Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 1

For a separated harmonic function $R(r)\Theta (\theta )$ in polar coordinates, single-valuedness forces integer angular frequencies. The radial factors for frequency $m\ge 1$ are $r^m,r^{-m}$, while frequency zero has a constant and $\ln r$. Finiteness at the origin excludes $r^{-m}$ and $\ln r$.

The source's angular factor is $\cos \theta$ in every term, as printed, rather than $\cos (n\theta )$. The power-law angular mode for the planar Poisson equation givesHence, assuming the series permits termwise differentiation, the general finite, single-valued solution isThe arbitrary harmonic series must itself converge to a finite function on the prescribed disk. If ${\alpha }_0\ne 0$, the source has a direction-dependent limit at the origin; the displayed potential still extends finitely there, but the equation at that point is understood weakly rather than as a smooth classical source.

For the special source, a particular potential is $r^3\cos \theta /8$. Its boundary cosine term is canceled by $-a^{2}r\cos \theta /8$, and the boundary sine term is supplied by $(r/a{)}^2\sin 2\theta$. The resulting Dirichlet boundary conditions has solutionIt is finite at zero, has the required boundary values, and its Laplacian is $r\cos \theta$.

The far radial derivative selects the growing frequency-one term $Ur\cos \theta$. The matching decaying harmonic is $B\cos \theta /r$. The no-penetration condition gives ${\partial }_r\varphi =(U-B/r^2)\cos \theta =0$ at $r=a$, so $B=U{a}^2$. Thus the potential flow around a circular cylinder isThe remaining single-valued exterior harmonics are excluded by the zero radial derivative at the cylinder and the prescribed derivative at infinity, apart from the arbitrary additive constant. In particular the logarithmic radial term would have a nonzero radial derivative on the cylinder. A circulation term proportional to $\theta$ is excluded by single-valuedness of the potential.