ThreeDimensional Doublet
Consider a sink and source of equal but opposite strength located at points О and A, as sketched in Figure 6.2. The distance between the source and sink is l. Consider an arbitrary point P located a distance r from the sink and a distance rx from the source. From Equation (6.7), the velocity potential at P is
Let the source approach the sink as their strengths become infinite; that is, let
In the limit, as / —> 0, r — rx —> OB = l cosd, and rrx —»■ r2. Thus, in the limit, Equation (6.8) becomes
, X rrx
Ф = — hm————–
An rrx




where д = Л/. The flow field produced by Equation (6.9) is a threedimensional doublet’ д is defined as the strength of the doublet. Compare Equation (6.9) with its twodimensional counterpart given in Equation (3.88). Note that the threedimensional effects lead to an inverse rsquared variation and introduce a factor 47Г, versus 2тг for the twodimensional case.
From Equations (2.18) and (6.9), we find
The streamlines of this velocity field are sketched in Figure 6.3. Shown are the streamlines in the zr plane; they are the same in all the zr planes (i. e., for all values of Ф). Hence, the flow induced by the threedimensional doublet is a series of stream surfaces generated by revolving the streamlines in Figure 6.3 about the г axis. Compare these streamlines with the twodimensional case illustrated in Figure 3.18; they are qualitatively similar but quantitatively different.
Note that the flow in Figure 6.3 is independent of Ф; indeed, Equation (6.10) clearly shows that the velocity field depends only on r and 9. Such a flow is defined as axisymmetric flow. Once again, we have a flow with two independent variables. For this reason, axisymmetric flow is sometimes labeled “twodimensional” flow. However, it is quite different from the twodimensional planar flows discussed earlier. In reality, axisymmetric flow is a degenerate threedimensional flow, and it is somewhat misleading to refer to it as “twodimensional.” Mathematically, it has only two independent variables, but it exhibits some of the same physical characteristics as general threedimensional flows, such as the threedimensional relieving effect to be discussed later.

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