$($a$)$ Sketch a eutectic phase diagram for a binary system A$-$B$.$ Consider the idealized case of

solidification with no diffusion in the solid and diffusional mixing in the liquid for a one$-$

dimensional bar and an A$-$rich alloy where the concentration of B is below its solubility limit

at the eutectic temperature. Draw and explain the concentration profiles of B in the partially

$(50\%)$ und fully solidified bar. $(10)$