A singular perturbation solution of viscous incompressible fluid flow between two eccentric rotating cylinders

Abstract

The flow inside two concentric cylinders is one dimensional and an exact solution for quantities is easily found. However, when the cylinders axes are displaced by a small distance, two dimensional effects become obvious. In this research, the equations governing an incompressible viscous flow between two rotating cylinders are considered in polar coordinates that can be simplified by introducing vorticity and stream functions. By taking the curl of the vector form of the momentum equation, the pressure term is omitted. Because of the boundary conditions being in terms of perturbation parameter, a modified bi-polar coordinate system is introduced. This transforms the two eccentric cylinders into two concentric ones. By expanding the quantities in terms of the perturbation parameter up to second-order accuracy and by substitution into the vorticity-stream function, sets of differential equations are obtained to be solved for these functions. At the end, the closed-form velocity components are determined.