Effect of Low Rotation Rate on Steady Convection During the Solidification of a Ternary Alloy

Abstract

We consider the problem of steady convective flow during the directional solidification of a horizontal ternary alloy system rotating at a constant and low rate about a vertical axis. Under the limit of large far-field temperature, the flow region is modeled to be composed of two horizontal mushy layers, which are referred to here as a primary layer over a secondary layer. We first determine the basic state solution and then carry out linear stability analysis to calculate the neutral stability boundary and the critical conditions at the onset of motion. We find, in particular, that there are two flow solutions and each solution exhibits two neutral stability boundaries, and the flow can be multi-modal in the low rotating rate case with local minima on each neutral boundary. The critical Rayleigh number and the wave number as well as the vertical volume flux increase with the rotation rate, but the flow is found to be less stabilizing as compared to the binary alloy counterpart flow. The effects of low rotation rate increase the solid fraction and the liquid fraction at certain vertically oriented fluid lines, and the highest value of such increase is at a horizontal level close to the interface between the two mushy layers.

Appendix

The coefficients $b_{m} (m=1, {\ldots }, 8)$ introduced in (8) satisfy the following set of linear algebraic equations:

$$\begin{aligned}&b_{1}{\exp }[(\gamma )^{0.5} \; l]+b_{2}{\exp }[-(\gamma )^{0.5} \; l]+b_{3}{ exp}[(\gamma _{0})^{0.5} \; l]\nonumber \\&\qquad + b_{4}{\exp }[-(\gamma _{0})^{0.5} \; l]=0, \end{aligned}$$

(14a)

$$\begin{aligned}&[{\alpha }^{2} \; R_{p0})]b_{5}-(\tau ^{2}+\gamma -\alpha ^{2})b_{1} =0, \end{aligned}$$

(14b)

$$\begin{aligned}&[{\alpha }^{2} \; R_{p0})]b_{6} -(\tau ^{2}+\gamma -\alpha ^{2})b_{2}=0, \end{aligned}$$

(14c)

$$\begin{aligned}&[{\alpha }^{2} \; R_{p0})]b_{7} -(\tau ^{2}+\gamma _{0}-\alpha ^{2})b_{3} =0, \end{aligned}$$

(14d)

$$\begin{aligned}&[{\alpha }^{2} \; R_{p0})]b_{8} -(\tau ^{2}+\gamma _{0}-\alpha ^{2})b_{4} =0, \end{aligned}$$

(14e)

$$\begin{aligned}&b_{5}{\exp }[(\gamma )^{0.5} ]+ b_{6}{\exp }[-(\gamma )^{0.5} ]+b_{7}{\exp }[(\gamma _{0})^{0.5}]+b_{8}{\exp }[-(\gamma _{0})^{0.5}]=0, \end{aligned}$$

(14f)

$$\begin{aligned}&b_{5}{\exp }[(\gamma )^{0.5} \; l]+b_{6}{\exp }[-(\gamma )^{0.5} \; l]+b_{7}{\exp }[(\gamma _{0})^{0.5} \; l]\nonumber \\&\qquad +b_{8}{\exp }[-(\gamma _{0})^{0.5} \; l]=0. \end{aligned}$$

(14g)

The coefficients $b_{m}\,(m=9, {\ldots }, 16)$ introduced in (12) satisfy the following set of linear algebraic equations:

$$\begin{aligned}&b_{9}+b_{10}+b_{11}+b_{12} =0, \end{aligned}$$

(15a)

$$\begin{aligned}&[({\tau }^{2} +1)\gamma _{1}-\alpha ^{2}](b_{9}, b_{10}) -(R_{{s}0} \; \alpha ^{2})(b_{13}, b_{14})=0, \end{aligned}$$

(15b)

$$\begin{aligned}&[({\tau }^{2} +1)\gamma _{2}-\alpha ^{2}](b_{11}, b_{12})-(R_{{s}0} \; \alpha ^{2})(b_{15}, b_{16}) =0, \end{aligned}$$

(15c)

$$\begin{aligned}&B_{13}+b_{14}+b_{15}+b_{16} =0, \end{aligned}$$

(15d)

$$\begin{aligned}&B_{13} {\exp }[(\gamma _{1})^{0.5}]+b_{14} {\exp }[-(\gamma _{1})^{0.5}]+b_{15} {\exp }[(\gamma _{2})^{0.5}]\nonumber \\&\quad +b_{16} {\exp }[-(\gamma _{2})^{0.5}]=0, \end{aligned}$$

(15e)

$$\begin{aligned}&\quad -b_{1} {\exp }[(\gamma )^{0.5}]-b_{2} {\exp }[-(\gamma )^{0.5}]-b_{3} {\exp }[(\gamma _{0})^{0.5}]\nonumber \\&\quad -b_{4} {\exp }[-(\gamma _{0})^{0.5}]+b_{9} {\exp }[(\gamma _{1})^{0.5}]\nonumber \\&\quad +b_{10} {\exp }[-(\gamma _{1})^{0.5} ]+b_{11} {\exp }[(\gamma _{2})^{0.5}]+b_{12} {\exp }[-(\gamma _{2})^{0.5}]=0, \end{aligned}$$

(15f)

$$\begin{aligned}&\quad -(\gamma )^{0.5}\{ b_{5} {\exp }[(\gamma )^{0.5}]-b_{6} {\exp }[-(\gamma )^{0.5}]\}\nonumber \\&\quad -(\gamma _{0})^{0.5}\{b_{7} {\exp }[(\gamma _{0})^{0.5}]-b_{8} {\exp }[-(\gamma _{0})^{0.5}]\}\nonumber \\&\quad +(\gamma _{1})^{0.5} \{b_{13} {\exp }[(\gamma _{1})^{0.5}]-b_{14} {\exp }[-(\gamma _{1})^{0.5}]\}+(\gamma _{2})^{0.5}\{b_{15} {\exp }[(\gamma _{2})^{0.5}]\nonumber \\&\quad -b_{16} {\exp }[-(\gamma _{2})^{0.5}]\}=0. \end{aligned}$$

(15g)