= V-shaped channel gravity current
{c}
{title2=$(h^2)_t=C(h^4h_x)_x$}
For a fixed half-angle $0<\alpha<\pi/2$, a slow slender <gravity current> in a V-shaped channel has an almost horizontal <free surface> across each section. Its area is $h^2\tan\alpha$, and its axial <volume flux> is $-gQ(\alpha)h^4h_x/\nu$. Here $Q$ is the integral of the dimensionless cross-sectional <Poisson equation> solution: $\Delta\phi=-1$, with zero velocity on the two rigid slopes and zero normal velocity derivative on the horizontal shear-free surface. <Conservation of mass> gives $C=gQ/(\nu\tan\alpha)$. This treats the full two-dimensional section and does not require a shallow V angle.
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