The Representation of Ф in Terms of Boundary Values
Let us consider a motion of the type treated in the foregoing sections and examine the question of finding Фр at a certain point (x, y, z) arbitrarily located in the flow field. For this purpose we turn to the reciprocal theorem (2-12), making Ф the velocity potential for the actual flow and
r лДзГ— Xi)2 + (у — Ух)2 + (z — Zx)2
Here (xi, yi, Z) is any other point, one on th; boundary for instance. See Fig. 2-1. By direct substitution in Laplace’s equation, it is easy to prove that
vV = v2
everywhere except in the immediate vicinity of r = 0, where the La – placian has an impulsive behavior corresponding to the local violation of the requirement of continuity.
Fig. 2-і. Arbitrary point P in liquid flow produced by general motion of an inner boundary S.
If we are to use the reciprocal theorem, which requires that the Laplacians of both members vanish, we must exclude the point P from the volume V. This we do by centering a small spherical surface a around the point, and we obtain
§i%ds + §*%^-§i’!ids + §t,!iAr- <2-20>
<f S-f-2 a
Inserting the value of Ф’, we rewrite this
If we let <7 become a very small sphere, then
da = r2 da, (2-22)
where dQ is an element of solid angle such that
Over the surface of a,
By the mean value theorem of integration, it is possible to replace the finite, continuous Ф by Фр in the vicinity of the point P to an acceptable degree of approximation. These considerations lead to
<r—*0 J J P ^/mean r—»0 J J
Moreover, although дФ/дп varies rapidly over <r, it will always be possible to find a bounded average value (дФ/дп)т(.ап such that
* = -§£s;ds + §*iQ*)ds-
In the limit as the outer boundary goes to infinity, with the liquid at rest there, one can show that the integrals over 2 vanish, leaving
All the manipulations in (2-27) and (2-28) on the right-hand sides should be carried out using the variables aq, уb zi. In particular, the normal derivative is expressed in terms of these dummy variables.
The reader will be rewarded by a careful examination of some of the deductions from these results that appear in Sections 57 and 58 of Lamb (1945). For instance, by imagining an artificial fluid motion which goes on in the interior of the bounding surface S, it is possible to reexpress (2-28) entirely in terms of either the boundary values of Ф or its normal derivative. Thus, the determinate nature of the problem when one or the other of these quantities is given from the boundary conditions becomes evident.
The quantities —1/47гг and д/дп(1/4тгг), which appear in the integrands of (2-27) and (2-28), are fundamental solutions of Laplace’s equation that play the role of Green’s functions in the representation of the velocity potential. Their names and physical significances are as follows.
The Point Source
where r may be regarded as the radial coordinate in a set of spherical coordinates having the origin at the center of the source. The radial velocity component is whereas the other velocity components vanish. Evidently we have a spherically symmetric outflow with radial streamlines. The volume efflux from the center is easily shown to equal unity. The equipotential surfaces are concentric spheres.
A negative source is referred to as a point sink and has symmetric inflow. When the strength or efflux of the source is different from unity, a strength factor H [dimensions (length)3 (time)-1] is normally applied in (2-29) and (2-30).
2. The Doublet. The derivative of a source in any arbitrary direction s is called a doublet,
In particular, a doublet centered at the point (aq, ylt zi) and oriented in the г-direction would have the velocity potential
By examining the physical interpretation of the directional derivative, we see that a doublet may be regarded as a source-sink pair of equal strengths, with the line between them oriented in the direction of the doublet’s axis, and carried to the limit of infinitesimal separation between them. As this limit is taken, the individual strengths of the source and sink must be allowed to increase in inverse proportion to the separation.
Sources and doublets have familiar two-dimensional counterparts, whose potentials can be constructed by appropriate superposition of the threedimensional singular solutions. There also exist more complicated and more highly singular solutions of Laplace’s equation, which are obtained by taking additional directional derivatives. One involving two differentiations is known as a quadrupole, one with three differentiations an octu – pole, and so forth.