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# Point Sources

Consider a point source, coincident with the origin, that emits fluid isotropically at the steady rate of volumes per unit time. By symmetry, we expect the associated steady flow pattern to be isotropic, and everywhere directed radially away from the source. In other words,

 (7.29)

where is a spherical coordinate. Consider a spherical surface of radius whose center coincides with the source. In a steady state, the rate at which fluid crosses this surface must be equal to the rate at which the source emits fluid. Hence,

 (7.30)

which implies that

 (7.31)

Of course, .

According to Equations (7.4), the Stokes stream function associated with a point source at the origin is such that , and is obtained by integrating

 (7.32)

It follows that

 (7.33)

It is clear, from a comparison of Equations (7.10) and (7.33), that the previously specified flow pattern is irrotational. Hence, this pattern can also be derived from a velocity potential. In fact, by symmetry, we expect that . The potential itself is obtained by integrating

 (7.34)

It follows that

 (7.35)

Next: Dipole Point Sources Up: Axisymmetric Incompressible Inviscid Flow Previous: Uniform Flow
Richard Fitzpatrick 2016-03-31