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# Conformal Maps

As we saw in Section 6.7, conformal maps are extremely useful in the theory of two-dimensional, irrotational, incompressible flows. It turns out that such maps also have applications to the theory of axisymmetric, irrotational, incompressible flows. Consider the general coordinate transformation

 (7.79)

where is an analytic function. The Cauchy-Riemann relations (see Section 6.3) yield

 (7.80) (7.81)

It follows, from the previous two expressions, that . In other words, and are orthogonal coordinates in the meridian plane. [Incidentally, we are assuming that are a right-handed set of coordinates.] Furthermore, it can also be shown from the Cauchy-Riemann relations that

 (7.82)

where and . Writing the flow velocity in terms of a velocity potential, so that , or, alternatively, in terms of a Stokes stream function, so that , we get

 (7.83) (7.84)

Of course, writing the velocity field in terms of a Stokes stream function ensures that the field is incompressible, which also implies that . The additional requirement that the field be irrotational yields . Making use of the analysis of Appendix C, this requirement reduces to

 (7.85)

or

 (7.86)

Let represent the surface of an axisymmetric solid body moving with velocity through an incompressible irrotational fluid that is at rest a long way from the body. Let the fluid occupy the region , where far from the body. (See Figure 7.5.) Let be an angular coordinate such that on the positive -axis, and on the negative -axis. The fact that the fluid is at rest at infinity implies that asymptotes to a constant a long way from the body. Without loss of generality, we can chose this constant to be zero. Thus, one constraint on the system is that

 (7.87)

The appropriate constraint at the surface of the body is that

 (7.88)

where . However, we can write , where . (See Section 7.6.) Hence, from Equation (7.83), the previous constraint becomes

 (7.89)

when . Integrating, making use of the constraint (7.87) (which implies that on the -axis, where is constant, by symmetry), we obtain

 (7.90)

We can also set the velocity potential, , to zero at infinity, and on the -axis.

The total kinetic energy of the fluid surrounding the moving body is

 (7.91)

where we have made use of the fact that . Here, is the fluid mass density, and is an element of the volume obtained by rotating the area , shown in Figure 7.5, about the -axis. Making use of the divergence theorem, we obtain

 (7.92)

where is an element of the curve , and is an outward pointing, unit, normal vector to the area . Here, we have made use of the fact that the velocity potential is zero at infinity (i.e., on ), and also on the -axis (i.e., on and ). On the curve , we can write . Furthermore, it follows from Equations (7.82) and (7.83) that , and . Thus,

 (7.93)

or

 (7.94)

As a simple example, consider the conformal map

 (7.95)

where is real and positive. It follows that

 (7.96) (7.97)

which implies that

 (7.98)

where

 (7.99)

Thus, the constant- surfaces are concentric spheres of radius . If we set

 (7.100)

then the problem reduces to that of a sphere, of radius , moving through a fluid that is at rest at infinity. This problem was solved, via different methods, in Section 7.10. The constraints (7.87) and (7.90) yield

 (7.101) (7.102)

where use has been made of Equation (7.97). This suggests that we can write

 (7.103)

Substitution into the governing equation, (7.86), gives

 (7.104)

whose most general solution is

 (7.105)

The constraints (7.101) and (7.102) yield

 (7.106) (7.107)

respectively. Thus, we obtain

 (7.108)

Now, from Equations (7.84) and (7.97),

 (7.109)

which can be integrated to give

 (7.110)

Note that the previous expression is formally the same as expression (7.63), as long as we make the identifications , , and .

On the surface of the sphere, , we obtain

 (7.111) (7.112)

Thus,

 (7.113)

As is clear from the analysis of Section 7.10, the sphere's added mass can be written

 (7.114)

Hence, we arrive at the standard result that the added mass is half the displaced mass [i.e., half of ].

Next: Flow Around a Submerged Up: Axisymmetric Incompressible Inviscid Flow Previous: Motion of a Submerged
Richard Fitzpatrick 2016-03-31