Finite Difference Investigation of a Polluted Non-Isothermal Variable-Viscosity Porous Media Flow

Abstract:

We extend previous studies of channel flows to porous media flows with combined effects ofboth heat and mass transfer. We consider a temperaturedependent viscosity fluid and a concentrationdependent diffusivity in an unsteady and pressuredriven nonisothermal Brinkman flow. This leads to the governing equations for velocity, concentration and temperature. By lagging nonlinear coefficients, in time, a convergent finite difference scheme is formulated. We adopt the method of manufactured solutions to verify the convergence and second order spatial accuracy of the scheme. The impact of the flow parameters on the flow fields are numerically investigated. The results show that increase in the Darcy number and temperature parameter both increase the velocity while the increase in the pollutant diffusion parameter decreases the pollutant concentration.

• Periodical: Diffusion Foundations (Volume 26)
• Pages: 145-156
• DOI: https://doi.org/10.4028/www.scientific.net/DF.26.145
• Online since: March 2020
• Authors: Chinedu Nwaigwe*, Oluwole Daniel Makinde
• Keywords: Experimental Order of Convergence, Finite Difference Scheme, Method of Manufactured Solutions, Non-Newtonian Fluid, Porous Media, Semi-Implicit Scheme, Temperature-Dependent Viscosity

References

[1] Simón Martı́nez­Martı́nez, Rubén D Leal­Garza, Fausto A Sánchez­ Cruz, Esteban Baez Villar­ real, and Miguel Amado­Covarrubias. Cfd analysis of the effect of the exhaust manifold design on the close­coupled catalytic converter performance. Journal of KONES, 17:303-311, (2010).

[2] Tao Zhu. Unsteady porous­media flows. PhD thesis, Technische Universität München, (2016).

[3] Vadim S Ermolaev, Kirill O Gryaznov, Eduard B Mitberg, Vladimir Z Mordkovich, and Valentin F Tretyakov. Laboratory and pilot plant fixed­bed reactors for fischer-tropsch synthesis: Mathe­ matical modeling and experimental investigation. Chemical Engineering Science, 138:1-8, (2015).

DOI: 10.1016/j.ces.2015.07.036

[4] Giovanni Cerri, Ambra Giovannelli, Lorenzo Battisti, and Roberto Fedrizzi. Advances in effusive cooling techniques of gas turbines. Ap­ plied Thermal Engineering, 27(4):692-698, (2007).

DOI: 10.1016/j.applthermaleng.2006.10.012

[5] A­RA Khaled and K Vafai. The role of porous media in modeling flow and heat transfer in bio­ logical tissues. International Journal of Heat and Mass Transfer, 46(26):4989-5003, (2003).

DOI: 10.1016/s0017-9310(03)00301-6

[6] Zbigniew Żmudka and Stefan Postrzednik. Inverse aspects of the three­ way catalytic converter operation in the spark ignition engine. Journal of KONES, 18:509-516, (2011).

[7] K Mohan Laxmi, V Ranjith Kumar, and YV Hanumantha Rao. Modeling and simulation of gas flow velocity in catalytic converter with porous. (2013).

[8] Wolfgang Dahmen, Thomas Gotzen, Siegfried Müller, and M Rom. Numerical simulation of transpiration cooling through porous material. International journal for numerical methods in fluids, 76(6):331-365, (2014).

DOI: 10.1002/fld.3935

[9] Zheng Huang, Yin­Hai Zhu, Pei­Xue Jiang, and Yan­Bin Xiong. Investigation of transpiration cooling with local thermal non­equilibrium model: Effects of different thermal boundary condi­ tions at the porous­fluid interface. (2014).

[10] Jing Fan and Liqiu Wang. Analytical theory of bioheat transport. Journal of Applied Physics, 109(10):104702, (2011).

DOI: 10.1063/1.3580330

[11] C Israel­Cookey and C Nwaigwe. Unsteady mhd flow of a radiating fluid over a moving heated porous plate with time­dependent suction. American Journal of Scientific and Industrial Re­ search, 1(1):88-95, 2010.[12] Jawali C Umavathi, J Prathap Kumar, and Mikhail A Sheremet. Heat and mass transfer in a vertical double passage channel filled with electrically conducting fluid. Physica A: Statistical Mechanics and its Applications, 465:195-216, (2017).

DOI: 10.1016/j.physa.2016.07.073

[13] JC Umavathi, Mikhail A Sheremet, and Syed Mohiuddin. Combined effect of variable viscosity and thermal conductivity on mixed convection flow of a viscous fluid in a vertical channel in the presence of first order chemical reaction. European Journal of Mechanics­B/Fluids, 58:98-108, (2016).

DOI: 10.1016/j.euromechflu.2016.04.003

[14] JC Umavathi and Mikhail Alexandrovich Sheremet. Mixed convection flow of an electrically conducting fluid in a vertical channel using robin boundary conditions with heat source/sink. European Journal of Mechanics­B/Fluids, 55:132-145, (2016).

DOI: 10.1016/j.euromechflu.2015.08.013

[15] JC Umavathi and Mikhail Alexandrovich Sheremet. Influence of temperature dependent conduc­ tivity of a nanofluid in a vertical rectangular duct. International Journal of Non­Linear Mechanics, 78:17-28, (2016).

DOI: 10.1016/j.ijnonlinmec.2015.09.018

[16] Taza Gul, Inayatur Rehman, S Islam, Muhammad Altaf Khan, Wajid Ullah, and Zahir Shah. Unsteady third order fluid flow with heat trans­ fer between two vertical oscillating plates. J. Appl. Environ. Biol. Sci., 5(4):72-79, (2015).

[17] O. D. Makinde. On steady flow of a reactive variable viscosity fluid in a cylindrical pipe with an isothermal wall. International Journal of Numerical Methods for Heat & Fluid Flow, 17(2):187- 194, (2007).

DOI: 10.1108/09615530710723957

[18] Shriram Srinivasan and KB Nakshatrala. A stabilized mixed formulation for unsteady brinkman equation based on the method of horizontal lines. International Journal for Numerical Methods in Fluids, 68(5):642-670, (2012).

DOI: 10.1002/fld.2544

[19] T. Chinyoka and O. D. Makinde. Analysis of nonlinear dispersion ofpollutant ejected by an ex­ ternal source into a channel flow. Mathematical Problems in Engineering, (2010).

DOI: 10.1155/2010/827363

[20] Oluwole D Makinde and T Chinyoka. Numerical investigation of transient heat transfer to hydro­ magnetic channel flow with radiative heat and convective cooling. Communications in Nonlinear Science and Numerical Simulation, 15(12):3919-3930, (2010).

DOI: 10.1016/j.cnsns.2010.01.013

[21] T Chinyoka and Oluwole D Makinde. Computational dynamics of unsteady flow of a variable viscosity reactive fluid in a porous pipe. Mechanics Research Communications, 37(3):347-353, (2010).

DOI: 10.1016/j.mechrescom.2010.02.007

[22] Chinedu Nwaigwe. Sequential implicit numerical scheme for pollutant and heat transport in a plane­poiseuille flow. Journal of Applied and Computational Mechanics, 6(1):13-25, (2020).

[23] E Kumaresan, AG Vijaya Kumar, and J Prakash. Exact solution of unsteady mhd free convective chemically reacting visco­elastic fluid flow past a moving vertical plate through porous medium. International Journal of Applied and Computational Mathematics, 3(4):3897-3923, (2017).

DOI: 10.1007/s40819-017-0327-3

[24] KV Prasad, K Vajravelu, Hanumesh Vaidya, MM Rashidi, and Z Basha Neelufer. Flow and heat transfer of a casson liquid over a vertical stretching surface: Optimal solution. American Journal of Heat and Mass Transfer, 5(1):1-22, (2018).

DOI: 10.1166/jon.2016.1255

[25] Morteza Heydari and Hossein Shokouhmand. Numerical study on the effects of variable prop­ erties and nanoparticle diameter on nanofluid flow and heat transfer through micro­annulus. In­ ternational Journal of Numerical Methods for Heat & Fluid Flow, 27(8):1851-1869, 2017.[26] O. Anwar Bég and Oluwole D Makinde. Viscoelastic flow and species transfer in a darcian high­ permeability channel. Journal of Petroleum Science and Engineering, 76(3­4):93-99, (2011).

DOI: 10.1108/hff-04-2016-0164

[27] T. Chinyoka and OD Makinde. Analysis of non­newtonian flow with reacting species in a channel filled with a saturated porous medium. Journal of Petroleum Science and Engineering, 121:1-8, (2014).

DOI: 10.1016/j.petrol.2014.07.004

[28] Tirivanhu Chinyoka and Daniel Oluwole Makinde. Unsteady and porous media flow of reac­ tive non­newtonian fluids subjected to buoyancy and suction/injection. International Journal of Numerical Methods for Heat & Fluid Flow, 25(7):1682-1704, (2015).

DOI: 10.1108/hff-10-2014-0329

[29] Ram Prakash Sharma, M.C. Raju, O.D. Makinde, P.R. Krishna Reddy, and P. Chandra Reddy. Buoyancy effects on unsteady mhd chemically reacting and rotating fluid flow past a plate in a porous medium. Defect and Diffusion Forum, 392:1-9, (2018).

DOI: 10.4028/www.scientific.net/ddf.392.1

[30] William L Oberkampf and Timothy G Trucano. Verification and validation benchmarks. Nuclear engineering and Design, 238(3):716-743, (2008).

DOI: 10.1016/j.nucengdes.2007.02.032

[31] Li Wang, Wei­jiang Zhou, and Chu­qun Ji. Verification of a chemical nonequilibrium flows solver using the method of manufactured solutions. Procedia Engineering, 99:713-722, (2015).

DOI: 10.1016/j.proeng.2014.12.593

[32] Benjamin Grier, Richard Figliola, Edward Alyanak, and José Camberos. Discontinuous solutions using the method of manufactured solutions on finite volume solvers. AIAA Journal, 53(8):2369- 2378, (2015).

DOI: 10.2514/1.j053725

[33] O. D. Makinde, P. O. Olanrewaju, and W. M. Charles. Unsteady convection with chemical reac­ tion and radiative heat transfer past a flat porous plate moving through a binary mixture. Journal of African Mathematical Union, 22:65-78, (2011).

DOI: 10.1007/s13370-011-0008-z

[34] Aleksander Andreevich Samarskii. The theory of difference schemes. New York: Marcel Dekker, (2001).

[35] Piotr Matus, Lubin G Vulkov, et al. Analysis of second order difference schemes on non­uniform grids for quasilinear parabolic equations. Journal of Computational and Applied Mathematics, 310:186-199, (2017).

DOI: 10.1016/j.cam.2016.04.006

[36] Jipu Wang, William Martin, and Benjamin Collins. The application of method of manufactured solutions to method of characteristics in planar geometry. Annals of Nuclear Energy, 121:295- 304, (2018).

DOI: 10.1016/j.anucene.2018.07.041

[37] Patrick J Roache. The method of manufactured solutions for code verification. In Computer Simulation Validation, pages 295-318. Springer, (2019).