Faxens Law for a Body in an Unbounded Fluid

As a first application of the reciprocal theorem, we consider its use in calculating the hydrodynamic force on a body in an undisturbed flow, u00(x). 

In particular, let us suppose that we have obtained the solution of the creeping-motion equations for uniform flow U past a stationary body 3 D. Equivalently, we may consider the case of a particle that translates with velocity —U in a fluid at rest at infinity. We denote the solution of this problem as u and the corresponding surface-force vector on 3 D as f. Then, on applying the reciprocal theorem, we find that 

But this simple formula provides an example of the idea of something for nothing. For if we have actually solved the uniform-flow problem, we can immediately deduce the force on the same body held fixed in any undisturbed flow u°°(x) that satisfies the creeping-flow equations. In particular, 

Hence we can obtain the force F for the undisturbed flow u°°(x) by means of a simple integration of the right-hand side of (8-215), with f known from the solution of the uniform-flow problem but without any need to actually solve the creeping-flow problem involving u°°(x). To obtain all three components of F, we require a solution for uniform flow past the same body along any three mutually perpendicular directions. 

An especially powerful result is obtained if we apply the formula (8-215) to a stationary solid sphere of radius a in the undisturbed flow u00(x). In this case, the solution for uniform flow past a stationary solid sphere yields the general result 

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