 Conical sink boundary layer

An axisymmetric sink inside a cone has outer speed $U=-A/x^2$. With local distance $x$ along the wall and normal distance $y$, leading continuity is $u_x+u/x+v_y=0$ because the circumference is proportional to $x$. Balancing convection with normal viscous diffusion gives the displayed thickness. With streamfunction $\Psi =-\sqrt{A\nu }{x}^{1/2}F(y/\delta )$, $xu={\Psi }_y$ and $xv=-{\Psi }_x$, the similarity equation is $F^{\prime \prime \prime }-F{F}^{\prime \prime }/2+2(1-F^{\prime 2})=0$, with $F(0)=F^{\prime }(0)=0$ and $F^{\prime }(\infty )=1$. The displacement-flux correction scales as $\sqrt{A\nu }{R}^{1/2}$, and the corresponding outer velocity correction as $\sqrt{A\nu }{R}^{-3/2}$.