Figure 1

A two-dimensional mushy layer formed between two heat baths pulled at velocity $V$ aligned with the gravitational acceleration $g\mathbf{k}$. The overlying liquid has constant far-field temperature $T_{\infty }$ and concentration $C_0$, and the mush solidifies at eutectic temperature $T_E$ and concentration $C_E$ at $z=0$. For a periodic array of chimneys, of dimensional width $2\widehat{a}(z,t)$ and imposed spacing $l$, we exploit symmetry to solve for the properties of a mushy layer of thickness $h(x,t)$ within the dashed outline. The specific heat capacity $c_p$ and thermal diffusivity $\kappa$ are assumed constant across solid and liquid phases, and the fluid has dynamic viscosity $\mu$ and density ${\rho }_{0}g\beta (C-C_E)$ for constant solutal coefficient $\beta$ and reference density ${\rho }_0$. This describes aqueous ${\mathrm{NH}}_4\mathrm{Cl}$ solidified from below, among other systems.Reuse & Permissions

Figure 2

(a) Steady-state solute fluxes $F_S(\lambda )$ exiting a mushy layer vary with the chimney spacing $\lambda$. Two branches of solutions are determined and exhibit hysteresis; the upper branch results from convection with chimneys (red crosses with $\lambda >{\lambda }_s$), and the lower branch results from a state of no flow and hence $F_S(\lambda )=0$ (blue squares) which is unstable for $\lambda >{\lambda }_u$. Red and blue dashed lines indicate the trajectories at ${\lambda }_s$ and ${\lambda }_u$, respectively. A maximal solute flux $F_{S\mathcal{O}}$ is attained for a chimney spacing $\lambda ={\lambda }_{\mathcal{O}}$, with weaker solute drainage at $\lambda >{\lambda }_{\mathcal{O}}$. The inset shows the upper stable branch (red curve) and intermediate unstable branch (black dashed curve) in the vicinity of the stabilization point confirmed by arclength continuation. Here $R_m=40$, $\mathcal{C}=15$, $\mathcal{S}=5$, ${\theta }_{\infty }=0.4$, and $\mathrm{Da}=5\times {10}^{-3}$ for consistency with previous studies [10] which use properties for aqueous ${\mathrm{NH}}_4\mathrm{Cl}$. The optimal solute flux is approximated by (7, 8), with $\gamma =0.03$ and $R_c=20$. (b) Stability curves tracing the variation of ${\lambda }_s$ (red points) and ${\lambda }_u$ (blue points) with $R_m$; all other parameters fixed. In I only the lower branch is stable, yielding no flow. In II only the upper branch is stable, yielding convection with chimneys. Hysteresis is observed with two steady states in III. Both no flow and chimney convection states are unstable in IV, yielding weak convection with no chimney.Reuse & Permissions

Figure 3

Comparison of mushy region profiles at a long-wavelength chimney spacing $\lambda =1.0>{\lambda }_{\mathcal{O}}$ (a),(b) with those at the optimal spacing $\lambda ={\lambda }_{\mathcal{O}}=0.46$ (c),(d). Here $\theta$ is shown by the color scale in (a),(c) with isotherms shown as solid black curves; $\varphi$ is shown by the color scale in (b),(d), with constant $\varphi$ contours as black curves, and Darcy velocity streamlines as magenta curves. Other parameters are as in Fig. 2.Reuse & Permissions