An Analytical Approximation for the Local Flow Between Counter-Rotating Cylinder Pair

Abstract

Flow around a pair of cylinders is of interest for various applications. Especially with the development of self-propelled, autonomous vehicles, this topic has gained further importance. There have been experimental, numerical and, to a much less extent, theoretical studies of this flow and its implications on the cylinders. However, the complexity of the problem does not allow a general solution, and case-by-case solutions have been produced. In this study, a theoretical approach was embraced, in which differential geometry is employed to model the local flow between a pair of counter-rotating cylinders. In order to obtain practical formulae for a steady, laminar, incompressible flow, Navier-Stokes equations were simplified and expressed in a new, parabolic coordinate system. Then, further simplifications due to the symmetrical nature of the problem were applied. Finally, boundary conditions were imposed while performing the integrals, and the desired equations for velocity and pressure were reached. Unlike the previous studies that are limited to very slow flows, the equations obtained in this study are applicable in the entire incompressible, laminar regime, albeit only between the cylinders. Using these equations, the effects of the rotation speed, cylinder spacing and cylinder radius were studied and presented graphically.