Infinite-blade gravity pile
= Infinite-blade gravity pile
{title2=$X(h)=x_0-x$}
The steady upstream film of an infinite blade has $h_x=-3(h-1)/h^3$. Thus $X(h)=h^3/9+h^2/6+h/3+\log(h-1)/3$. The head is fixed by under-blade leakage balancing the incoming film. The deep pile has a cubic profile, while the actual upstream transition has an exponential tail towards $h=1$.