Finite-blade side-drainage pile ($h(0^{+},y)=[126({\alpha }^2-y^2){]}^{1/7}$)

A long blade with negligible under-blade leakage diverts incoming fluid laterally. A deep-pile profile $h^3=9[x_N(y)-x]$ and integrated flux conservation give $d(\int h^{4}dx)/dy=-12y$, hence $x_N=[126({\alpha }^2-y^2){]}^{3/7}/9$. This leading approximation excludes the nose and blade-tip transition regions.