Axisymmetric MHD viscous flow about a solid sphere translating along the axis of a solid and motionless cylindrical tube

This work considers the translation of a solid sphere in a conducting Newtonian liquid, bounded by a cylindrical solid and motionless tube with a radius R, subject to a prescribed ambient magnetic field B. The sphere, with a radius a, has its center located on the tube axis which is parallel to both B and the sphere velocity. Assuming vanishing Reynolds and magnetic Reynolds numbers, the liquid flow about the sphere, axisymmetric and without swirl, obeys quasi-steady Stokes equations with a Lorentz body force. The stress arising on the sphere surface and the liquid flow are here obtained by truncating the fluid domain, solving coupled boundary-integral equations for the stress axial and radial components and using integral representations for the flow pressure and axial and radial velocity components. A boundary element method is employed to numerically get the drag exerted on the sphere and the flow about it. Both depend on the tube normalized radius R/a and the problem Hartmann number Ha = a/d, where d is the Hartmann layer thickness. The numerical implementation is presented and the computed drag and flow patterns are reported for some settings (R/a,Ha). It is found that, in contrast to the unbounded liquid case, the drag is weakly sensitive to Ha for small Ha and a region of reverse flow takes place near the tube boundary. Tables 3, Figs 5, Refs 14.