= Solution
Seek a steady downslope solution with $u>0$ constant. The <volume conservation> equation reduces to $h_x=E$, so the <constant-speed entraining gravity current on a slope> has
$$
\boxed{h=h_0+Ex.}
$$
The conserved <buoyancy flux> per unit width is $B=ug'h$, giving
$$
\boxed{g'(x)=\frac{B}{u(h_0+Ex)}.}
$$
Substitute these expressions into the momentum equation. Its advective term is $Eu^2$ and its hydrostatic term is $\tfrac12E(B/u)\cos\theta$. The balance is therefore
$$
Eu^2+\frac{EB}{2u}\cos\theta=\frac Bu\sin\theta-cu^2,
$$
which gives
$$
\boxed{u=\beta_*B^{1/3},\qquad\beta_*^3=\frac{\sin\theta-\tfrac12E\cos\theta}{c+E}.}
$$
The assumed slope inequality makes $\beta_*>0$. The increasing thickness dilutes <reduced gravity> while leaving $g'h$ and the <buoyancy flux> constant. The denominator includes both bottom drag and the momentum needed to accelerate newly entrained ambient fluid.
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