 Solution

In the heat-exchanger frame the crystal velocity is ${\mathbf{v}}_s=-V{\mathbf{e}}_z$, while the liquid velocity is ${\mathbf{v}}_l={\mathbf{v}}_s+\mathbf{u}/(1-\phi )$. The steady vertical mass flux is constant and equals $-{\rho }_{s}V$ in the fully solid eutectic material. HenceThis gives the density-change flow in a pulled mushy layerThe laboratory mixture volume flux is $-V[r+(1-r)\phi ]{\mathbf{e}}_z$. In the pure liquid $\mathbf{u}=(1-r)V{\mathbf{e}}_z$, while in the fully solid region $\mathbf{u}=0$ and the material moves at $-V{\mathbf{e}}_z$. The liquid velocity formula applies wherever liquid exists; at solid fraction one there is no liquid phase to assign a velocity.