= Constant-volume similarity in a V-shaped channel
{title2=$h=[3(X^2-x^2)/(7Ct)]_+^{1/3},\quad X\propto t^{2/7}$}
The symmetric <similarity solution> for a localized fixed volume $V$ in a <V-shaped channel gravity current> has $h(x,t)=[3(X^2-x^2)/(7Ct)]_+^{1/3}$. Integrating the similarity <conservation of mass> equation once gives $Ch^4h_x=-2xh^2/(7t)$, and vanishing thickness at the two noses gives the profile. Its volume fixes $X=[V/(I\tan\alpha)]^{3/7}(7Ct/3)^{2/7}$, where $I=\int_{-1}^1(1-s^2)^{2/3}\,ds$. At any fixed nonzero position, the thickness first rises after arrival and then falls; its maximum occurs when $X^2=7x^2/3$.
Back to article page
