Journal of Applied Nonlinear Dynamics
Mathematical Model of Flow in a Doubly Constricted Permeable Channel with Effect of Slip Velocity
Journal of Applied Nonlinear Dynamics 8(4) (2019) 655666  DOI:10.5890/JAND.2019.12.010
P. Muthu, M. Varunkumar
Department of Mathematics, National Institute of Technology, Warangal506004, Telangana, India
Download Full Text PDF
Abstract
A mathematical model of steady laminar flow in a channel of varying cross section is studied under the effect of slip velocity at the permeable boundary. The fluid reabsorption at walls is taken care by the assumption of flow rate as a function of the axial coordinate at each cross section. An analytical solution of NavierStokes equations is determined by employing the perturbation technique. The graphical results are presented to illustrate the significance of slip velocity and various parameters on the velocity profiles, mean pressure drop, wall shear stress and stream function.
References

[1]  Apelblat, A., Katchasky, A.K., and Silberberg, A. (1974), A mathematical anaylsis of capillary tissue fluid exchange, Biorheology, 11, 149. 

[2]  Berman, A.S. (1953), Laminar flow in channels with porous walls, Journal of Applied Physics, 24, 12321235. 

[3]  Berman, A.S. (1958), Laminar flow in an annulus with porous walls, Journal of Applied Physics, 29, 7175. 

[4]  Sinha, A. and Mishra, J.C. (2012), Influence of slip velocity on blood flow through an artery with permeable wall: a theoretical study, International Journal of Biomathematics, 5(5), 1250042 120. 

[5]  Yuan, S.W. and Finkelstein, A.B. (1956), Laminar pipe flow with injection and suction through a porous wall, Trans. ASME, 78, 719724. 

[6]  Kelman, R.B. (1962), A theoretical note on exponential flow in the proximal part of the mammalian nephron, Bulletin of Mathematical Biophysics, 24, 303317. 

[7]  Macey, R.I. (1963), Pressure flow patterns in a cylinder with reabsorbing walls, Bulletin of Mathematical Biophysics, 25, 19. 

[8]  Macey, R.I. (1965), Hydrodynamics of renal tubule, Bulletin of Mathematical Biophysics, 27, 117124. 

[9]  Kozinski, A.A., Schmidt, F.P., and Lightfoot, E.N. (1970), Velocity profiles in porouswalled ducts, Industrial and Engineering Chemistry fundamentals, 9(3), 502505. 

[10]  Marshall, E.A. and Trowbridge, E.A. (1974), Flow of a Newtonian fluid through a permeable tube: The application to the proximal renal tubule, Bulletin of Mathematical Biophysics, 36, 457476. 

[11]  Palatt, J.P., Henry, S., and Roger, I.T. (1974), A hydrodynamical model of a permeable tubule, Journal of Theoretical Biology, 44, 287303. 

[12]  Salathe, E.P. and An, K.N., (1976), A mathematical analysis of fluid movement across capillary walls, Microvascular Research, 11, 123. 

[13]  Oka, S. and Murata, T. (1970), A theoretical study of the flow of blood in a capillary with permeable wall, Japan Journal of Applied Physics, 9(4), 345352. 

[14]  Mariamma, N.K. and Majhi, S.N., (2000), Flow of a Newtonian fluid in blood vessel with permeable wall  a theoretical model, Computers and Mathematics with Applicatons, 40, 14191432. 

[15]  Haroon, T., Siddiqui, A.M., and Shahzad, A. (2016), Stokes flow through a slit with periodic reabsorption: An application to renal tubule, Alexandria Engineering Journal, 55, 17991810. 

[16]  Radhakrisnamacharya, G., Peeyush, C., and Kaimal, M.R. (1981), A hydrodynamical study of the flow in renal tubules, Bulletin of Mathematical Biology, 43, 151163. 

[17]  Chaturani, P. and Ranganatha, T.R. (1991), Flow of Newtonian fluid in nonuniform tubes with variable wall permeability with application to flow in renal tubules, Acta Mechanica, 88, 1126. 

[18]  Muthu, P. and Tesfahun, B., (2010), Mathematical model of flow in renal tubules, International Journal of Applied Mathematics and Mechanics, 6, 94107. 

[19]  Nadeem, S. and Ijaz, S. (2015), Study of Radially VaryingMagnetic Field on Blood Flow through Catheterized Tapered Elastic Artery with Overlapping Stenosis, Communications in Theoretical Physics, 64, 537?46. 

[20]  Reza, H.A, Shahbazi, A.M., and Kiyasatfar, M. (2015), Mathematical modeling of unsteady blood flow through elastic tapered artery with overlapping stenosis, Journal of Brazilian Society Mechanical Sciences and Engineering, 37(2), 571578. 

[21]  Muthu, P. and Varunkumar, M. (2016), Flow in a channel with an overlapping constriction and permeability, International Journal of Fluid Mechanics Research, 43(2), 141160. 

[22]  Beavers, G.S. and Joshep, D.D. (1967), Boundary conditions at a naturally permeable wall, Journal Fluid Mechanics, 30, 197207. 

[23]  Saffman, P.G. (1971), On the Boundary Condition at the Surface of a Porous Medium, Studies in Mathematics, 50(2), 93101. 

[24]  Cox, B.J. and Hill, J.M. (2011), Flow through a circular tube with a permeable Navier slip boundary, Nanoscale Research Letters, 6, 389. 

[25]  Misra, J.C. and Shit, G.C. (2007), Role of slip velocity in blood flow through stenosed arteries: A non Newtonian model, Journal of Mechanics in Medicine and Biology, 7, 337353. 

[26]  Moustafa, E. (2004), Blood flow in capillary under starling hypothesis, Applied Mathematics and Computation, 149, 431?39. 

[27]  Shankararaman, C., Mark, R.W., and Clint, D. (1992), Slip at uniformly porous boundary: effect on fluid flow and mass transfer, Journal of Engineering Mathematics, 26, 481492. 

[28]  Muthu, P. and Tesfahun, B. (2012), Flow through nonuniform channel with permeable wall and slip effect, Special Topics and Reviews in Porous Media, 3(4), 321328. 

[29]  Chakravarthy, S. and Mandal, P.K. (1994), Mathematical modelling of blood flow through an overlapping stenosis, Mathematical and Computer Modelling, 19, 5973. 

[30]  Sutradhar, A., Mondal, J.K., Murthy, P.V.N.S., and Rama Subba Reddy, G., (2016), Influence of Starlingâ€™s hypothesis and Joule heating on peristaltic flow of an electrically conducting casson fluid in a permeable microvessel, Journal of Fluids Engineering, 138, 111106113. 