Potential flow

Potential flow is irrotational flow represented as the gradient of a scalar potential.

The region bounded by the dividing streamlines of a two-dimensional point source superposed on uniform flow. Its stagnation point is $-Q/(2\pi U)$ upstream and its far downstream width is $Q/U$. The boundary encloses fluid originating at the source rather than a rigid obstacle; the upstream stagnation point is reached only asymptotically by the upstream source trajectory.

A two-dimensional incompressible line source with area flux $Q$ has radial velocity $Q/(2\pi r)$, obtained from flux conservation around a circle. It is irrotational away from the source, with potential $(Q/(2\pi ))\ln r$. For a circular expanding interface, $Q=2\pi a{a}_t$ and $u_r=a{a}_t/r$, whose radial strain is $-a{a}_t/r^2$. That strain supplies a normal viscous stress even though the velocity is a potential flow.

A potential dipole in three dimensions is the potential flow $\mathbf{u}=\nabla \varphi$ with $\varphi =\mathbf{d}\cdot \mathbf{x}/r^3$. Away from the origin its potential is a harmonic function, so it is also a constant-pressure Stokes flow. Its velocity decays as $r^{-3}$ and its vorticity is zero. It is distinct from a stresslet, whose leading velocity decays as $r^{-2}$.

A two-dimensional point source of strength $q$ has velocity potential $\varphi =q\log r/(2\pi )$ and outward velocity $q{\mathbf{e}}_r/(2\pi r)$. Every closed contour enclosing it has volume flux $q$.

A line sink of strength $\alpha Q$ at the vertex of a sector $0<\theta <\alpha$ produces the irrotational outer flowIt satisfies no penetration at the radial walls but violates no slip there, so viscous boundary layers form along both walls.

Near a radial wall, let $x$ measure distance from the sink along the wall and $y$ point into the fluid. Withthe Prandtl boundary-layer equation reduces toOne physical solution has

For a sphere of radius $a$ translating at speed $U$ along the polar axis through fluid at rest at infinity, the laboratory-frame potential isIt gives $u_r=U{a}^3\cos \theta /r^3$ and $u_{\theta }=U{a}^3\sin \theta /(2r^3)$.

For radial flow outside a sphere, incompressible flow makesindependent of radius. Its velocity potential is $\varphi =-Q/(4\pi r)$.

A vacuum cavity of radius $a(t)$ in an infinite inviscid incompressible fluid of density $\rho$ and far-field pressure $p_0$ obeysIf $a(0)=a_0$ and $\dot{a}(0)=0$, its collapsing branch satisfies

Inviscid flow between closing hinged plates is a quadratic potential flow that expels fluid radially.