$3-1$ A Ge$-$Ga crystal is it$-$own by normal freezing. Initial melt composition is $10$ ppm Ga$.$ Growth rate is $8$ x $10-3$ cm$/$s$.$ Assume k $=0\mathrm{.}1,$ ml$.=-4$C$/\%,$ DL $=5$ x $10^-5$ cm$2/$s$.$

$($a$)$ If convection is completely absent$,$ what thermal gradient is required to maintain a plane front when the ingot is $50$ percent solidified?

$($b$)$ If convection is sufficiently vigorous that k$\text{'}=$ k$,$ what thermal gradient is required to maintain a plane front when the ingot is $50$ percent solidified?

$($c$)$ If S $=0\mathrm{.}005$ cm $($S$=$delta$),$ what thermal gradient is required to maintain a plane front when the ingot is $50$ percent solidified?