Steady, Viscous Flow within a Circular Boundary

The steady two‐dimensional flow of a viscous incompressible fluid within a circular vessel with prescribed wall velocity is studied both analytically and numerically in order to see how the flow pattern varies with the Reynolds number $R$ . First, an algorithm to determine the Stokes' type successive approximation for any stage of approximation from the preceding one, when the wall velocity is given as a Fourier series, is established and carried out on a digital computer to the eighth approximation (correct to $R^8$ ). Second, a finite‐difference method is applied to solve the Navier‐Stokes' equation numerically. Three typical cases of flow pattern are treated: flow without separation, flow consisting of two symmetric recirculating regions of semicircular form, and flow consisting of two unsymmetrical recirculating regions. It is found that the Stokes' type successive approximation is convergent for $R$ below 32. It is also found that for larger Reynolds number the flow pattern agrees well with Batchelor's uniform vorticity model.