Keywords
Hartmann magnetic number
exponential permeability
magnetorheologicaal fluid
How to Cite
Downloads
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright (c) 2021 Roberto Silva, Mohammad H. Hamdan, Romel Erazo-Bone, Fidel Chuchuca-Aguilar, Kenny Escobar-Segovia
Abstract
This article involves the study and analysis of the fully developed flow of a magnetorheological fluid through a non-isotropic porous medium under the effect of an external, uniform, and transversal magnetic field. Permeability is taken as an exponential distribution function of the transverse direction. The Darcy-Brinkman-Lapwood-Lorentz equation for the fluid flow in porous media has been used and solved under non-slip boundary conditions by Modified Homotopy Perturbation Method and the results validated by the Numerical Shooting Method. Finally, the analysis of results is made of the influence on the velocity, volumetric flow, and wall shear stress.
References
Nield, D. A., & Bejan, A. (2006). Mechanics of Fluid Flow Through Porous Media. In D. A. Nield, & A. Bejan, Convection in Porous Media (3 ed., pp. 14-16). USA: Springer.
Rudraiah, N. (1986). Flow past porous layers and their stability. In N. P. Cheremisinoff (Ed.), Encyclopedia of Fluid Mechanics, Slurry Flow Technology (Vol. 8, pp. 567-647). Houston, Texas, USA: Gulf Publishing. Retrieved from https://archive.org/details/in.ernet.dli.2015.148666/page/n3/mode/2up
Alzahrani, S., Gadoura, I., & Hamdan, M. (2016, May-June). Nield- Kuznetsov Functions of the First- and Second Kind. IOSR Journal of Applied Physics, 8(3 Version III), 47-56. Retrieved from http://www.iosrjournals.org/iosr-jap/papers/Vol8-issue3/Version-3/H0803034756.pdf
Attia, H., & Abdeen, M. (2012). Unsteady MHD Flow and Heat Transfer Between Parallel Porous Plates with Exponential Decaying Pressure Gradient. Kragujevac Journal of Science(34), 15-22. Retrieved from https://www.pmf.kg.ac.rs/KJS/images/volumes/vol34/kjs34attiaabdeen15revision.pdf
Barcena, C., Sra, A., & Gao, J. (2009). Applications of Magnetic Nanoparticles in Biomedicine. In J. Ping Liu, E. Fullerton, O. Gutfleisch, & D. Sellmyer (Eds.), Nanoscale Magnetic Materials and Applications (pp. 591-626). Springer. doi: https://doi.org/10.1007/978-0-387-85600-1
Mishra, S., Baag, S., Dash, G., & Acharya, M. (2019). Numerical approach to MHD flow of power-law fluid on a stretching sheet with non-uniform heat source. Nonlinear Engineering, 9(1), 81-93. doi: https://doi.org/10.1515/nleng-2018-0026
Shehzad, S., & Hayat, T. A. (2015). Influence of convective heat and mass conditions in MHD flow of nanofluid. Bull. PolishAcadSc. Tech. Sci., 63(2), 465-474. doi: https://doi.org/10.1515/bpasts-2015-0053
Hamdan, M., Kamel, M., & Siyyam, H. (2009). A permeability function for Brinkman’s equation. Proceedings of the 11th Conf. on Mathematical Methods, System Theory and Control (pp. 198-205). WSEAS Publications. Retrieved from https://pdfs.semanticscholar.org/b2b7/8e0741d09e511c6a461d3b22a01a5d4151e7.pdf
Brinkman, H. C. (1947). A Calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Scientific Res., A1, 27-34. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.454.3769&rep=rep1&type=pdf
Hamdan, M., & Kamel, M. (2011). Flow through Variable Permeability Porous Layers. Adv. Theor. Appl. Mech., 4(3), 135 - 145. Retrieved from http://m-hikari.com/atam/atam2011/atam1-4-2011/hamdanATAM1-4-2011-2.pdf
Auriault, J.-L. (2009, September). On the Domain of Validity of Brinkman’s Equation. Transport in Porous Media, 79(2), 215-223. doi: https://doi.org/10.1007/s11242-008-9308-7
Parvazinia, M., Nassehi, V., Wakeman, R. J., & Ghoreishy, M. H. (2006, April). Finite element modelling of flow through a porous medium between two parallel plates using the Brinkman equation. Transport in Porous Media(63), 71-90. doi: https://doi.org/10.1007/s11242-005-2721-2
Nield, D. A. (1991, September). The Limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface. International Journal of Heat and Fluid Flow, 12(3), 269-272. doi: https://doi.org/10.1016/0142-727X(91)90062-Z
Sahraoui, M., & Kaviany, M. (1992). Slip and no-slip velocity boundary conditions at interface of porous, plain media. International Journal of Heat and Mass Transfer, 35(4), 927-943. doi: http://dx.doi.org/10.1016/0017-9310(92)90258-T
Kaviany, M. (1995). Part I Single Phase Flow. Fluid Mechanics. In M. Kaviany, Principles of Heat Transfer inPorous Media (2 ed., pp. 95-100). Spinger, Mechanical Engineering Series. doi: https://doi.org/10.1007/978-1-4684-0412-8
Lundgren, T. S. (1972). Slow Flow Through Stationary Random Beds and Suspensions of Spheres. Journal of Fluid Mechanics(51), 273-299. doi: http://dx.doi.org/10.1017/S002211207200120X
Cheng, A. H.-D. (1984, July). Darcy’s Flow With Variable Permeability: A Boundary Integral Solution. (A. G. Union, Ed.) Water Resources Research, 20(7), 980-984. doi: https://doi.org/10.1029/WR020i007p00980
Hamdan, M. H., & Abu Zaytoon, M. S. (2017, May-June). Flow over a Finite Forchheimer Porous Layer with Variable Permeability. IOSR Journal of Mechanical and Civil Engineering, 14(3), 15-22. doi: https://doi.org/10.9790/1684-1403041522
Elaiw, A., Ibrahim, F., & Bakr, A. (2009). Variable permeability and inertia effect on vortex instability of natural convection flow over horizontal permeable plates in porous media. Commun Nonlinear Sci Numer Simulat, 14, 21902201. doi: https://doi.org/10.1016/j.cnsns.2008.06.022
Hassanien, I. (2003). Variable permeability effects on mixed convection along a vertical wedge embedded in a porous medium with variable surface heat flux. Applied Mathematics and Computation, 138, 41-59. doi: https://doi.org/S0096-3003(02)00098-X
Hassanien, I., Salama, A., & Elaiw, A. (2003, December). Variable permeability effect on vortex instability of mixed convection flow in a semi-infinite porous medium bounded by a horizontal surface. Applied Mathematics and Computation, 146(2-3), 829-847. doi: https://doi.org/10.1016/S0096-3003(02)00635-5
Jang, J.-Y., & Chen, J.-L. (1993). Variable porosity effect on vortex instability of a horizontal mixed convection flow in a saturated porous medium. International Journal of Heat and Mass Transfer, 36(6), 1573-1582.
Chandrasekhara, B., Namboodiri, P., & Hanumanthappa, A. (1984, March 01). Similarity solutions for buoyancy induced flows in a saturated porous medium adjacent to impermeable horizontal surfaces. Wärme-und Stoffübertragung, 18(1), 17-23.
Abu Zaytoon, M. (2015). Flow through and over porous layers of variable thickness and permeability. Canada: University of New Brunswick. Retrieved from https://unbscholar.lib.unb.ca/islandora/object/unbscholar%3A7625
Chandrasekhara, B., Hanumanthappa, A., & Chandranna, S. (1984, January 01). Influence of Variable Permeability on the Basic Flows in Porous Media. Indian Journal of Technology, 22(8), 281-283.
Goldstein, R., Ibele, W., Patankar, S., Simon, T., Kuehn, T., Strykowski, P., . . . Srinivasan, V. (2006). Heat transfer—A review of 2003 literature. International Journal of Heat and Mass Transfer, 49, 451-534. doi: https://doi.org/10.1016/j.ijheatmasstransfer.2005.11.001
Schiffman, R., & Gibson, R. (1964). Consolidation of Nonhomogeneous Clay Layers. Journal of the Soil Mechanics and Foundations Division, 90(5), 1-30.
Mahmoud, M. S., & Deresiewicz, H. (1980, January-March). Settlement of inhomogeneous consolidating soils—I: The single-drained layer under confined compression. International Journal for Numerical and Analytical Methods in Geomechanics, 4(1), 57-72. doi: https://doi.org/10.1002/nag.1610040105
Rees, D. A., & Pop, I. (2000). Vertical free convection in a porous medium with variable permeability effects. International Journal of Heat and Mass Transfer, 43, 2565-2571. Retrieved from https://people.bath.ac.uk/ensdasr/PAPERS/PAPERS.bho/paper51.pdf
Alloui, Z., Bennacer, R., & Vasseur, P. (2009, June). Variable permeability effect on convection in binary mixtures saturating a porous layer. Heat and Mass Transfer, 45(8), 1117-1127. doi: https://doi.org/10.1007/s00231-009-0488-7
Choukairy, K., & Bennacer, R. (2012). Numerical and Analytical Analysis of the Thermosolutal Convection in an Heterogeneous Porous Cavity. FDMP-Fluid Dynamics & Materials Processing, 8(2), 155-172. doi: https://doi.org/10.3970/fdmp.2012.008.155
Hartmann, J., & Lazarus, F. (1937). Experimental investigations on the flow of mercury in a homogeneous magnetic field. Matematisk-fysiske meddelelser Kongelige Danske Videnskabernes Selskab, 15(7), 1-45.
He, J.-H. (2003, February 15). Homotopy perturbation method: A new nonlinear analytical technique. Applied Mathematics and Computation, 135(1), 73-79. doi: https://doi.org/10.1016/S0096-3003(01)00312-5
Müller, U., & Bühler, L. (2001). Analytical solutions for MHD channel flow. In U. Müller, & L. Bühler, Magnetofluiddynamics in Channels and Containers (pp. 37-56). Berlin-Heidelberg, Germany: Springer.
Rothmayer, A. P. (2014). Magnetohydrodynamic channel flows with weak transverse magnetic fields. Phil. Trans. R. Soc. A. (372), 1-12. doi: https://doi.org/10.1098/rsta.2013.0344
Awartani, M., & Hamdan, M. (2005, October). Fully developed flow through a porous channel bounded by flat plates. Applied Mathematics and Computation, 2(169), 749-757. doi: https://doi.org/10.1016/j.amc.2004.09.087
Fu, W.-S., Huang, H.-C., & Liou, W.-Y. (1996). Thermal enhancement in laminar channel flow with a porous block. International Journal of Heat and Mass Transfer, 39(10), 2165 2175.
Harwin, D. A. (2007). Flows in Porous Channels. Bath, United Kingdom: University of Bath. Retrieved from https://people.bath.ac.uk/masjde/Theses/Thesis-Harwin.pdf
Kaviany, M. (1985). Laminar flow through a porous channel bounded by isothermal parallel plates. International Journal of Heat and Mass Transfer, 28(4), 851-858.
Liu, S., Afacan, A., & Masliyah, J. (1994). Steady Incompressible Laminar Flow in Porous Media. Chmdml Engineering Science, 3565-3586.
Pillai, K., Varma, S., & Babu, M. S. (1987, August). Aligned magnetic effects through varying permeable bed. Proc. Indian Acad. Sci. (Math. Sci.), 96(1), 61-69. Retrieved from https://www.ias.ac.in/article/fulltext/pmsc/096/01/0061-0069
Sinha, A., & Chadda, G. (1984, September). Steady Laminar Viscous Flow Down an Open Inclined Channel with a Bed of Varying Permeability. Indian J. Pure Appl. Math., 15(9), 1004-1013. Retrieved from https://insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a6e_1004.pdf
Narasimha Murthy, S., & Feyen, J. (1989). Influence of variable permeability on the dispersion of a chemically reacting solute in porous media. International Journal of Engineering Science, 27(12), 1661-1671. doi: https://doi.org/10.1016/0020-7225(89)90159-6
Mathew, S. (2005). Mathematical Analysis. In S. Mathew, MHD Convective Heat Transfer Through a Porous Medium in a Vertical Channel with periodic permeability (pp. 7-10). India: Sri Krishnadevaraya University Anantapur. Retrieved from http://hdl.handle.net/10603/64940
Srivastava, B., & Deo, S. (2013, May). Effect of magnetic field on the viscous fluid flow in a channel filled with porous medium of variable permeability. Applied Mathematics and Computation, 219(17), 8959-8964. doi: https://doi.org/10.1016/j.amc.2013.03.065
He, J. H. (2008). Recent Development of the Homotopy Perturbation Method. Topological Methods in Nonlinear Analysis, 31, 205-209. Retrieved from https://projecteuclid.org/download/pdf_1/euclid.tmna/1463150264
Geindreau, C., & Auriault, J.-L. (2002, September). Magnetohydrodynamic flows in porous media. Journal of Fluid Mechanics, 466, 343-363. doi: https://doi.org/10.1017/S0022112002001404
Rudraiah, N., Ramaiah, B., & Rajasekhar, B. (1975, January). Hartmann flow over a permeable bed. International Journal of Engineering Science, 13(1), 1-24. doi: https://doi.org/10.1016/0020-7225(75)90070-1
Merabet, N., Siyyam, H., & Hamdan, M. (2008). Analytical approach to the Darcy-Lapwood-Brinkman equation. Applied Mathematics and Computation, 196, 679-685. doi: https://doi.org/10.1016/j.amc.2007.07.003
Abu Zaytoon, M., Alderson, T., & Hamdan, M. (2016, June). Flow through a Layered Porous Configuration with Generalized Variable Permeability. International. Journal of Enhanced Research in Science, Technology & Engineering, 5(6), 1-21. Retrieved from http://www.erpublications.com/uploaded_files/download/download_05_06_2016_16_55_43.pdf
Seyf, H., & Rassoulinejad-Mousavi, S. (2011). He’s Homotopy Method for Investigation of Flow and Heat Transfer in a Fluid Saturated Porous Medium. World Applied Sciences Journal, 15(12), 1791-1799. Retrieved from http://www.idosi.org/wasj/wasj15(12)11/21.pdf
He, J. H. (2006). New interpretation of homotopy perturbation. International Journal of Modern Physics B, 20(18), 2561-2568. Retrieved from https://works.bepress.com/ji_huan_he/3/
He, J. H. (2006, April). Some Asymptotic Methods for Strongly Nonlinear Equations. International Journal of Modern Physics B, 20(10), 1141-1199. doi: https://doi.org/10.1142/S0217979206033796
He, J.-H. (1999, August). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178(3-4), 257-262. doi: https://doi.org/10.1016/S0045-7825(99)00018-3
He, J.-H. (2000). A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics, 35(1), 37-43. doi: https://doi.org/10.1016/S0020-7462(98)00085-7
He, J.-H. (2003, February 15). Homotopy perturbation method: A new nonlinear analytical technique. Applied Mathematics and Computation, 135(1), 73-79. doi: https://doi.org/10.1016/S0096-3003(01)00312-5
He, J.-H. (2004, March 30). The homotopy perturbation method for non-linear oscillators with discontinuities. Applied Mathematics and Computation, 151(1), 287-292. doi: https://doi.org/10.1016/S0096-3003(03)00341-2
Jazbi, B., & Moini, M. (2008, February). Application of He’s homotopy perturbation method for Schrodinger equation. Iranian Journal of Mathematical Sciences and Informatics, 3(2), 13-19. https://ijmsi.ir/browse.php?a_id=51&sid=1&slc_lang=en
Usman, M., Naheed, Z., Nazir, A., & Mohyud-Din, S. (2014, July). On MHD flow of an incompressible viscous fluid. Journal of the Egyptian Mathematical Society, 22(2), 214-219. doi: https://doi.org/10.1016/j.joems.2013.07.003
Ochoa-Tapia, J. A., & Whitaker, S. (1995, September 1). Momentum transfer at the boundary between a porous medium and a homogeneous fluid — I. Theoretical development. International Journal of Heat and Mass Transfer, 38(14), 2635-2646. doi: https://doi.org/10.1016/0017-9310(94)00346-W
Nield, D. (2000). Modelling Fluid Flow and Heat Transfer in a Saturated Porous Medium. Journal of Applied Mathematics & Decision Sciences, 4(2), 165-173. Retrieved from http://www.kurims.kyoto-u.ac.jp/EMIS/journals/HOA/JAMDS/Volume4_2/173.pdf
Givler, R., & Altobelli, S. (1994, January 10). A determination of the effective viscosity for the Brinkman-Forchheimer flow model. Journal of Fluid Mechanics, 258, 355-370. doi: https://doi.org/10.1017/S0022112094003368
Liu, S., & Masliyah, J. (2005). Dispersion in Porous Media. In K. Vafai (Ed.), Handbook of Porous Media (2 ed., pp. 110111). Boca Raton, Florida, USA: Taylor & Francis.