Unsteady Stokes Flow through a Porous Pipe with Periodic Suction and Injection with Slip Conditions

Document Type : Research Paper


1 Department of Mathematics and Social Sciences, Sukkur IBA University, Airport Road Sukkur, 65200, Pakistan

2 Department of Mathematics (York Campus), Pennsylvania State University, York, PA 17403, USA


The problem of unsteady Stokes flow of certain Newtonian fluids in a circular pipe of uniform cross section is discussed. The pipe is uniformly porous. The unsteady Navier-Stokes equations for the system in cylindrical polar coordinates have been solved analytically to obtain a complete description of the flow. The solution of the flow equations subject to the slip boundary conditions leads to the detailed expressions for axial and radial components of velocity and the pressure distribution depending on position coordinates and time. As a special case we have presented the situation when no-slip boundary conditions are implemented. The velocity profile is analyzed for different values of the flow parameters like Womersley number, slip length and time.


Main Subjects

[1] Berman, A. S., Laminar flow in channels with porous walls, Journal of Applied Physics, 24(9), 1953, 1232–1235.
[2] Kirubhashankar, C. K., Ganesh, S., Ismail, A. M., An exact solution of the problem of unsteady MHD flow through parallel porous plates, International Journal of Advances in Mechanical & Automobile Engineering, 1, 2014, 39–42.
[3] Khaled, A.-R. A., Wafai, K., The effect of slip condition on Stokes and Couette flows due to an oscillating wall: Exact solutions, International Journal of Non-Linear Mechanics, 39(1), 2004, 795-809.
[4] Bano, Z., Bhatti, K., Siddiqui, A. M., Unsteady incompressible Stokes flow through porous pipe of uniform circular cross section with periodic suction and injection, Sukkur IBA Journal of Computing and Mathematical Sciences, 1(1), 2017, 13–22.
[5] Narasimhan, M. N. L., Laminar non-Newtonian flow in a porous pipe, Applied Scientific Research, 10, 1961, 393-409.
[6] Ganesh, S., Krishnambal, S., Unsteady Stokes flow of viscous fluid between two parallel porous plates, Journal on Information Sciences and Computing, 1(1), 2007, 63–66.
[7] Zaman, H., Hall effects on the unsteady incompressible MHD fluid flow with slip conditions and porous walls. Applied Mathematics and Physics, 1(2), 2013, 31–38.
[8] Tsangaris, S., Kondaxakis, D., Vlachakis, N. W., Exact solution for flow in a porous pipe with unsteady wall suction and/or injection, Communications in Nonlinear Science and Numerical Simulation, 12, 2007, 1181-1189.
[9] Sidnawi, B., Santhanam S., Wu Q., Analytical and numerical study of pulsatile flow in a porous tube, Journal of Fluid Engineering, 141, 2019, 121205-1.
[10] Silvia, J. B., Bogdan, V., Derivation of Navier slip and slip length for viscous flows over a rough boundary, Physics of Fluids, 29, 2017, 057103.
[11] Womersley, J. R., Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, Journal of Physiology, 127, 1955, 553-563.