Solution
= Solution
With streamwise and normal <velocities> $u,v$, zero exterior <pressure> <gradient> gives the steady boundary-layer equations
$$
u_x+v_y=0,
\qquad uu_x+vu_y=\nu u_{yy}.
$$
At the organism,
$$
v(x,0)=0,
\qquad \rho\nu u_y(x,0)=Sx^{-1/2},
$$
and matching to the quiescent exterior requires $u(x,y)\to0$ as $y\to\infty$.