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Let $x$ increase away from the tip along the wall and $y$ point normally into the fluid. Put $u=u_x$, $v=u_y$ for velocity components and $U(x)=-A/x^2$ for the outer tangential velocity. The local Cartesian momentum and pressure equations areFor a two-dimensional flat layer the usual continuity equation is ${\partial }_{x}u+{\partial }_{y}v=0$. For this axisymmetric cone, however, the circumference is proportional to $x$, so its leading continuity equation isThe circumference term is of the same order as ${\partial }_{x}u$ and cannot be omitted when deriving the cone's similarity equation. Conditions at the stationary wall are $u=v=0$; at the layer edge $u\to U$.

Balance $U^2/x\sim \nu |U|/{\delta }^2$. With $|U|=A/x^2$ this givesThe choice of multiplicative constant fixes the normalization of the similarity variable. To enforce axisymmetric continuity set $xu={\Psi }_y$, $xv=-{\Psi }_x$, withThenDifferentiation gives $u{\partial }_{x}u+v{\partial }_{y}u=-(A^2/x^5)[2F^{\prime 2}+F{F}^{\prime \prime }/2]$, whereas $U{U}^{\prime }=-2A^2/x^5$ and $\nu u_{yy}=-A^2{F}^{\prime \prime \prime }/x^5$. Consequently the conical sink boundary layer satisfiesThe first condition is no penetration, the second no slip, and the last matches the inward core flow. If one instead makes a strictly planar approximation and omits the circumference term, the streamfunction is $-\sqrt{A\nu }{x}^{-1/2}F$ and the equation becomes $F^{\prime \prime \prime }+F{F}^{\prime \prime }/2+2(1-F^{\prime 2})=0$. The thickness exponent is the same, but that planar equation is not the axisymmetric cone equation.

Across a cap near $R=x$, the inward flux in a layer truncated at $y=Y$ is, to leading order,To match to the core, use the finite displacement-flux deficit rather than integrate the uniform outer flow to infinite $Y$. If $I={\int }_0^{\infty }(1-F^{\prime })d\eta ={\lim }_{\eta \to \infty }(\eta -F)$, thenConservation of the fixed total extraction rate requires an extra core flux of this order. Dividing by cap area, proportional to $R^2$, provesThe correction is a relative $O(\sqrt{\nu R/A})$ contribution. Its angular structure requires outer matching; the requested radial power follows from the displacement flux alone. Its cap-average direction increases the inward core speed to compensate for the wall deficit.