that the angular behaviour at infinity is $\propto \sin\theta$ we make the standard assumption and seek solutions $$\psi_i =  g_i(r) \sin\theta\ \ {\rm for}\ \ i=0,1$$ The potential flow part (as usual) will be taken to be $$g_0 = 1 + B/r$$ for some constant $B$. Our analysis will differ from Stokes’ classic (Stokes, 1851) treatment in that $g_1$ does not satisfy the separated biharmonic equation. Instead we use the Helmholtz equation, which in separated form is just $$g_1''(r) +\frac{1}{r}g_1'(r)-\left(\epsilon^2+\frac{1}{r^2}\right)
   g_1(r)=0$$ where $\epsilon = \sqrt{\lambda}R_e^{\beta}$. The general solution to this ODE is given by $$g_1(r) = \alpha K_1(\epsilon r) + \beta I_1(\epsilon r)$$ where $I_1, K_1$ are standard “modified” Bessel functions. It is now quite clear that we can construct $g_1$ so as the preserve the boundary condition at infinity by setting $\beta = 0$. The $K_1$ function decays exponentially if $\epsilon>0$. We are left with two arbitrary constants that can be determined by satisfying the boundary conditions on the circle given by Eq. (\[circlebc\]). This is a matter of elementary algebra using some standard Bessel function identities. The final result for the total stream function can be simplified to: $$\psi = \left[r
-\biggl(1+
\frac{2
   K_1(\epsilon )}{\epsilon  K_0(\epsilon
   )}
\biggr) 
