Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / K / 3

For an analytic complex potential $f$ whose singularities lie outside $|z|=a$, the circle theorem adds the reflected potentialThe added term is analytic outside the circle wherever the reflected original is regular; in particular no new exterior singularities occur when $f$ is regular inside the circle. On $|z|=a$, the relation $a^2/\bar{z}=z$ makes $F=f+\bar{f}$ real. Thus its imaginary part, the stream function, is constant on the circle and the normal velocity vanishes. This proves the theorem, with any harmless additive constant left free. Its usual decay condition at infinity selects the surrounding flow inherited from $f$.

Write $a_1=z_1$ for the dipole position to distinguish it from the circle radius, now one. Substitution givesThe image singularity is at $1/{\bar{a}}_1$, strictly inside the solid circle. This exterior potential has the original dipole and an impermeable cylinder boundary.

Across a straight impermeable wall the reflected potential is obtained by conjugating the potential after reflecting its argument. For the real axis the operation is $f(z)\mapsto \overline{f(\bar{z})}$; the sum is real on that axis. For the imaginary axis it is $f(z)\mapsto \overline{f(-\bar{z})}$. Applying both commuting reflections to the dipole producesThe other three poles are outside the first quadrant. The sum is invariant under each reflected-conjugation operation, hence real on each boundary axis. Both walls are therefore streamlines.

On the line $y=-x$ one has $\overline{iz}=z$. Hence the reflected term in the printed formula is $\overline{w(z)}$, and the total is real on the line. This proves that its stream function is constant there. The nested conjugation is important: it makes the added term analytic as a function of $z$.

For the planar potential dipole its image contribution isThusThe image is the reflected point $-i\bar{a}$, which lies in the opposite half-plane when the original dipole lies strictly inside the fluid half-plane. The wall itself is a limiting boundary, not an admissible position for the singularity.

For wall-induced motion of a planar potential dipole, the complex velocity convention is $F^{\prime }(z)=v_x-i{v}_y$. A singularity transported by the surrounding flow therefore satisfies $\dot{\bar{a}}=F_{\mathrm{reg}}^{\prime }(a)$, where its own term $m/(z-a)$ is removed before differentiating. The images at a given instant depend on that instantaneous position, but the spatial derivative treats their parameters as fixed.

For the circular wall, the derivative of the regular image is $\bar{m}/(1-\bar{a}z{)}^2$. For the quadrant, differentiate its three image terms. For the sloping wall, differentiate its single image term. This gives respectivelyThese are equations for $d{\bar{z}}_1/dt$, as printed. Conjugating gives equations for $d{z}_1/dt$ if desired; no self-induced singular velocity is included.