Analysis of Laminar Flow in a Porous Pipe with Slip Velocity
Ashwini Bhat†, Nagaraj N. Katagi†*, N. M. Bujurke‡
‡Department of Mathematics, Karnatak University, Dharwad, Karnataka, India
ABSTRACT: In this manuscript, we present the semi-numerical solution for laminar flow in a porous pipe with velocity slip. The flow is analyzed by employing Computer extended series method(CES) and Homotopy analysis method(HAM). The primary objective is to study the influence of non-zero tangential slip velocity on the velocity field and shear stress. The convergence region of the obtained solutions are examined by Domb-Sykes plot and -curve. The validity of the series solution is further extended to a larger value of Reynolds number for different slip coefficients. Finally, we compared the results obtained by the proposed methods, and are presented in the form of graphs. The above methods admits desired accuracy and are validated with the available numerical results.
Keywords : Computer extended series; Homotopy analysis method; Reynold number; Domb-Sykes plot; curve; Velocity profiles; Pressure gradient.
 A. S. Berman, “Laminar flow in channles with porous walls,” Journal of Applied Physics, vol. 24, pp. 1232-1235, 1953.
 J. R. Sellars, “Laminar Flow in Channels with Porous Walls at High Suction Reynolds number,” J. appl. Phys., vol. 26, p. 489, 1955.
 S. Yuan and A. Finklestein, “Laminar pipe flow with injection and Suction through a porous wall,” Trans. ASME, vol. 78, p. 719, 1956.
 F. White, “Laminar flow in a uniformly porous tube,” Jl. Of Applied Mechanics Trans. ASME, p. 201, 1962.
 R. Terrill, “Laminar flow in a uniformly porous channel,” Aeronautical Quarterly, vol. 16, pp. 299-310, 1964.
 R. Terrill, “The flow between two parallel circular disks, one of which is subject to a normal sinusoidal oscillation,” ASME Trans. J. Lub. Tech., vol. 91, pp. 126-131, 1969.
 W. A. Robinson, “The existence of Multiple Solutions for the Laminar Flow in a Uniformly Porous Channel with Suction at Both Walls,” Journal of Engg. Math., vol. 10, pp. 23-40, 1976.
 J. F. Brady, “Flow development in a Porous Channel and Tube,” Phys. Fluids, vol. 27, pp. 1061-1067, 1984.
 M. B. Zaturska, P. G. Drazin and W. H. H. Banks, “On the flow of a viscous fluid driven along a channel by suction at porous walls,” Fluid Dynamics Research, vol. 4, p. 151, 1988.
 S. Gordon Beavers and D. Daniel Joseph, “Boundary conditions at a naturally permeable wall,” Journal of Fluid Mechanics, vol. 30, pp. 197-207, 1967.
 E. Sparrow, G. Beavers and L. Hung, “Channel and Tube flows with surface mass transfer and velocity slip,” Physics Fluids, vol. 14(7), p. 1312, 1971.
 R. Singh and L. Laurence Robert, “Influence of slip velocity at a membrane surface on ultrafiltration performance – II. Tube flow system,” Int. J. Heat and Mass Transfer, vol. 12, pp. 731-737, 1979.
 M. Van Dyke, “Analysis and improvement of Perturbation Series,” Q. Jl. Mech., vol. 27(4), pp. 423-450, 1974.
 M. Van Dyke, Perturbation methods in Fluid Mechanics, Parabolic Press, 1975.
 M. Van Dyke, “Computer Extended Series Method,” Ann Rev. Fluid Mech., vol. 16, pp. 287-309, 1984.
 N. M. Bujurke, T. J. Pedley and O. R. Tutty, “Comparison of Series Expansion and Finite-Difference Computations of Internal Laminar Flow Separation,” Phil. Trans. R. Soc. Lond. , vol. 354, pp. 1751-1773, 1996.
 N. M. Bujurke, N. N. Katagi and V. B. Awati, “Analysis of steady viscous flow in slender tubes,” Journal of Advanced Mathematics and Physics, vol. 56, pp. 836-851, 2005.
 S. J. Liao, “The proposed homotopy analysis technique for the solution of nonlinear problems,” 1992.
 S. Asghar, Z. Abbas, M. Mushtaq and T. Hayat, “Flow and Heat Transfer Analysis in a Deformable Channel,” Journal of Engineering Physics and Thermophysics, vol. 89, pp. 927-939, 2016.
 M. D. G. Rashidi and S. Dinarvand, “The homotopy analysis method for explicit analytical solutions of Jaulent- Miodek equations,” Numerical Methods for Partial Differential Equations, vol. 25, pp. 430-439, 2008.
 C. Domb and M. F. Sykes, “On the susceptibility of a ferromagnetic above the curic point,” Proc. Roy. Lond. Ser., vol. A 240, pp. 214-228, 1957.
 L. Schwartz, “Computer extension and analytic continuation of Stokes expansion for gravity waves,” Fluid Mech., vol. 62, p. 553, 1974.
 C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Mc Graw-Hill Book Co., 1987.
 S. J. Liao, Beyond Perturbation: Introduction to homotopy analysis method, Boca Raton: Chapman and Hall/ CRC Press, 2003.
 S. J. Liao, “The Homotopy Analysis Method in Nonlinear Differential Equations,” @Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, 2012.
 S. W. Yuan, “Further investigations of laminar flow in channelswith porous walls,” J. appl. Phys., vol. 27, p. 267, 1956.
 R. Singh and L. Laurence Robert, “Influence of slip velocity at a membrane surface on ultrafiltration performance – I. Channel flow system,” Int. J. Heat and Mass Transfer, vol. 22, pp. 721-729, 1979.
 S. A. Shehzad, T. Hayat and S. A. A. Asghar, “Stagnation Point Flow of Thixotropic Fluid over aStretching Sheet with Mass Transfer and Chemical Reaction,” Journal of Applied Fluid Mechanics, vol. 8(3), pp. 465-471, 2015.
 D. A. Nield, “The Beavers-Joseph Boundary Condition and Related Matters: A Historical and Critical Note,” Transport in Porous Media , vol. 78, pp. 537-540, 2009.
 Nayfeh, Introduction to Perturbation Techniques, John Wiley and Sons, New York, 1981.
 J. R. King and S. M. Cox, “Asymptotic Analysis of The Steady-State and Time-Dependent Berman Problem,” Journal of Engg. Math., vol. 39, pp. 87-130, 2001.
 S. Goldstein, Modern Developments in Fluid Dynamics, Oxford University Press, 1938.
 S. M. Cox, “Analysis of steady Flow in a channel with one Porous Wall, or with Accelerating Walls,” SIAM J. Appl. Math., vol. 51, pp. 429-438, 1991.
 A. Aziz and T. Y. Na, Perturbation Methods in Heat transfer, Hemisphere Publishing Corporation Springer Verlag, 1984.