= Solution
Write $D/Dt=\partial_t+\mathbf u\cdot\nabla$ for the <material derivative> and $q=\mathbf u\cdot\mathbf B$ for the <cross-helicity> density, so $H_c=\int_V q\,dV$. The <ideal magnetohydrodynamic induction equation> of <ideal magnetohydrodynamics>, with $\nabla\cdot\mathbf B=0$, gives
$$
\frac{D\mathbf B}{Dt}=(\mathbf B\cdot\nabla)\mathbf u-\mathbf B\,\nabla\cdot\mathbf u.
$$
The <dot product> of the <Euler momentum equation> with the <magnetic field> has no <Lorentz force> contribution, because $\mathbf B\cdot[(\nabla\times\mathbf B)\times\mathbf B]=0$. Consequently,
$$
\frac{Dq}{Dt}=\mathbf B\cdot\nabla\left(\frac{u^2}{2}-\Phi\right)-\frac{\mathbf B\cdot\nabla p}{\rho}-q\,\nabla\cdot\mathbf u.
$$
Here $h=e+p/\rho$ is the <specific enthalpy>, $T$ the <temperature>, and $s$ the <specific entropy>; $e$ is the <internal energy> per unit mass. The <first law of thermodynamics> in this convention is $de=T\,ds-p\,d(1/\rho)$, so $dh=T\,ds+dp/\rho$. Substitution, followed by $\nabla\cdot\mathbf B=0$, yields the <cross-helicity conservation law>
$$
\partial_tq+\nabla\cdot\left[\mathbf u q+\left(h+\Phi-\frac{u^2}{2}\right)\mathbf B\right]=T\mathbf B\cdot\nabla s.
$$
The <vector triple product> identity $\mathbf u\times(\mathbf u\times\mathbf B)=\mathbf u q-u^2\mathbf B$ puts this in the equivalent form
$$
\boxed{\partial_t(\mathbf u\cdot\mathbf B)+\nabla\cdot\left[\mathbf u\times(\mathbf u\times\mathbf B)+\left(\frac{u^2}{2}+\Phi+h\right)\mathbf B\right]=T\mathbf B\cdot\nabla s.}
$$
For the fixed volume, the <divergence theorem> gives the exact <cross-helicity> balance
$$
\frac{dH_c}{dt}=\int_V T\mathbf B\cdot\nabla s\,dV-\int_{\partial V}\left[\mathbf u q+\left(h+\Phi-\frac{u^2}{2}\right)\mathbf B\right]\cdot\mathbf n\,dS.
$$
Thus \b[<cross-helicity> is constant when the net source equals the outward flux]. Simple sufficient hypotheses are $\mathbf B\cdot\nabla s=0$ throughout the volume and $\mathbf u\cdot\mathbf n=\mathbf B\cdot\mathbf n=0$ on its boundary. The first condition includes a <homentropic flow>; the second prevents both relevant boundary fluxes. <Periodic boundary conditions>, or sufficiently rapid decay at infinity, provide alternatives. These statements concern smooth <ideal magnetohydrodynamics>: material <specific entropy> conservation alone, $Ds/Dt=0$, does not eliminate spatial <specific entropy> gradients along the <magnetic field lines>.
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