Line vortex ($\mathbf{u}=\frac{\Gamma }{2\pi }\frac{{\mathbf{e}}_z\times (\mathbf{r}-{\mathbf{r}}_0)}{|\mathbf{r}-{\mathbf{r}}_0{|}^2}$)

An ideal straight line vortex in a two-dimensional incompressible flow has circulation $\Gamma$ and induces velocityIts stream function, with $\mathbf{u}=({\psi }_y,-{\psi }_x)$, is $\psi =-\Gamma \log |\mathbf{r}-{\mathbf{r}}_0|/(2\pi )$. Positive $\Gamma$ means counterclockwise circulation. In an ideal point-vortex description, the vortex position is advected by the regular velocity produced by the other vortices and the boundaries; its own singular velocity is excluded.

A vortex of circulation $\Gamma$ distance $d$ from a plane wall has an opposite-sign image across the wall. The wall velocity has magnitude $|\Gamma |d/[\pi (d^2+y^2)]$, giving a pressure reduction $\rho {\Gamma }^2{d}^2/[2{\pi }^2(d^2+y^2{)}^2]$ by the Bernoulli equation. Integrating this deficit gives the displayed force per unit width, directed from the wall toward the fluid and vortex. The uniform ambient pressure is subtracted to isolate the finite vortex-induced force.

For a point vortex of circulation $\Gamma$ in the wedge $0<\theta <\pi /n$, with positive integer $n$, the conformal map $z\mapsto z^n$ sends the domain to the upper half-plane. An opposite image vortex gives the complex potentialRemove the singular self-induced velocity before setting $z=c$. The resulting complex velocity isThe finite first term follows from the expansion of $(z^n-c^n)/(z-c)$ at $c$. In polar coordinates this becomes $\dot{r}=n\Gamma \cot (n\theta )/(4\pi r)$ and $\dot{\theta }=-\Gamma /(4\pi r^2)$. Therefore $r\sin (n\theta )$ is constant along a vortex trajectory. The case $n=1$ is motion parallel to a plane wall, while $n=3$ describes a sixty-degree wedge.

Let $\zeta =g(z)$ be a conformal map, and let a point vortex of circulation $\Gamma$ lie at $z=c$. Suppose the mapped complex potential of a two-dimensional potential flow iswhere $R$ is a holomorphic function near $g(c)$ and represents the other point vortices, boundaries, and any imposed flow. The physical complex velocity is $w^{\prime }=g^{\prime }{W}^{\prime }$. The point vortex follows its regular velocity, found by removing $\Gamma /[2\pi i(z-c)]$ before taking the limit. A Taylor expansion proveswhich gives the title formula. The complex conjugate occurs because the complex velocity convention is $w^{\prime }=u-iv$, whereas $\dot{c}=u+iv$.

For a mapped upper half-plane with no imposed flow, the method of images gives $R(\zeta )=-\Gamma \log (\zeta -\overline{g(c)})/(2\pi i)$, henceThe first term is essential: excluding a mapped singularity and then transforming its regular velocity does not in general equal subtracting the physical singularity after transformation.

Take vortex strength to mean circulation and $J(x,y)=(-y,x)$. Each point vortex moves with the velocity induced by the other, so for $\mathbf{d}={\mathbf{x}}_1-{\mathbf{x}}_2$ the displayed equation holds. Since $\mathbf{d}\cdot J\mathbf{d}=0$, separation is constant; weighting the two velocities also gives $d({\kappa }_1{\mathbf{x}}_1+{\kappa }_2{\mathbf{x}}_2)/dt=0$. Equal strengths at opposite points of a circle of radius $a$ rotate with angular speed $\kappa /(4\pi a^2)$.

The method of images enforces impermeability at a plane wall by reflecting a line vortex across it and reversing its circulation. The two logarithmic stream functions cancel on the wall, which is consequently a streamline. For a vortex of circulation $\Gamma$ at $(0,b)$ above $y=0$, the image of circulation $-\Gamma$ at $(0,-b)$ induces velocity $(\Gamma /(4\pi b),0)$ at the real vortex. The signed drift parallel to the wall is therefore $\Gamma /(4\pi b)$.

In the half-plane $y>0$ with an impermeable wall, place a point vortex of circulation $\Gamma$ in the background velocity $(\kappa x,-\kappa y)$, where $\kappa >0$. The image vortex at a plane wall has circulation $-\Gamma$ at the reflected location and contributes horizontal velocity $\Gamma /(4\pi y)$. Excluding the vortex's own singular velocity gives the displayed motion equations.

Put $c=\Gamma /(4\pi )$. Their first integral is $\kappa xy+c\log y$, since its time derivative is $\kappa [(\kappa x+c/y)y-\kappa xy]+c(-\kappa )=0$. The explicit trajectory is $y=y_0{e}^{-\kappa t}$ and $x=e^{\kappa t}(x_0+ct/y_0)$. These formulas show how image-induced drift changes the hyperbolic trajectories of the background flow while preserving a modified stream-function-type invariant.

In the quadrant $x,y>0$, a line vortex of circulation $\Gamma$ at $(x,y)$ has three images: circulations $-\Gamma$ at $(-x,y)$ and $(x,-y)$, and $+\Gamma$ at $(-x,-y)$. Their induced velocity isIn polar coordinates, these equations give $\dot{r}=\Gamma \cot (2\theta )/(2\pi r)$ and $\dot{\theta }=-\Gamma /(4\pi r^2)$. Direct differentiation yields $\frac{d}{dt}(r\sin 2\theta )=0$, so the trajectory obeys $r\sin 2\theta =\text{constant}$.