Magnetohydrodynamic Bio-convective Casson Nanofluid Flow: A Numerical Simulation by Paired Quasilinearisation

Document Type : Research Paper


1 Department of Mathematics, School of Technology, Pandit Deendayal Petroleum University, India

2 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, South Africa


A study on the effects of gyrotactic microorganism and nanoparticles in the bio-convection magnetohydrodynamic flow of Casson fluid at the nonlinear stretching boundary is investigated. Irregular heat source/sink, Joule and viscous dissipations, Brownian motion, and thermophoresis are included in the energy equation. The model outlining the flow system is non-dimensionalised and retained in the same form. The equations are worked out by pairing, i.e. first pair momentum and gyrotactic micro-organism density equation and second pair energy and nanoparticle concentration equation. This technique is termed as a paired quasilinearisation method (PQLM). Convergence and accuracy of PQLM are shown. Obtained numerical results are depicted in graphs in order to observe further insight into the flow pattern. Interesting aspects of various controlling parameters in flow, heat, nanoparticle concentration and microorganism density are discussed.


Main Subjects

[1] Srivastava, V. P., Saxena, M., Two-layered model of Casson fluid flow through stenotic blood vessels: applications to the cardiovascular system, Journal of Biomechanics, 27(7), 1994, 921-928.
[2] Nadeem, S., UlHaq, R., Lee, C., MHD flow of a Casson fluid over an exponentially shrinking sheet, Scientia Iranica, 19(6), 2012, 1550-1553.
[3] Shehzad, S. A., Hayat, T., Qasim, M., Asghar, S., Effects of mass transfer on MHD flow of Casson fluid with chemical reaction and suction, Brazilian Journal of Chemical Engineering, 30(1), 2013, 187-195.
[4] Hayat, T., Shehzad, S.A., Alsaedi, A., Alhothuali, M.S., Mixed Convection Stagnation Point Flow of Casson Fluid with Convective Boundary Conditions, Chinese Physics Letters, 22(11), 2012, 114704.
[5] Ullah, I., Bhattacharyya, K., Shafie, S., Khan, I., Unsteady MHD mixed convection slip flow of Casson fluid over nonlinearly stretching sheet embedded in a porous medium with chemical reaction, thermal radiation, heat generation/absorption, and convective boundary conditions, PLoS One, 11(10), 2016, 135.
[6] Mustafa, M., Hayat, T., Pop, I., Aziz, A., Unsteady boundary layer flow of a Casson fluid due to an impulsively started moving at the plate, Heat Transfer - Asian Reserach, 40(6), 2011, 563-576.
[7] Venkatesan, J., Sankar, D. S., Hemalatha, K., Yatim, Y., Mathematical analysis of Casson fluid model for blood rheology in stenosed narrow arteries, Journal of Applied Mathematics, 2013, Article ID 583809, 11p.
[8] Zubair, M., Waqas, M, Hayat, T., Ayub, M., Alsaedi. Simulation of nonlinear convectivethixotropic liquid with Cattaneo-Christov heat flux, Results in Physics, 22(8), 2018, 1023-1017.
[9] Zubair, M., Waqas, M., Hayat, T., Alsaedi, A., Ayub, M., Stagnation point flow of third-grade liquid due to variable thickness: A useful application to non-Fourier heat flux approach, Results in Physics, 8, 2018, 1010-1016.
[10] Zubair, M., Ijaz, M., Abbas, T., Riaz, A., Analysis of modified Fourier law in flow of ferromagnetic Powell-Eyringfluid considering two equal magnetic dipoles, Canadian Journal of Physics, 97(7), 2019, 772-776.
[11] Choi, S. U. S., Enhancing thermal conductivity of fluids with nanoparticles, Proc. 1995 ASME Int. Mech. Eng. Congr. Expo., 66, 1995, 99-105.
[12] Eastman, J. A., Choi, S. U. S., Li, S., Yu, W., Thompson, L. J., Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Applied Physics Letters, 78(6), 2001, 718-720.
[13] Hayat, T., Bilal Ashraf, M., Shehzad, S. A., Alsaedi, A., Mixed convection flow of Casson nanofluid over a stretching sheet with convectively heated chemical reaction and heat source/sink, Journal of Applied Fluid Mechanics, 8(4), 2015, 803-813.
[14] Anwar, M.I., Tanveer, N., Salleh, M.Z., Shafie, S., Diffusive effects on hydrodynamic Cassonnanofluid boundary layer flow over a stretching surface, Journal of Physics: Conference Series, 890, 2017, 1-8.
[15] Sulochana, C., Ashwinkumar, G.P., Sandeep, N., Similarity solution of 3D Casson nanofluid flow over a stretching sheet with convective boundary conditions, Journal of the Nigerian Mathematical Society, 35(1), 2016, 128-141.
[16] Madhu, M., Kishan, N., MHD flow and heat transfer of Casson nanofluid over a wedge, Mechanics & Industry, 18(2), 2017, 210.
[17] Ahmad, K., Hanouf, Z., Ishak, A., MHD Casson nanofluid flow past a wedge with Newtonian heating, European Physical Journal Plus, 132(2), 2017, 87.
[18] Abbas, T., Bhatti, M. M., Ayub, M., Aiding and opposing of mixed convection Cassonnanofluid flow with chemical reactions through a porous Riga plate, Proc. Inst. Mech. Eng. Part E J. Process Mech. Eng., 232(5), 2018, 519-527.
[19] Makinde, O. D., Nagendramma, V., Raju, C. S. K., Leelarathnam, A., Effects of Cattaneo-Christov Heat Flux on Casson Nanofluid Flow Past a Stretching Cylinder, Defect and Diffusion Forum, 378, 2017, 28-38.
[20] Ijaz, M., Ayub, M., Zubair, M., Riaz, A., On stratified flow of ferromagnetic nanofluid with heat generation/absorption, Physica Scripta, 94(4), 2018, 045206.
[21] Alsabery, A. I., Sheremet, M.A., Chamkha, A.J. and Hashim, I.J.S.R., MHD convective heat transfer in a discretely heated square cavity with conductive inner block using two-phase nanofluid model, Scientific Reports, 8(1), 2018, 1-23.
[22] Lu, D, Ramzan, M. Ahmad, S., Chung, J.D. and Farooq, U., A numerical treatment of MHD radiative flow of Micropolar nanofluid with homogeneous-heterogeneous reactions past a nonlinear stretched surface, Scientific Reports, 8(1), 2018, 12431.
[23] Sheikholeslami, M., Khan, I., and Tlili, I., Non-equilibrium model for nanofluid free convection inside a porous cavity considering Lorentz forces, Scientific Reports, 8(1), 2018, 16881.
[24] Ramzan, M., Sheikholeslami, M., Saeed, M. and Chung, J.D., On the convective heat and zero nanoparticle mass flux conditions in the flow of 3D MHD Couple Stress nanofluid over an exponentially stretched surface, Scientific Reports, 9(1), 2019, 562.
[25] Sheikholeslami, M., Shah, Z. Shafee, A. Khan, I. and Tlili, I., Uniform magnetic force impact on water-based nanofluid thermal behavior in a porous enclosure with ellipse-shaped obstacle, Scientific Reports, 9(1), 2019, 1196.
[26] Nguyen-Thoi, T., Sheikholeslami, M. Shah, Z., Kumam, P. and Shafee, A., Magnetohydrodynamic nanofluid radiative thermal behavior by means of Darcy law inside a porous media, Scientific Reports, 9(1), 2019, 1-11.
[27] Kuznetsov, A. V., The onset of nanofluid bioconvection in a suspension containing both nanoparticles and gyrotactic microorganisms, International Communications in Heat and Mass Transfer, 37(10), 2010, 1421-1425.
[28] Platt, J. R., Bioconvection patterns in cultures of free-swimming organisms, Science, 133(3466), 1961, 1766-1767.
[29] Sivaraj, R., Animasaun, I. L., Olabiyi, A. S., Saleem, S., Sandeep, N., Gyrotactic microorganisms and thermoelectric effects on the dynamics of 29nm CuO-water nanofluid over an upper horizontal surface of a paraboloid of revolution, Multidiscipline Modeling in Materials and Structures, 14(4), 2018, 695-721.
[30] Kuznetsov, A. V., The onset of thermo-bioconvection in a shallow fluid-saturated porous layer heated from below in a suspension of oxytactic microorganisms, European Journal of Mechanics - B/Fluids, 25(2), 2006, 223-233.
[31] Hopkins, M. M., Fauci, L. J., A computational model of the collective fluid dynamics of motile micro-organisms, Journal of Fluid Mechanics, 455, 2002, 149-174.
[32] Tarakaramu, N, Narayana, P.V.S., Chemical Reaction Effects on Bio-Convection Nanofluid flow between two Parallel Plates in Rotating System with Variable Viscosity: A numerical study, Journal of Applied and Computational Mechanics, 5(4), 2019, 791-803.
[33] Chen, Z., Feng, Q., Liu, R., Chen, J., Ni, F., Development, and application of thermophilic microorganism species in oil recovery, Acta Petrolei Sinica, 22(6), 2001, 59-62.
[34] Wu, X., Hou, Z., Shi, M., Wang, Y., Research on using microorganisms to improve chemical flooding effect, Acta Petrolei Sinica, 27(1), 2006, 91-94.
[35] Nadeem, S., Khan, M.N., Muhammad, N., Ahmad, S., Mathematical analysis of bio-convectivemicropolar nanofluid, Journal of Computational Design and Engineering, 6, 2019, 233-242.
[36] Zilman, G., Novak, J., Liberzon, A., Perkol-Finkel, S., Benayahu, Y., The hydrodynamics of contact of a marine larva, Bugula neritina, with a cylinder, Journal of Experimental Biology, 216(15), 2013, 2789-2797.
[37] Khan, W. A., Makinde, O. D., MHD nanofluid bioconvection due to gyrotactic microorganisms over a convectively heat stretching sheet, International Journal of Thermal Sciences, 81(1), 2014, 118-124.
[38] Shaw, S., Sibanda, P., Sutradhar, A., Murthy, P. V. S. N., Magnetohydrodynamics and Soret Effects on Bioconvection in a Porous Medium Saturated With a Nanofluid ContainingGyrotactic Microorganisms, Journal of Heat Transfer, 136(5), 2014, 052601.
[39] Das, K., Duari, P. R., Kundu, P. K., Nanofluid bioconvection in presence of gyrotactic microorganisms and chemical reaction in a porous medium, Journal of Mechanical Science and Technology, 29(11), 2015, 4841-4849.
[40] Makinde O. D., Animasaun, I. L., Thermophoresis and Brownian motion effects on MHDbioconvection of nanofluid with nonlinear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution, Journal of Molecular Liquids, 221, 2016,733-743.
[41] Latiff, N.A.A., Yahya, E., Ismail, A.I.M., Amirsom, A., Basir, F., The effect of velocity slipand multiple convective boundary conditions in a Darcian porous media with microorganismpast a vertical stretching/shrinking sheet, AIP Conf. Proc., 1870, 2017, 10p.
[42] Alsaedi, A., Khan, M.I., Farooq, M., Gull, N., Hayat, T., Magnetohydrodynamic (MHD) stratified bioconvective flow of nanofluid due to gyrotactic microorganisms, Advanced Powder Technology, 28(1), 2017, 288-298.
[43] Aman, F., Khazim, W.N.H.W.M., Mansur, S., Mixed convection flow of a nanofluid containing gyrotactic microorganisms over a stretching/shrinking sheet in the presence of magnetic field, Journal of Physics: Conference Series, 890(1), 2017, 11p.
[44] Khan, M.I., Waqas, M., Hayat, T., Khan, M.I., Alsaedi, A., Behaviour of stratification phenomenon in flow of Maxwell nanomaterial with motile gyrotactic microorganisms in the presence of magnetic field, International Journal of Mechanical Sciences, 131, 2017, 426-434.
[45] Nagendramma, V., Raju, C.S.K., Mallikarjuna, B., Shehzad, S.A., Leelarathnam, A.,3DCasson nanofluid flow over slendering surface in a suspension of gyrotactic microorganisms with Cattaneo-Christov heat flux, Applied Mathematics and Mechanics (English Edition), 39(5), 2018, 623-638.
[46] Rehman, K.U., Malik, A.A., Tahir, M., Malik, M.Y., Undersized description on motilegyrotactic micro-organisms individualities in MHD stratified water-based Newtonian nanofluid, Results in Physics, 8, 2018, 981-987.
[47] Chen, H., Chen, J., Geng, Y., Chen, K., Three-dimensional boundary layer flow over a rotating disk with power-law stretching in a nanofluid containing gyrotactic microorganisms, Heat Transfer - Asian Research, 47(3), 2018, 569-582.
[48] Bhatti, M.M., Mishra, S.R., Abbas, T., Rashidi, M.M., A mathematical model of MHDnanofluid flow having gyrotactic microorganisms with thermal radiation and chemical reaction effects, Neural Computing and Applications, 30(4), 2018, 1237-1249.
[49] Sulaiman, M., Ali, A., Islam, S., Heat and Mass Transfer in Three-Dimensional Flow of an Oldroyd-B Nanofluid with Gyrotactic Micro-Organisms, Mathematical Problems in Engineering, 2018, Article ID 6790420, 1-15.
[50] Rashad, A.M., Chamkha, A.J., Mallikarjuna, B., Abdou, M.M.M., Mixed bioconvection flow of a nanofluid containing gyrotactic microorganisms past a vertical slender cylinder, Frontiers in Heat and Mass Transfer, 10, 2018, 21p.
[51] Ramzan, M., Mutaz M., Fares H., Magnetized suspended carbon nanotubes based nanofluid flow with bio-convection and entropy generation past a vertical cone, Scientific Reports, 9(1), 2019, 1-15.
[52] Oyelakin, I.S., Mondal, S., Sibanda, P., Nonlinear Radiation in Bioconvective Casson Nanofluid Flow, International Journal of Applied and Computational Mathematics, 5, 2019, 124.
[53] Kessler, J.O., Co-operative and concentrative phenomena of swimming micro-organisms, Contemporary Physics, 26(2), 1985, 147-166.
[54] Kessler, J.O., Hydrodynamic focusing of motile algal cells, Nature, 313(5999), 1985, 218.
[55] Pedley, T.J., Kessler, J.O., Hydrodynamic phenomena in suspensions of swimming microorganisms, Annual Review of Fluid Mechanics, 24(1), 1992, 313-358.
[56] Metcalfe, A.M., Pedley, T.J., Falling plumes in bacterial bioconvection, Journal of Fluid Mechanics, 445, 2001, 121-149.
[57] Kumar, P.B.S., Gireesha, B.J., Mahanthesh, B., Chamkha, A.J., Thermal analysis of nanofluid flow containing gyrotactic microorganisms in bioconvection and second-order slip with convective condition, Journal of Thermal Analysis and Calorimetry, 136, 2019, 19-47.
[58] Khan N.S., Bioconvection in Second Grade Nanofluid Flow Containing Nanoparticles and Gyrotactic Microorganisms, Brazilian Journal of Physics, 48(3), 2018, 227-241.
[59] Ali, A., Sulaiman, M., Islam, S., Shah, Z., Bonyah, E., Three-dimensional Magnetohydrodynamic (MHD) flow of Maxwell nanofluid containing gyrotactic micro-organisms with heat source/sink, AIP Advances, 8, 2018, 085303.
[60] Atif, S.M., Hussain, S., Sagheer, M., Magnetohydrodynamic stratified bioconvectiveflow of micropolar nanofluid due to gyrotactic microorganisms, AIP Advances, 9, 2019, 025208.
[61] Xu, H., Pop, I., Mixed convection flow of a nanofluid over a stretching surface with uniform free stream in the presence of both nanoparticles and gyrotactic microorganisms, International Journal of Heat and Mass Transfer, 75, 2014, 610-623.
[62] Ansari, M.S., Motsa S.S., Trivedi, M., Flow and heat transfer of a nanofluid by mixed convection with non-uniform heat source/sink and magnetic field effect: A numerical approach, Computational Thermal Sciences, 11(3), 2019, 189-203.
[63] Abbasbandy, S., Ghehsareh, H.R., Solutions of the Magnetohydrodynamic Flow over a nonlinear Stretching Sheet and Nano Boundary Layer over Stretching Surfaces, International Journal for Numerical Methods in Fluids, 70, 2012, 1324-1340.
[64] Otegbeye, O., Motsa S.S., Ansari, Md. S., An application of paired quasi-linearization on double-diffusive convection flow over a cone embedded in a porous medium in the presence of nanoparticles, Heat Transfer-Asian Research, 48, 2019, 1413-1439.
[65] Trivedi, M., Otegbeye, O., Ansari, M.S., Motsa, S.S., A paired quasi-linearization on magnetohydrodynamic flow and heat transfer of Casson nanofluid with Hall effects, Journal of Applied and Computational Mechanics, 5(5), 2019, 849-860.
[66] Chen, H., Chen, J., Geng, Y., Chen, K., Three-dimensional boundary layer flow over a rotating disk with power-law stretching in a nanofluid containing gyrotactic microorganisms, Heat Transfer-Asian Research, 47(3), 2018, 569-582.