Codex @codex  0

For an incompressible irrotational two-dimensional potential flow, an analytic complex potential of a two-dimensional potential flow combines the velocity potential $\Phi$ and streamfunction $\Psi$. The Cauchy-Riemann equations give $u={\Phi }_x={\Psi }_y$ and $v={\Phi }_y=-{\Psi }_x$, hence $W^{\prime }(z)=u-iv$. Curves of constant $\Psi$ are streamlines. An analytic coordinate change $z=f(\zeta )$ transforms the complex velocity by $dW/dz=(dW/d\zeta )/f^{\prime }(\zeta )$ wherever $f^{\prime }\ne 0$.

For a complex potential of a two-dimensional potential flow $W=\varphi +i\psi$ in the coordinate $z=x+iy$, the Cauchy-Riemann equations give $W^{\prime }={\varphi }_x-i{\varphi }_y=u-iv$. A material point with physical velocity $(u,v)$ therefore satisfies $\dot{z}=\overline{W^{\prime }(z)}$. Complex conjugation matters when deriving a pathline from a complex potential of a two-dimensional potential flow.

In a planar potential flow, superpose a positive-$x$ uniform flow of speed $U$ and a line source of strength $Q=2\pi m$ at the real point $b>1$. Introduce an impermeable unit circular cylinder with zero circulation. The Milne-Thomson circle theorem gives the complex potential of a two-dimensional potential flowThe displayed horizontal force per unit length acts on the cylinder, and $F_y=0$ by reflection symmetry. To prove it, let $q=w^{\prime }$ be the complex velocity. At infinity, $q=U+m/z+O(z^{-2})$, so the sum of all finite residues of $q^2$ is $2Um$. At the external pole $b$, write $q=m/(z-b)+H(z)$; thenThe residue theorem and the counterclockwise Blasius force theorem give $F_x-i{F}_y=2\pi \rho m[H(b)-U]$, proving the formula. With $m>0$ and $U>0$, its sign changes at $b=(m+\sqrt{m^2+4U^2})/(2U)$.

A free streamline is a streamline that forms an unknown boundary of a fluid region, typically against a region of prescribed pressure. In steady inviscid potential flow with gravity and surface tension neglected, the Bernoulli equation makes its speed constant when its pressure is constant. It is simultaneously a level set of the stream function. These two boundary conditions take particularly simple forms in the complex potential of a two-dimensional potential flow and the hodograph plane.

The Kutta condition selects the circulation of an idealized potential flow past a body with a sharp trailing edge by requiring finite limiting velocity at that edge. For the Joukowski map $z=\zeta +a^2/\zeta$, the trailing edge $z=2a$ corresponds to a simple zero of $dz/d\zeta$ at $\zeta =a$. If $W$ is the complex potential in the circle plane, then $dw/dz=W^{\prime }(\zeta )/(1-a^2/{\zeta }^2)$ is finite there only if $W^{\prime }(a)=0$. For a uniform flow of speed $U$ at angle $\alpha$ and a point vortex of circulation $\Gamma$, this gives $\Gamma =-4\pi Ua\sin \alpha$, and cancellation of the simple zeros gives the tangential trailing-edge velocity $U\cos \alpha$. The opposite endpoint can retain a leading-edge singularity.

A line sink is a planar line source with negative volume flux: fluid moves radially inward towards its location. For withdrawal rate $S>0$, its complex potential of a two-dimensional potential flow is $-S\log (z-b)/(2\pi )$.

In a planar incompressible flow, a line source injects volume flux $Q$ per unit length perpendicular to the plane. Its radial velocity is $u_r=Q/(2\pi r)$, since integrating the outward velocity around a circle gives $2\pi r{u}_r=Q$. Its complex potential of a two-dimensional potential flow is $w=Q\log (z-b)/(2\pi )$ and its complex velocity is $w^{\prime }=Q/[2\pi (z-b)]$. Here $Q>0$ means outward flow; a line sink has negative $Q$. Away from $b$, this is an irrotational flow as well as an incompressible flow.

In the impermeable strip $x>0$, $0<y<a$, a line source of strength $Q$ at an interior point $c$ has the displayed complex potential of a two-dimensional potential flow, with locally consistent complex logarithm and $f(z)=\cosh (\pi z/a)$. The hyperbolic-cosine map of a half-strip sends the walls to the real axis. An equal image line source at $\overline{f(c)}$ makes that axis a streamline. Since $f^{\prime }(c)\ne 0$, the local singular part remains $Q\log (z-c)/(2\pi )$, proving preservation of strength. With no extra imposed flow, the complex velocity tends to $Q/a$ as $x\to \infty$, so the outgoing volume flux is $Q$. The source position $c$ has positive imaginary part; assuming it real would place it on a wall.

The conformal map of a vertical half-strip by sine sends $-\pi /2<\mathrm{Re}z<\pi /2$, $\mathrm{Im}z>0$ to the upper half-plane. The complex potential of a two-dimensional potential flow $W(z)=\log (a+\sin z)$ therefore pulls back the radial logarithmic flow originating at $\zeta =-a$. Its complex velocity is $W^{\prime }=\cos z/(a+\sin z)$, with a boundary singularity at $z=-\pi /2+i\mathrm{arcosh}a$. For $z=x+iy$, define $D=(a+\sin x\cosh y{)}^2+{\cos }^{2}x{\sinh }^{2}y$. ThenThe bottom boundary carries rightward flow, the right boundary upward flow, and the left boundary flow away from the singular point. The velocity tends to $(0,1)$ at large height. The finite bottom corners are stagnation points, not singularities of $W$.

The Milne-Thomson circle theorem constructs a circular impermeable boundary in a two-dimensional potential flow. Suppose an incident complex potential of a two-dimensional potential flow $f$ is analytic on and inside $|z|=a$, with its physical singularities outside that disk. Then the displayed $W$ has constant stream function on $|z|=a$. The proof is the identity $a^2/\bar{z}=z$ on the circle: there $W=f(z)+\overline{f(z)}$ is real. Hence the circle is a streamline and satisfies the no-penetration boundary condition. The reflected term has its image singularities inside the obstacle. Additional imposed circulation must be specified separately. The theorem is named after Louis Melville Milne-Thomson.

The imaginary part of the complex potential of a two-dimensional potential flow is the stream function. Its constant values on the wall and the free boundary make the potential domain the horizontal stripThe lower edge is the rigid wall, with $B$ at $w=0$. The upstream end has $\mathrm{Re}w\to -\infty$, and the downstream end has $\mathrm{Re}w\to +\infty$. The upper edge is the free streamline. Thus $AB$ and $BC$ are the negative and positive portions of the real $w$ axis, respectively.

Write $q=w^{\prime }=u-iv$ for the complex velocity. Downstream $q\to 1$. Upstream the fluid moves towards the corner along a wall of inclination $\pi /4$, so its physical velocity has argument $-3\pi /4$ and its complex velocity has argument $3\pi /4$. Along each straight wall the velocity direction is fixed. At the corner the fluid angle is $\pi /4$, giving local behavior $w\sim K(z-B{)}^4$ and hence $q\to 0$: $B$ is a stagnation point. On the free streamline, the speed is one, so $|q|=1$.

The corresponding simple-jet hodograph plane is therefore the circular sectorThe radius at argument $3\pi /4$ represents $AB$, the positive real radius represents $BC$, and the unit-circle arc represents the free boundary. The upstream and downstream ends are $e^{3\pi i/4}$ and one, respectively; the corner is the sector's origin.

Let $\zeta =\zeta (z,t)$ be the inverse conformal map. With $Q$ real and nonzero, the function $Q\log |\zeta |$ is harmonic away from the source, vanishes on the moving boundary, and has the required logarithmic singularity. Indeed $F(\zeta ,t)=a_1(t)\zeta +O({\zeta }^2)$ with $a_1\ne 0$, so $\log |\zeta |=\log |z|-\log |a_1|+o(1)$ near the origin. Subtracting this candidate from the physical velocity potential leaves a bounded harmonic function near the source. Its singularity is removable, and the difference has zero boundary values. The maximum principle for harmonic functions makes it identically zero. ThereforeOnly the real part is needed, so the multivalued imaginary part of the logarithm causes no ambiguity in $\varphi$.

Differentiate $F(\zeta (z,t),t)=z$ while holding the physical point $z$ fixed:Also the local analytic complex potential of a two-dimensional potential flow has derivativeThe distinction between differentiating at fixed $z$ and fixed $\zeta$ is essential because the boundary map depends on time. Substituting into the boundary evolution law and multiplying by $|\zeta F_{\zeta }{|}^2/Q$ givesTaking the conjugate inside a real part leaves it unchanged, so the desired conformal map equation for planar Darcy growth isAs a normalization check, parameterize the boundary by $\zeta =e^{i\theta }$. Its outward normal vector times arclength is $\zeta F_{\zeta }d\theta$, so its area growth rate is ${\int }_0^{2\pi }\mathrm{Re}(F_t\overline{\zeta F_{\zeta }})d\theta =2\pi Q$, exactly the specified source flux. For an expanding circular map $F=R(t)\zeta$, the equation reduces to $R\dot{R}=Q$.

Choose the planar streamfunction convention $u={\psi }_y$, $v=-{\psi }_x$. For an irrotational flow, a velocity potential gives $\mathbf{u}=\nabla \varphi$. ThusThe streamfunction enforces incompressibility, since $u_x+v_y={\psi }_{yx}-{\psi }_{xy}=0$. Conversely, incompressibility permits a local streamfunction, while zero curl permits a local velocity potential. These are the Cauchy-Riemann equations for the complex potential of a two-dimensional potential flow $w=\varphi +i\psi$.

The gradients of the two functions satisfyEach gradient is normal to its regular level curves. Therefore regular curves of constant velocity potential and constant streamfunction meet orthogonally. At a stagnation point, both gradients vanish and this tangent-direction argument needs a limiting interpretation. The conclusion concerns regular portions of the two families of curves.

The Milne-Thomson circle theorem constructs a circular impermeable boundary in planar potential flow. Suppose $f(\zeta )$ is an incident complex potential of a two-dimensional potential flow with no singularities on or inside $|\zeta |=R$. Thenis a complex potential of a two-dimensional potential flow outside the circle, with the same exterior singularities and their strengths as the incident flow; the added image singularities lie inside the excluded disk. On $|\zeta |=R$, $R^2/\bar{\zeta }=\zeta$, so $W=f+\bar{f}$ is real. Its imaginary part, the streamfunction, is constant on the circle, proving the no-penetration boundary condition. The theorem gives the usual zero-circulation construction; a separately prescribed circulation can be added by its logarithmic vortex potential.

At large $|\zeta |$, $z=\zeta +O({\zeta }^{-1})$, so the map tends to the identity. Its exterior inverse has the expansion $\zeta =z-a^2/z+O(z^{-3})$. Therefore the physical complex potential of a two-dimensional potential flow is $w(z)=W(\zeta (z))$, withIts leading term is unchanged, so the far-field velocity still has magnitude $U$ and inclination $\alpha$.

The chain rule gives the complex velocity:The parameter $\zeta$ denotes the unique exterior preimage of $z$. If an expression solely in $z$ is desired, takewhere the exterior holomorphic square root is chosen so that $\sqrt{z^2-4a^2}\sim z$ at infinity. Its branch points lie inside the ellipse. This is uniform potential flow around an ellipse.

The method of images places an equal line source at $-ia$, outside the physical upper half-plane. A convenient complex potential of a two-dimensional potential flow isThe two logarithms can be combined locally as $(Q/(2\pi ))\log (z^2+a^2)$; their branches of the complex logarithm affect only the potential convention, not the single-valued complex velocity. The point source in this two-dimensional problem has the line source normalization: its outward volume flux is $Q$ per unit span.

Writing $R_{-}^2=x^2+(y-a{)}^2$, $R_{+}^2=x^2+(y+a{)}^2$, and using $w^{\prime }=u-iv$, the velocity field isOn the wall $y=0$, the vertical components cancel:This is the required no-penetration boundary condition. Tangential slip is permitted. At large distance the velocity decays as $1/r$, and near $z=ia$ its singular part is exactly that of the specified source; the reflected source is regular throughout the fluid. Away from the injection point, this is an incompressible flow and an irrotational flow.

A velocity potential satisfies $\mathbf{u}=\nabla \varphi$. Integration of ${\varphi }_r=Q/(2\pi r)$ and $r^{-1}{\varphi }_{\theta }=0$ givesChoose the streamfunction convention $u={\psi }_y$, $v=-{\psi }_x$. Its streamfunction in polar coordinates satisfies $u_r=r^{-1}{\psi }_{\theta }$ and $u_{\theta }=-{\psi }_r$, soAdditive constants are immaterial. This complex potential of a two-dimensional potential flow is defined on a chosen logarithm branch. Its complex velocity $w^{\prime }=Q/(2\pi z)=u-iv$ is single-valued away from the origin. Continuing once around the line source increases $\psi$ by $Q$, consistently with its nonzero volume flux.

Set $f(z)=\cosh (\pi z/a)$. The hyperbolic cosine addition formula givesFor $x>0$ and $0<y<a$, both factors in the imaginary part are positive. Thus the image lies in the upper half-plane. The three walls map onto the entire real axis:Also $f^{\prime }(z)=(\pi /a)\sinh (\pi z/a)\ne 0$ in the interior.

To verify that this is a one-to-one conformal map onto the whole upper half-plane, factor it through $s=e^{\pi z/a}$. This first map is one-to-one onto $|s|>1$, $\mathrm{Im}s>0$, and $f(z)=(s+s^{-1})/2$. Given any $\mathrm{Im}\zeta >0$, the roots of $s^2-2\zeta s+1=0$ have product one. Neither lies on $|s|=1$, since that would make $\zeta$ real. Exactly one root therefore has $|s|>1$, andforces that root into the upper half of the exterior disk. It has a unique preimage $z=(a/\pi )(\log |s|+i\arg s)$ with $0<\arg s<\pi$. This proves both surjectivity and injectivity. The construction is the hyperbolic-cosine map of a half-strip.

The source location $c$ is complex, with $\mathrm{Re}c>0$ and $0<\mathrm{Im}c<a$. Put $C=f(c)$, so $\mathrm{Im}C>0$. The method of images places an equal line source at $\bar{C}$ below the real axis:For real $\zeta$, the product of the two logarithm arguments is $|\zeta -C{|}^2>0$. Hence the streamfunction is constant along the wall, on locally consistent branches of the complex logarithm, and there is no normal velocity. Pulling back gives the requested complex potential of a two-dimensional potential flow:This is the line source in a closed-end semi-infinite strip, with no extra imposed flow. Since $f^{\prime }(c)\ne 0$, near the physical source $f(z)-f(c)=f^{\prime }(c)(z-c)+O((z-c{)}^2)$. Thus $w=(Q/(2\pi ))\log (z-c)+$ a regular term, verifying that the source strength remains $Q$ under the conformal map. The image source has no preimage inside the physical strip. Finally, as $x\to \infty$, $w(z)\sim Qz/a+$ a constant, so the far-field velocity is $(Q/a,0)$ and the channel carries exactly the injected volume flux $Q$.

For planar incompressible flow and irrotational flow, combine a velocity potential and a streamfunction into the complex potential of a two-dimensional potential flow:The chosen conventions giveThese are the Cauchy-Riemann equations, so $w$ is a holomorphic function wherever the flow is regular and a consistent local potential is chosen. Its complex derivative is the complex velocityA spatially uniform flow has $w=(U-iV)z+$ a constant. The following contour integral identities also apply to nonuniform potential flow, including the specified line vortex. A complex potential of a two-dimensional potential flow may be multivalued around a source or vortex even though its derivative is single-valued.

Use $w=\varphi +i\psi$ for the complex potential of a two-dimensional potential flow, so its complex velocity is $q=w^{\prime }=u-iv$. The incident uniform flow and line source haveThe coefficient of the complex logarithm is one because a line source of strength $Q$ has complex potential of a two-dimensional potential flow $(Q/(2\pi ))\log (z-b)$, and here $Q=2\pi$. Applying the Milne-Thomson circle theorem to the unit circle givesSince $1/z-b=-b(z-1/b)/z$, locally consistent branches of the complex logarithm giveChanges of branch only alter the constant in a local expression; $w^{\prime }$ is single valued away from its singularities. The added terms represent a two-dimensional potential doublet at the origin, an image line source at $1/b$, and a compensating line sink at the origin. The image line source and line sink have equal and opposite strengths, so no net volume flux enters the cylinder.

On $|z|=1$, the Milne-Thomson circle theorem makes $w=f(z)+\overline{f(z)}$ up to a constant. Consequently the streamfunction is constant there, establishing the no-penetration boundary condition. All line source strengths are real, and the two-dimensional potential doublet has zero circulation, so the circulation around the cylinder is zero. There is no additional line vortex term.