Phase-field investigation of non-isothermal solidification coupled with melt flow dynamics

Abstract

Solidification, coupled with melt flow, plays a critical role in determining the microstructure and properties of materials in several manufacturing processes. Phase-field models coupled with the Navier-Stokes equations are widely used to model and simulate these dynamics. However, most existing models neglect essential thermodynamic couplings, particularly the capillary (Korteweg) stress in the momentum equation. This stress, which arises from the coupling between the phase field and the melt flow, accounts for thermal capillary effects during non-isothermal solidification. Neglecting it leads to models inconsistent with non-equilibrium thermodynamics and incapable of capturing capillarity-driven melt flow. In this work, we present a thermodynamically consistent, non-isothermal phase-field model for solidification coupled with melt flow, incorporating cross-coupling terms and explicitly including the Korteweg stress in the momentum equation. Model validation is performed for solidification-only cases, followed by simulations of dendritic growth under melt flow. The results show that thermal capillary effects induce flow near the interface, influencing dendrite tip velocity and morphology. Simulations under forced convection further demonstrate asymmetric dendrite growth due to the imposed flow field. Additionally, we numerically demonstrate the influence of viscosity interpolation schemes on enforcing the no-slip boundary condition in phase-field models with melt flow.

Figures (14)

• Figure 1: (a) Comparison of analytical predictions and simulated interface velocities for various interface widths during 1D planar solidification. Here, $v_a$ represents the analytically predicted velocity, while $v_s$ denotes the simulated velocity; (b) Comparison of the 2D Ivantsov solution for the Peclet number ($Pe$) with the Peclet number obtained from simulation results, both plotted as functions of undercooling ($\Delta$).
• Figure 2: Schematic illustrations of the simulation setups for dendritic growth, showing the specified initial and boundary conditions for (a) conventional dendrite growth without imposed flow; (b) simulations with imposed forced flow. Note that the temperature boundary conditions from (a) also apply to (b).
• Figure 3: Simulated dendritic morphologies, temperature isolines, and melt flow velocity fields for (a) the case without thermal capillary effect; (b) the case with thermal capillary effect.
• Figure 4: (a) Temporal evolution of the dendrite tip velocity comparing simulations with and without thermal capillary effects. $t^* = 500$ units; (b) Comparison of the final north dendrite tip positions for both cases.
• Figure 5: Simulated dendritic morphologies, temperature isolines, and melt flow velocity fields for a case with an imposed temperature gradient: (a) without thermal capillary effect; (b) with thermal capillary effects.