Point source

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 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 upper half-plane, a line source of outward volume flux $Q$ at height $a$ above an impermeable wall has an equal image source below it. On the wall $u=Qx/[\pi (x^2+a^2)]$ and the normal velocity vanishes. The steady Bernoulli equation gives $p-p_{\infty }=-\rho Q^2{x}^2/[2{\pi }^2(x^2+a^2{)}^2]$. Sincethe ambient-relative force per unit length pulls the wall toward the source with the displayed magnitude. The torque about the source's projection onto the wall is zero in the symmetric Cauchy principal value sense. Individual one-sided moments diverge logarithmically. This planar result differs from the three-dimensional hydrodynamic attraction of a plane wall to a point source.

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.

A source dipole is the limit of an equal source and sink brought together with their source-strength times separation vector fixed at $\mathbf{M}$. Differentiating the three-dimensional point source field gives $\mathbf{u}=-(\mathbf{M}\cdot \nabla )[\mathbf{r}/(4\pi r^3)]$. It has velocity potential $-\mathbf{M}\cdot \mathbf{r}/(4\pi r^3)$, zero total volume flux, and $r^{-3}$ far-field decay. Unlike a stresslet, it is an irrotational potential singularity rather than a force-dipole flow.

A point injection of volume at rate $Q$ produces the displayed radial potential flow in three dimensions. Its velocity potential is $-Q/(4\pi r)$, and every enclosing sphere has outward volume flux $Q$. The divergence vanishes away from the source and is $Q$ times the three-dimensional Dirac delta distribution at it. The field is also a Stokes flow with constant pressure away from its singularity.

An incompressible point source of volume flux $Q$ at height $a$ above an impermeable plane has velocity potential $\varphi =-Q(1/r_{+}+1/r_{-})/(4\pi )$, with an equal image source below the plane. On the wall, $u_s=Qs/[2\pi (s^2+a^2{)}^{3/2}]$. The Bernoulli equation gives $p-p_{\infty }=-\rho Q^2{s}^2/[8{\pi }^2(s^2+a^2{)}^3]$. Integrating the pressure deficit over the plane gives the displayed attraction force relative to ambient pressure. If source strength instead denotes the coefficient $Q/(4\pi )$ of radial inverse-square velocity, the force is $\pi \rho m^2/a^2$.