cf3e25a4-92bc-4f8a-ae59-7192ab6a579620220301074312161wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON FLUID MECHANICS2224-347X1790-508710.37394/232013http://wseas.org/wseas/cms.action?id=40363120223120221710.37394/232013.2022.17https://wseas.com/journals/fluids/2022.phpParallel Flow of a Pressure-Dependent Viscosity Fluid through Composite Porous LayersM. S. AbuZaytoonDepartment of Mathematics and Statistics University of New Brunswick 100 Tucker Park Road, Saint John, N.B., CANADAM. H.HamdanDepartment of Mathematics and Statistics University of New Brunswick 100 Tucker Park Road, Saint John, N.B., CANADAFlow of a fluid with pressure-dependent viscosity through a composite of two porous layers is considered in this work in an attempt to validate velocity and shear stress continuity conditions at the interface, and are popular in the study of flow over porous layers and through composite layers when viscosity of the fluid is constant. For the current problem, conditions at the interface between the porous layers reflect continuity assumptions of velocity and shear stress, with additional continuity assumptions on pressure and viscosity. Viscosity is assumed to vary continuously and exponentially across the layers as a function of pressure. Analytical solutions are obtained to illustrate the effects of flow and media parameters (Darcy numbers, layer thicknesses, angle of inclination, and viscosity adjustment parameter) on the dynamic behaviour of pressure-dependent viscosity fluids in porous structures. All computations, simulations and graphs in this work have been carried out and obtained using Maple 2020 software package.312022312022191https://wseas.com/journals/fluids/2022/a025113-001(2022).pdf10.37394/232013.2022.17.1https://wseas.com/journals/fluids/2022/a025113-001(2022).pdf10.1016/s0096-3003(01)00016-9Allan, F.M. and Hamdan, M.H., Fluid mechanics of the interface region between porous layers, J. Applied Math and Computation, Vol. 128, 2002, pp. 37-43. 10.1016/s0096-3003(97)10141-2Ford, R.A. and Hamdan, M.H., Coupled parallel flow through composite porous layers, J. Applied Math and Computation, Vol. 97, 1998, pp. 261-271. 10.1016/0017-9310(87)90171-2Vafai, K. and R. Thiyagaraja, Analysis of flow and heat transfer at the interface region of a porous medium. Int. J. Heat and Mass Transfer, Vol. 30(7), 1987, pp. 1391-1405. Vergne, P.H., Pressure viscosity behavior of various fluids, High Press. Res., Vol. 8, 1991, pp. 451–454. Szeri, A.Z., Fluid Film Lubrication: Theory and Design, Cambridge University Press, 1998. 10.1021/ef200958vMartinez-Boza, F.J., Martin-Alfonso, M.J., Callegos, C. and Fernandez, M., High-pressure behavior of intermediate fuel oils, Energy Fuels, Vol. 25, 2011, pp. 138-5144. Bridgman, P.W., The Physics of High Pressure, MacMillan, New York, 1931. 10.1080/10402000308982628Bair, S. and Kottke, P., Pressure-Viscosity relationship for elstohydrodynamics. Tribology Trans. Vol 46, 2003, pp. 289-295. 10.1002/zamm.201000141Savatorova, V.L. and Rajagopal, K.R., Homogenization of a generalization of Brinkman’s equation for the flow of a fluid with pressure dependent viscosity through a rigid porous solid, ZAMM, Vol. 91(8), 2011, pp. 630-648. 10.46300/9104.2021.15.15Abu Zaytoon, M.S., Jayyousi-Dajani, S. and Hamdan, M.H., Effects of the porous microstructure on the drag coefficient in flow of a fluid with pressure-dependent viscosity, International Journal of Mechanics, Vol. 15, 2021, pp. 136-144. Abu Zaytoon, M.S., Allan, F.M., Alderson, T.L. and Hamdan, M.H., Averaged equations of flow of fluid with pressure-dependent viscosity through porous media, Elixir Appl. Math. Vol. 96, 2016, pp. 41336-41340. 10.12988/atam.2016.51212Alharbi, S.O., Alderson, T.L. and Hamdan, M.H., Flow of a fluid with pressure-dependent viscosity through porous media, Advances in Theoretical and Appl. Mechanics, Vol. 9(1), 2016, pp. 1-9. 10.1016/j.amc.2007.10.038Kannan, K. and Rajagopal, K.R., Flow through porous media due to high pressure gradients, Applied Mathematics and Computation, Vol. 199, 2008, pp. 748-759. 10.1017/jfm.2012.244Rajagopal, K.R., Saccomandi, G. and Vergori, L., Flow of fluids with pressure- and shear-dependent viscosity down an inclined plane, Journal of Fluid Mechanics, Vol. 706, 2012, pp. 173-189. Singh, A.K., Sharma, P.K. and Singh, N.P., Free convection flow with variable viscosity through horizontal channel embedded in porous medium, The Open Applied Physics Journal, Vol. 2, 2009, pp. 11- 19. 10.37394/232012.2021.16.19Abu Zaytoon, M.S. and Hamdan, M.H., Fluid Mechanics at the Interface between a variable viscosity fluid layer and a variable permeability porous medium, WSEAS Transactions on Heat and Mass Transfer, Vol. 16, 2021, pp. 159-169. 10.1002/fld.2358Nakshatrala, K.B. and Rajagopal, K.R., A numerical study of fluids with pressure-dependent viscosity flowing through a rigid porous medium, Int. J. Numer. Meth. Fluids, Vol. 67, 2011, pp. 342-368. Stokes, G.G., On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids, Trans. Camb. Philos. Soc., Vol. 8, 1845, pp. 287-305. Barus, C.J., Note on dependence of viscosity on pressure and temperature, Proceedings of the American Academy, Vol. 27, 1891, pp. 13-19. 10.2475/ajs.s3-45.266.87Barus, C.J., Isothermals, isopiestics and isometrics relative to viscosity, American Journal of Science, Vol. 45, 1893, pp. 87–96. 10.1016/s1874-5792(07)80011-5Málek, J. and Rajagopal, K.R., Mathematical properties of the solutions to the equations governing the flow of fluids with pressure and shear rate dependent viscosities, in: Handbook of Mathematical Fluid Dynamics, Elsevier, 2007. 10.1016/j.ijengsci.2014.09.004Housiadas, K.D., Georgiou, G.C. and Tanner, R.I., A note on the unbounded creeping flow past a sphere for Newtonian fluids with pressure-dependent viscosity, International Journal of Engineering Science, Vol. 86, 2015, pp. 1–9. 10.1016/s0893-9659(02)00070-8Málek, J., Necas, J. and Rajagopal, K.R., Global existence of solutions for flows of fluids with pressure and shear dependent viscosities. Applied Mathematics Letters, Vol. 15, 2002, pp. 961-967. 10.1016/j.camwa.2006.02.023Subramanian, S.C. and Rajagopal, K.R., A note on the flow through porous solids at high pressures, Computers and Mathematics with Applications, Vol. 53, 2007, pp. 260–275. 10.1016/j.ijengsci.2014.11.007Fusi, L., Farina, A. and Rosso, F., Mathematical models for fluids with pressure-dependent viscosity flowing in porous media, International Journal of Engineering Science, Vol. 87, 2015, pp. 110-118. 10.1016/j.ijengsci.2016.07.001Housiadas, K.D. and Georgiou, G.C., New analytical solutions for weakly compressible Newtonian Poiseuille flows with pressure-dependent viscosity, International Journal of Engineering Science, Vol. 107, 2016, pp. 13-27. Alzahrani, S.M., I. Gadoura, I. and Hamdan, M.H., A note on the flow of a fluid with pressuredependent viscosity through a porous medium with variable permeability, Journal of Modern Technology and Engineering, Vol. 2 (1), 2017, pp. 21-33. Alzahrani, S.M., I. Gadoura, I. and Hamdan, M.H., Flow down an inclined plane of a fluid with pressure-dependent viscosity through a porous medium with constant permeability, Journal of Modern Technology and Engineering, Vol. 2(2), 2017, pp. 155-166. Pozrikidis, C., Multifilm flow down an inclined plane: Simulations based on the lubrication approximation and normal-mode decomposition of linear waves. In Fluid Dynamics at Interfaces, 112- 128, Wei Shyy and Ranga Narayanan eds. Cambridge University Press, 1999.