Three-Dimensional Source

Return to Laplace’s equation written in spherical coordinates, as (3.43). Consider the velocity potential given by

given by Equation

C

ф =——

r

[6.1]

where C is a constant and r is the radial coordinate from the origin. Equation (6.1) satisfies Equation (3.43), and hence it describes a physically possible incompressible, irrotational three-dimensional flow. Combining Equation (6.1) with the definition of the gradient in spherical coordinates, Equation (2.18), we obtain

V = V0 =

rL

[6.2]

In terms of the velocity components, we have

C

V’ = ~ r1

[6.3a]

■ о

II

s?

[6.3b]

II

О

[6.3c]

Clearly, Equation (6.2), or Equations (6.3a to c), describes a flow with straight stream­lines emanating from the origin, as sketched in Figure 6.1. Moreover, from Equation

(6.2) or (6.3a), the velocity varies inversely as the square of the distance from the origin. Such a flow is defined as a three-dimensional source. Sometimes it is called

Подпись: Figure 6.1 Three-dimensional (point) source.

simply a point source, in contrast to the two-dimensional line source discussed in Section 3.10.

Подпись: Mass flow Подпись: pV-dS

To evaluate the constant C in Equation (6.3a), consider a sphere of radius r and surface S centered at the origin. From Equation (2.46), the mass flow across the surface of this sphere is

Подпись: A = Подпись: V • dS Подпись: [6.4]

Hence, the volume flow, denoted by A., is

On the surface of the sphere, the velocity is a constant value equal to Vr = С/г2 and is normal to the surface. Hence, Equation (6.4) becomes

Подпись: A =C 7

—г 4тсг2 = 4лС rz

Подпись:TT A

Hence, C = —

47Г

Three-Dimensional Source Подпись: [6.6]

Substituting Equation (6.5) into (6.3a), we find

Compare Equation (6.6) with its counterpart for a two-dimensional source given by Equation (3.62). Note that the three-dimensional effect is to cause an inverse r – squared variation and that the quantity 4л appears rather than 2л. Also, substituting

Подпись: X ф = A nr Подпись: [6.7]

Equation (6.5) into (6.1), we obtain, for a point source,

In the above equations, X is defined as the strength of the source. When X is a negative quantity, we have a point sink.

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