A magnesium alloy with $5wt\%$ aluminum is continuously solidified, for example in a

zone refining furnace. Temperature and composition in the liquid and growing solid

follow the red trace below on the phase diagram.

$($a$)$ Sketch the concentration profile from the solid across the interface into the liquid

phase.

$($b$)$ In the frame of reference of the moving interface, write the equation for aluminum

concentration in the liquid phase, assuming the liquid is quiescent $($no turbulence

etc.$)$ and aluminum diffusivity is uniform.

$($c$)$ If solidification rate is limited by aluminum diffusion away from the interface in

the liquid phase, is this interface stable or unstable? Briefly explain.