The Steady Flow Around A Rotating Sphere with Variable Viscosity

The steady flow around a rotating sphere at large Reynolds numbers, up to 105, is obtained and analysed which has also been adapted with a temperature dependent viscosity. The Navier-Stokes problem is reduced to solving simpler PDE systems in distinct regions around the sphere where it is determined that the boundary layer approximation is once again an excellent model of the flow close to the sphere and far above the equator. Stewartson’s (1958) model of the equatorial flow is solved for the first time using a multi-grid method but it is found that this gives arise to unobserved behaviour compared to Calabretto et al (2019). A new model, based on Stewartson (1958), but amended with added azimuthal terms qualitatively describes the flow well compared to Calabretto et al (2019) and suggests that the flow in this area is more complicated than once thought. The radial jet is subsequently determined and displays the hallmark characteristics of jet flow however as the Reynolds number increases, the length of the jet slightly decreases. This is unexpected as one would expect the distance traversed by the jet to increase with Reynolds number, however this is suspected to be due to the position of the inlet but further tests are required for confirmation. The full flow has been compiled and is ready to be used in stability calculations or for any other use required for Reynolds numbers less than 105. Additionally, a temperature dependent viscosity has been successfully added to the steady flow where similar behaviour, with respect to the sensitivity parameter λ, is observed to that of the heated rotating disk flow examined by Miller et al (2020) and the heated sphere boundary layer by Molla & Hossain (2006); namely, that reducing outflow is favourable to the transfer of heat.