisymmetric state with azimuthal wavenumber 2, for a zone with aspect ratio 1. The critical Reynolds number for this bifurcation is 1960. This three dimensional steady solution loses stability to an oscillatory state at a Reynolds number of 6250. For small Prandtl numbers, both bifurcations are seen to be quite insensitive to changes in the Prandtl number, and are thus hydrodynamic in nature. An analogy to the instability of thin vortex rings is made. This analogy suggests a physical mechanism behind the instability and also gives an explanation of how the azimuthal wavenumber of the bifurcated solution is selected. The implications of this for the floating-zone crystal growth process are discussed.

In this paper we present simulations of dynamic wetting far from equilibrium based on phase field theory. In direct simulations of recent experiments ͓J. C. Bird, S. Mandre, and H. A. Stone, Phys. Rev. Lett. 100, 234501 ͑2008͔͒, we show that in order to correctly capture the dynamics of rapid wetting, it is crucial to account for nonequilibrium at the contact line, where the gas, liquid, and solid meet. A term in the boundary condition at the solid surface that naturally arises in the phase field theory is interpreted as allowing for the establishment of a local structure in the immediate vicinity of the contact line. A direct qualitative and quantitative match with experimental data of spontaneously wetting liquid droplets is shown.

An existing phase-field model of two immiscible fluids with a single soluble surfactant present is discussed in detail. We analyze the well-posedness of the model and provide strong evidence that it is mathematically ill-posed for a large set of physically relevant parameters. As a consequence, critical modifications to the model are suggested that substantially increase the domain of validity. Carefully designed numerical simulations offer informative demonstrations as to the sharpness of our theoretical results and the qualities of the physical model. A fully coupled hydrodynamic test-case demonstrates the potential to capture also non-trivial effects on the overall flow.

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Dynamic wetting of a solid surface is a process that is ubiquitous in Nature, and also of increasing technological importance. The underlying dissipative mechanisms are, however, still unclear. We present here short-time dynamic wetting experiments and numerical simulations, based on a phase field approach, of a droplet on a dry solid surface, where direct comparison of the two allows us to evaluate the different contributions from the numerics. We find that an important part of the dissipation may arise from a friction related to the motion of the contact line itself, and that this may be dominating both inertia and viscous friction in the flow adjacent to the contact line. A contact line friction factor appears in the theoretical formulation that can be distinguished and quantified, also in room temperature where other sources of dissipation are present. Water and glycerin-water mixtures on various surfaces have been investigated where we show the dependency of the friction factor on the nature of the surface, and the viscosity of the liquid.

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