36d1e200-eba2-46d4-b978-72341bd0eb7420210901053355567naun:naunmdt@crossref.orgMDT DepositInternational Journal of Mechanics1998-444810.46300/9104http://www.naun.org/cms.action?id=2828128202112820211510.46300/9104.2021.15https://www.naun.org/cms.action?id=23280Effects of the Porous Microstructure on the Drag Coefficient in Flow of a Fluid with Pressure-Dependent ViscosityM.S. AbuZaytoonDepartment of Mathematics, University of Petra, Amman, JordanS. JayyousiDajaniDepartment of Mathematics and Computer Science Lake Forest College, Lake Forest, IL 60045, USAM.H.HamdanDepartment of Mathematics and Statistics University of New Brunswick , Saint John, N.B., Canada, E2L 4L5Equations governing the flow of a fluid with pressure-dependent viscosity through an isotropic porous structure are derived using the method of intrinsic volume averaging. Viscosity of the fluid is assumed to be a variable function of pressure, and the effects of the porous microstructure are modelled and included in the pressure-dependent drag coefficient. Five friction factors relating to five different microstructures are used in this work91202191202113614415https://www.naun.org/main/NAUN/mechanics/2021/a302003-015(2021).pdf10.46300/9104.2021.15.15https://www.naun.org/main/NAUN/mechanics/2021/a302003-015(2021).pdf10.1615/jpormedia.v20.i3.60J. Chang, K.B. Nakashatrala and J.N. Reddy, “Modification to Darcy-Forchheimer model due to pressure-dependent viscosity: consequences and numerical solutions”, J. Porous Media, vol. 20(3), pp. 263-285, 2017. 10.1002/fld.2358K.B. Nakshatrala and K.R. Rajagopal, “A numerical study of fluids with pressure-dependent viscosity flowing through a rigid porous medium”, Int. J. Numer. Meth. Fluids, vol. 67, pp. 342-368, 2011. P.W. Bridgman, The Physics of High Pressure. MacMillan, New York, 1931. 10.1016/j.ijengsci.2014.11.007L. Fusi, A. Farina and F. Rosso, “Mathematical models for fluids with pressure- dependent viscosity flowing in porous media”, International Journal of Engineering Science, vol. 87, pp. 110-118, 2015. 10.1016/j.ijengsci.2014.09.004K.D. Housiadas, G.C. Georgiou and R.I. Tanner, “A note on the unbounded creeping flow past a sphere for Newtonian fluids with pressure-dependent viscosity”, Int. Journal of Engineering Science, vol. 86, pp. 1–9, 2015. 10.1021/ef200958vF.J. Martinez-Boza, M.J. Martin-Alfonso, C. Callegos and M. Fernandez, “High-pressure behavior of intermediate fuel oils”, Energy Fuels, vol. 25, pp. 5138-5144, 2011. A.Z. Szeri, Fluid Film Lubrication: Theory and Design, Cambridge University Press, 1998. P.H. Vergne, “Pressure viscosity behavior of various fluids”, High Press. Res., vol. 8, pp. 451–454, 1991. G.G. Stokes, “On the theories of the internal friction of fluids in motion, and of equilibrium and motion of elastic solids”, Trans. Camb. Philos. Soc., vol. 8, pp. 287-305, 1845. C.J. Barus, “Note on dependence of viscosity on pressure and temperature”. Proceedings of the American Academy, vol. 27, pp. 13-19, 1891. 10.2475/ajs.s3-45.266.87C.J. Barus, “Isothermals, isopiestics and isometrics relative to viscosity”, American Journal of Science, vol. 45, pp. 87–96, 1893. 10.1142/s0218202507001899K.R. Rajagopal, “On a hierarchy of approximate models for flows of incompressible fluids through porous solids”, Mathematical Models and Methods in the Applied Sciences, vol. 17 (2), pp. 215–252, 2007. 10.1098/rspa.2000.0723J. Hron, J., Malek, and K.R. Rajagopal, “Simple flows of fluids with pressure-dependent viscosities”, Proceedings of the Royal Society, vol. 457, pp. 1603– 1622, 2001. 10.1017/jfm.2012.244K.R. Rajagopal, G. Saccomandi and L. Vergori, “Flow of fluids with pressure- and shear-dependent viscosity down an inclined plane”, Journal of Fluid Mechanics, vol. 706, pp. 173–189, 2012. 10.1002/zamm.201000141V.L. Savatorova and K.R. Rajagopal, “Homogenization of a generalization of Brinkman’s equation for the flow of a fluid with pressure dependent viscosity through a rigid porous solid”, ZAMM, 91(8), pp. 630–648, 2011. 10.1615/jpormedia.v16.i3.20S. Srinivasan, A. Bonito and K.R. Rajagopal, “Flow of a fluid through a porous solid due to high pressure gradient”, Journal of Porous Media, vol. 16, pp. 193– 203, 2013. 10.1016/j.ijnonlinmec.2013.09.004S. Srinivasan and K.R. Rajagopal, “A thermodynamic basis for the derivation of the Darcy, Forchheimer and Brinkman models for flows through porous media and their generalizations”, International Journal of NonLinear Mechanics, vol. 58, pp. 162–166, 2014. 10.1016/j.amc.2007.10.038K. Kannan and K.R. Rajagopal, “Flow through porous media due to high pressure gradients”, J. Applied Mathematics and Computation, vol. 199, pp. 748-759, 2008. 10.1016/j.camwa.2006.02.023S.C. Subramanian and K.R. Rajagopal, “A note on the flow through porous solids at high Pressures”. Computers and Mathematics with Applications, vol. 53, pp. 260–275, 2007. 10.1115/1.3242486N. Rudraiah, “Coupled Parallel Flows in a Channel and a Bounding Porous Medium of Finite Thickness”, J. Fluids Engineering, ASME, vol. 107, pp. 322-329, 1985. 10.1007/bf02120313H.C. Brinkman, “A Calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles”, Appl. Scientific Res., vol. A1, pp. 27-34, 1947. 10.1007/bf00820342J. P. Du Plessis and J. H. Masliyah, “Mathematical modeling of flow through consolidated isotropic porous media,” Transport in Porous Media, vol. 3, pp. 145-161, 1988. 10.1007/bf00208950J. P. Du Plessis and J. H. Masliyah, “Flow through isotropic granular porous media,” Transport in Porous Media, vol. 6, pp. 207-221, 1991. 10.1007/bf00617551J. P. Du Plessis, 1994, “Analytical quantification of coefficients in the Ergun equation for fluid friction in a packed bed,” Transport in Porous Media, vol. 16, pp. 189-207, 1994. J. P. Du Plessis and G. P. J. Diedericks, “Pore-Scale Modeling of Interstitial Phenomena,” In: Fluid Transport in Porous Media, J.P. Du Plessis, ed., Computational Mechanics Publications, Southampton, 1997, pp. 61- 104. 10.12988/atam.2016.51212M.S. Abu Zaytoon, F.M. Allan, T.L. Alderson and M.H. Hamdan, “Averaged equations of flow of fluid with pressure-dependent viscosity through porous media”, Elixir Appl. Math., vol. 96, pp. 41336- 41340, 2016. 10.1017/s0022112005007998M. Le Bars and M. Grae Worster, “Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification”, Journal of Fluid Mechanics, vol. 550, pp.149-173, 2006. S. Whitaker, “Volume Averaging of Transport Equations,” In: Fluid Transport in Porous Media, J.P. Du Plessis, ed., Computational Mechanics Publications, Southampton, 1997, pp. 1-60. S. Whitaker, “The method of volume averaging,” Kluwer Academic Publishers, Dordrecht, 1999.