7.4 Potential flow in axisymmetric systems in spherical coordinates [potential flow top]

Axisymmetric systems in spherical coordinates are distinct from plane potential flows in that (a) isopotential contours and streamlines are not generally orthogonal and (b) the use of complex mathematics does not facilitate calculations. For axisymmetric flows, axisymmetric multipolar solutions (see Appendix F) can be used.

Axisymmetric flows are described using the Stokes stream function initially presented in Chapter 1; it is defined by

(7.76)

and

(7.77)

The velocity potential is given by

(7.78)

and

(7.79)

Axisymmetric potential flows include uniform flows as well as multipolar solutions such as sinks, sources, and doublets. Combinations of these describe flow over Rankine solids, suchas spheres and ellipsoids. For example, the potential flow over a sphere is given by the sum of a uniform flow (which has a velocity potential of ϕ_v = Urcosϑ) with a doublet of strength (1∕2)Ua³ (which has a velocity potential of (1∕2) cosϑ) to obtain

(7.80)

One immediate conclusion from this solution is that the peak velocity of potential flow around a sphere is 3 2 that of the freestream. The same, of course, can be said of the magnitude of the electric field around a sphere. Equation 7.80 is important when considering the relative motion (electrophoresis) of charged spheres.

For complicated geometries, we typically solve the Laplace equation numerically. Fortunately, the Laplace equation is a well behaved, elliptical equation, and these numerical simulations tend to be straightforward.

Three-dimensional potential flow is physically identical to two-dimensional potential flow, and the general nature of the flow behavior is identical. While the solution is still a solution of the Laplace equation, the three-dimensional nature of the solution changes the relevant solution techniques as well as the interpretation of streamlines and potential isocontours.

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EXAMPLE PROBLEM 7.5: 

Consider 2D potential flow. In the domain -20 < x < 20, -20 < y < 20, plot the streamlines for a uniform flow in the x-direction flowing over a cylinder of radius 5 located at the origin.

Solution: See Figure 7.8.

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