On the existence and determination of the incompressible laminar flow located in the polar plane of a porous annular pipe

Abstract

Despite the extensive research already done on laminar flows with porous boundaries, polar flows bounded by cylinders have not yet been studied. This paper investigates the polar laminar flow for an incompressible fluid located in a porous annular pipe and driven by suction-injection at the walls. The fluid being confined between the cylinders with zero axial velocity, it is proven that the flow takes place in the polar plane with conservation of mass if an incoming flow exists in the same plane to compensate the mass of fluid extorted by suction. So, one of the cylinders undergoes the suction, and the other the injection. Suitable boundary conditions for both cylinders are then found. The problem depends on the Reynolds number, the pipe gap ratio, and the pore density and surface ratios. The method of solution utilizes the shooting technique including the Runge-Kutta and Newton-Raphson algorithms. Radial flows are found as a possible solution. When the flow is not radial, the patterns of the acceptable streamlines highlight a particular zone in which the fluid is at rest, bounded by two singular streamlines and the downstream cylinder. The flow velocity is determined. Physical understandings of the flow are derived.

Keywords

• Polar laminar flows in porous annular pipes
• Conservation of mass with incoming and outcoming flow rates
• Vorticity equation in polar coordinates
• Similarity-solutions method
• Numerical shooting technique
• Optimisation by Newton-Raphson algorithm