= Solution
For uniform <specific entropy>, the <material conservation of cross-helicity density> criterion is
$$
\frac{Dh_c}{Dt}=\mathbf B\cdot\nabla\left(\frac{u^2}{2}-h-\Phi\right)-h_c\nabla\cdot\mathbf u.
$$
Thus the requested condition is
$$
\boxed{\mathbf B\cdot\nabla\left(\frac{u^2}{2}-h-\Phi\right)=(\mathbf u\cdot\mathbf B)\nabla\cdot\mathbf u.}
$$
In the <steady state>, <mass conservation> gives $\nabla\cdot\mathbf u=-\mathbf u\cdot\nabla\log\rho$. An equivalent form, expressed entirely in the flow and thermodynamic variables, is
$$
\boxed{\mathbf B\cdot\nabla\left(\frac{\gamma p}{(\gamma-1)\rho}+\Phi-\frac{u^2}{2}\right)
=(\mathbf u\cdot\mathbf B)\mathbf u\cdot\nabla\log\rho.}
$$
If “isentropic” means only constant <specific entropy> along individual trajectories, rather than a <homentropic flow>, the general criterion instead retains $T\mathbf B\cdot\nabla s$ on the right-hand side of the <material derivative> equation. The distinction matters because a source-free <cross-helicity conservation law> need not make $h_c$ constant along each trajectory.
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