= Solution
In the heat-exchanger frame the crystal velocity is $\mathbf v_s=-V\mathbf e_z$, while the liquid velocity is $\mathbf v_l=\mathbf v_s+\mathbf u/(1-\varphi)$. The steady vertical mass flux is constant and equals $-\rho_sV$ in the fully solid eutectic material. Hence
$$
-V[\rho_s\varphi+\rho_l(1-\varphi)]+\rho_l u_z=-\rho_sV.
$$
This gives the <density-change flow in a pulled mushy layer>
$$
\boxed{\mathbf u=(1-r)V(1-\varphi)\mathbf e_z,\qquad \mathbf v_l=-rV\mathbf e_z.}
$$
The laboratory mixture volume flux is $-V[r+(1-r)\varphi]\mathbf e_z$. 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.
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