Point source

A two-dimensional point source of strength

q

has velocity potential

ϕ = q lo g r / (2 π)

and outward velocity

q e_{r} / (2 π r)

. Every closed contour enclosing it has volume flux

q

.

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Line sink ( $Q < 0$ )

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 lo g (z - b) / (2 π)

.

Line source ( $w (z) = \frac{Q}{2 π} lo g (z - b)$ )

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 π r)

, since integrating the outward velocity around a circle gives

2 π r u_{r} = Q

. Its complex potential of a two-dimensional potential flow is

w = Q lo g (z - b) / (2 π)

and its complex velocity is

w^{'} = Q / [2 π (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.

Source dipole ( $u = [3 (M \cdot r) r / r^{5} - M / r^{3}] / (4 π)$ )

A source dipole is the limit of an equal source and sink brought together with their source-strength times separation vector fixed at

M

. Differentiating the three-dimensional point source field gives

u = - (M \cdot \nabla) [r / (4 π r^{3})]

. It has velocity potential

- M \cdot r / (4 π 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.

Three-dimensional point source ( $u = Q r / (4 π r^{3})$ )

A point injection of volume at rate

Q

produces the displayed radial potential flow in three dimensions. Its velocity potential is

- Q / (4 π 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.

Hydrodynamic attraction of a plane wall to a point source ( $F = ρ Q^{2} / (16 π a^{2})$ )

An incompressible point source of volume flux

Q

a

above an impermeable plane has velocity potential

ϕ = - Q (1/ r_{+} + 1/ r_{-}) / (4 π)

, with an equal image source below the plane. On the wall,

u_{s} = Q s / [2 π (s^{2} + a^{2})^{3/2}]

. The Bernoulli equation gives

p - p_{\infty} = - ρ Q^{2} s^{2} / [8 π^{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 π)

of radial inverse-square velocity, the force is

π ρ m^{2} / a^{2}

.

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Point source by

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A "point source" refers to a distinct, identifiable source of environmental pollution or emissions that can be pinpointed to a specific location. In various contexts, it may denote: 1. **Environmental Science**: A point source of pollution typically refers to contaminants that are discharged from a single, identifiable location such as a factory, sewage treatment plant, or a specific point along a river.

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