Perturbation Approach to Stokes-Whitehead Paradox: Oseen Approximation
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The Stokes equation describes low-Reynold’s number flow dynamics. Consider an uniform flow past a sphere. Although the solution to the Stokes equation satisfies both surface and far-field boundary conditions, the next order approximation to the Navier-Stokes equation does not. This is the Whitehead paradox. Instead, we consider the low Reynold’s number as an asymptotic parameter, and look at a perturbation problem to steady state, incompressible Navier Stokes equation. The Stokes equation is the unperturbed equation in this problem. Considering flow near the surface of the sphere as a boundary layer, a second order uniform approximation can be made to resolve the Whitehead paradox. This is the Oseen approximation.