Theoretical Study on Poiseuille Flow of Herschel-Bulkley Fluid in ‎Porous Media‎

Document Type : Research Paper


1 Applied Mathematics and Economics Programme, School of Applied Sciences and Mathematics, Universiti Teknologi Brunei,‎ Jalan Tungku Link, Bandar Seri Begawan, BE1410, Brunei Darussalam

2 UTM Centre for Industrial and Applied Mathematics, Ibnu Sina Institute for Scienntific and Industrial Research, Universiti Teknologi Malaysia,‎ ‎81310 Johor Bharu, Johor, Malaysia‎

3 Ship & Offshore Extreme Technology Industry-Academic Co-operation Research Center, Inha university, 100 Inha-ro, Incheon, 22212, South Korea

4 School of Mathematics, Computer Science and Engineering, Liverpool Hope University, Hope Park, Liverpool L16 9JD, United Kingdom‎

5 Department of Mathematical Sciences, Universiti Teknologi Malaysia, 81310 Johor Bharu, Johor, Malaysia‎

6 Department of Mathematics, National Institute of Technology, Yupia – 791112, Arunachal Pradesh, India‎


This theoretical study analyses the effects of geometrical and fluid parameters on the flow metrics in the Hagen-Poiseuille and plane-Poiseuille flows of Herschel-Bulkley fluid through porous medium which is considered as (i) single pipe/single channel and (ii) multi–pipes/multi-channels when the distribution of pores size in the flow medium are represented by each one of the four probability density functions: (i) Uniform distribution, (ii) Linear distribution of Type-I, (iii) Linear distribution of Type-II and (iv) Quadratic distribution. It is found that in Hagen-Poiseuille and plane-Poiseuille flows, Buckingham-Reiner function increases linearly when the pressure gradient increases in the range 1 - 2.5 and then it ascends slowly with the raise of pressure gradient in the range 2.5 - 5.In all of the four kinds of pores size distribution, the fluid’s mean velocity, flow medium’s porosity and permeability are substantially higher in Hagen-Poiseuille fluid rheology than in plane-Poiseuille fluid rheology and, these flow quantitiesascend considerably with the raise of pipe radius/channel width and a reverse characteristic is noted for these rheological measures when the power law index parameter increases.The flow medium’s porosity decreases rapidly when the period of the pipes/channels distribution rises from 1 to 2 and it drops very slowly when the period of the pipes/channels rises from 2 to 11.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Vafai, K., Porous media: Applications in biological systems and biotechnology, CRC Press: Taylor and Francis Group, 2011.
[2] Chevalier T., Talon, L., Moving line model and avalanche statistics of Bingham fluid flow in porous media, European Physical Journal E, 38, 2015, 76 - 81.
[3] Chen, M., Rossen, W., Yortsos, Y. C., The flow and displacement in porous media of fluids with yield stress, Chemical Engineering: Science, 60 2005, 4183–4202.
[4] Ho, C. K., Webb, S. W., Theory and Application of Transport in Porous Media, Gas Transport, Springer, Netherlands, 2006.
[5] J. Bleyer, J., Coussot, P., Breakage of non-Newtonian character in flow through porous medium: evidence from numerical simulation, Physical Review E -American Physical Society, 89, 2014, 063018.
[6] Dullien, F. A. L., Porous Media – Fluid Transport and pore structure, Second Edition, Academic Press, MIT, U.S.A., 1992.
[7] Heinemann, Z. E., Fluid flow in porous media, Textbook Ser. 1, 2003.
[8] Chevalier, T., Rodts, S., Chateau, X., Chevalier, C.,. Coussot, P., Breaking of non-Newtonian character in flows through porous medium, Physical Reviews Journal E, 89, 2014, 023002.
[9] Mehryan, S. A. M., Vaezi, M., Sheremet, M., Ghalambaz, M., Melting heat transfer of power-law-non-Newtonian phase change nano-enhanced n- octadecane-mesoporous silica (MPSiO2), Journal of Heat and Mass transfer, 151, 2020, 119385.
[10] Zadeh, S. M. H., Mehryan, S. A. M., Ghalambaz, M., Ghodrat, M., Young, J., Chamkha, A. J., Hybrid thermal performance enhancement of a circular latent heat storage system by utilizing partially filled copper foam and Cu/GO nano-additives, Energy, 213, 2020, 118761.
[11] Ghalambaz, M., Sheremet, M., Mehryan, S. A. M., Kashkooli, F. M., Pop, I., Local thermal non-equilibrium analysis of conjugate free convection within a porous enclosure occupied with Ag-MgO hybrid nanofluid, Journal of Thermal Analysis and Calorimetry, 135, 2019, 1381 – 1398.
[12] Ghalambaz, M., Zhang, J., Conjugate solid liquid-solid phase change heat transfer in heatsink filled with phase change material-metal foam, International Journal of Heat and Mass Transfer, 146, 2020, 118832.
[13] Mehryan, S. A. M., Ghalambaz, M., Chamkha, A. J., Mohsenlzadi, Numerical study on natural convection of Ag-MgO hybrid/water nanofluid inside a porous enclosure: A local thermal non-equilibrium model, Powder Technology, 367, 2020, 443–445.
[14] Coussot, P., Yield stress fluid flows: a review of experimental data, Journal of Non-Newtonian Fluid Mechanics, 211, 2014, 31–49.
[15] Moller, B. P., Fall, A., Chikkadi, Y., Deres, D., Bonn, D., An attempt to categorize yield stress fluid behaviour, Philosophical Transactions of Royal Society A, 367, 2009, 5139–5155.
[16] Kalogirou, A., Poyiadji, S., Georgiou, G. C., Incompressible Poiseuille flows of Newtonian liquids with pressure-dependent viscosity, Journal of Non-Newtonian Fluid Mechanics, 166, 2011, 413–419.
[17] Zhu, H., Kim, Y. D., Kee, D. D., Non-Newtonian fluids with yield stress, Journal of Non-Newtonian Fluid Mechanics, 129, 2015, 177–181.
[18] Pascal, H., Non-steady flow of non-Newtonian fluids through a porous medium, International Journal of Engineering. Science, 21, 1983, 199–210.
[19] Moller, P. C. F., Fall, A., Bonn, D., Origin of apparent viscosity in yield stress fluids below yielding, EPL, 87, 2009, 38004.
[20] Balmforth, N. J., Frigaard, I. A., Ovarlez, G., Yielding to stress: recent developments in viscoplastic fluid mechanics, Annual Reviews of Fluid Mechanics, 46, 2014, 121–146.
[21] Philippou, M., Kountouriotis, Z., Georgiou, G. C., Viscoplastic flow development in tubes and channels with wall slip, Journal of Non-Newtonian Fluid Mechanics, 234, 2016, 69–81.
[22] Chevalier, T., Chevalier, C., Clain, X., Dupla, J. C., Canou, J. Rodts S., Coussot, P., Darcy’s law for yield stress fluid flowing through a porous medium, Journal of Non-Newtonian Fluid Mechanics, 19, 2013, 57–66.
[23] Balhoff,M. T., Rivera, D. S., Kwok, A., Mehmani, Y., Prodanovic, M., Numerical algorithms for network modeling of yield stress and other non-Newtonian fluids in porous media, Transport in Porous Media, 93, 2012, 363–379.
[24] Papanastasiou,T. C., Flows of materials with yield stress, Journal of Rheology, 31, 1987, 385–404.
[25] Nagarani, P., Sarojamma, G., Effect of body acceleration on pulsatile flow of Casson fluid through a mild stenosed artery, Korea–Australia Rheololgy Journal, 20, 2008, 189–196.
[26] Sankar, D. S., Ismail, A. I. M., Effects of periodic body acceleration in blood flow through stenosed arteries – a theoretical model, International Journal of Nonlinear Sciences and Numerical Simulations, 11, 2010, 243 – 257.
[27] Tu, C., Deville, M., Pulsatile flow of non-Newtonian fluids through arterial stenosis, Journal of Biomechanics, 29, 1996, 899-908.
[28] Sankar, D. S., Two-phase non-linear model for blood flow in asymmetric and axisymmetric stenosed arteries, International Journal of Non-Linear Mechanics, 46, 2011, 296–305.
[29] Mitsoulis, E., Abdali, S. S., Flow simulation of Herschel–Bulkley fluids through extrusion dies, Canadian Journal of Chemical Engineering, 71, 1993, 147–160.
[30] Jaafar, N. A., Yatim, Y. M., Sankar, D. S., Effect of chemical reaction in solute dispersion in Herschel-Bulkey fluid flow with applications to blood flow, Advances and Applications in Fluid Mechanics, 20, 2017, 279–310.
[31] Sabiri, N. E., Comiti, J., Pressure drop in non-Newtonian purely viscous fluid flow through porous media, Chemical Engineering: Science, 50, 1995, 1193–1201.
[32] Lopez, X., Valvatne, P. H., Blunt, M. J., Predictive network modelling of single-phase non-Newtonian flow in porous media, Journal of Colloid Interface Science, 264, 2003, 256–265.
[33] Darcy, H., Les fontaines publiques de la ville de Dijon, Victor Dalmont, Paris, 1856.
[34] Prodanovic, M., Lindquist, W., Seright, R., Porous structure and fluid partitioning in polyethylene cores from 3D X-ray microtomographic imaging, Journal of Colloid Interface Science, 298, 2006, 282–297.
[35] Sochi, T., Modelling the flow of yield-stress fluids in porous media, Transport in Porous Media, 85, 2010, 489–503.
[36] Sanjari, A., Abbaszadah, M., Abassi, A., Lattice Boltzmann simulation of free convection in an inclined open ended cavity partially filled with fibrous porous media, Journal of Porous Media, 21, 2018, 1265–1281.
[37] Abbaszadeh, M., Salehi, A., Abbassi, A., Heat transfer enhancement in an asymmetrically heated channel partially filled with fibrous media – A LBM approach, Journal of Porous Media, 18, 2015, 1201–1220.
[38] Oukhlef, A., Champmartin, S., Ambari, A., Yield stress fluids method to determine the pore size distribution of a porous medium, Journal of Non-Newtonian Fluid Mechanics, 204, 2014, 87-93.
[39] Chevalier, T., Chevalier, C., Clain, X., Dupla, J. C., Canou, J., Rodts, S., Coussot, P., Darcy’s law for yield stress fluid flowing through a porous medium, Journal of Non-Newtonian Fluid Mechanics, 195, 2013, 57–66.
[40] Nash, S., Rees, D. A. S., The effect of microstructure on models for the flow of Bingham fluid in porous media: One dimensional flows, Transport in Porous Media, 116, 2017, 1073–1092.
[41] Sankar, D. S., Viswanathan, K. K., Mathematical analysis of Poiseuille flow of Casson fluid past porous medium, Journal of Applied and Computational Mechanics, 2019, DOI: 10.22055/JACM.2020.31961.1945.
[42] Purcell, W, R., Capillary pressures-their measurement using mercury and the calculation of permeability therefrom, Petroleum Transport, AIME, 186, 1949, 39–48.
[43] Falade, J. A., Ukaegbu, J. C., Egere, A. C., Adesanya, S. C., MHD oscillatory flow through a porous channel saturated with porous medium, Alexandria Engineering Journal, 56, 2017, 147–152.
[44] Chauhan, T. S., Chauhan, I. S., Shikka, S., Flow of viscous fluid through a porous circular pipe in presence of magnetic field, Mathematica Aeterna, 5, 2015, 395–402.