What is Stokes Paradox?

Answer 1

Stokes is the father of the modern fluid mechanics. He demonstrated the differential equations used nowdays that describes the fluid flow either internal flow (flow inside piepes of any shapes with or without obstacles) or external flows (flow arround veicles, falling objects etc...).

Stokes himself used his equations to describe some of the simplest cases of flow. One of them is the external flow arround an infinite cylinder such that in an infinite distance of it the flow is uniform and the direction is perpendicular to the axis of the cylinder. Stokes stated that the flow would be a 2-D flow and the fluid would be very viscous and the flow very slow. Although this situation is likelly to exist (approxmiatly) his equations stated that that such a situation will never occur.

Stokes himself stated that this paradox has happened because such a flow would never be a steady state flow. It would be a permanently transient flow. However, he has not been able to present a transient solution for the problem.

In a more recent work, Rayleigh studied the situation and stated that it is not possible to neglect the effect the inertial forces that were neglected by Stokes whan he stated that the fluid was very viscous and the flow very slow.

In an even more recent work, Villas Boas showed that the paradox happened because the problem was not well posed. It is not possible to state that something is infinitly distant from a cylinder of infinite height. So that one must say that the flow is uniform at a finite distance to the cylinder - than the problem has a solution - or to state that the heigh of the cylinder is finite. In this case, it is possible to state that the flow is uniform at an infinite distance of the cylinder. But the flow is no longer 2-D. Using 3-D analysis tools, it is possible to describe the flow past an short cylinder as showed by Villas Boas. His work has been published in http://congreso.pucp.edu.pe/cibim8/pdf/16/16-34.pdf .