# Bioconvection phenomenon for the boundary layer flow of magnetohydrodynamic Carreau liquid over a heated disk

Document Type : Article

Authors

1 Department of Applied Mathematics & Statistics, Institute of Space Technology Islamabad, P.O. Box 2750, Pakistan

2 - Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China. - Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China

3 Department of Mechanical Engineering, College of Engineering, Prince Muhammad bin Fahd University, Al-Khobar, Saudi Arabia

4 Renewable Energy Research Centre, Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut's University of Technology North Bangkok, 1518 Pracharat 1 Road, Bangsue, Bangkok 10800, Thailand

10.24200/sci.2021.53970.3518

Abstract

A numerical examination is conducted for the magnetohydrodynamics steady Carreau fluid flow on the transport of thermal energy and mass specie comprising nanoparticles with gyrotactic microorganisms through heated disk. The role of thermophoresis and Brownian motion are added in this flow problem. Governing equations are achieved by using the boundary layer theory in view of a coupled system of PDEs involving boundary conditions. The highly non-linear system of ODEs is generated using the concept of the transformation approach. Since the system of transformed equations is highly nonlinear, so, an approximate solution is estimated via optimal homotopy method. The role of prominent parameters on velocity, thermal energy, mass specie and motile density microorganisms examined graphically. Additionally, graphical observations regarding mass specie, thermal energy and velocities are discussed briefly. It has estimated that the motion of fluid particles is diminished because of the intensity of the magnetic field while mass specie and fluid temperature rise versus enhancement the values of the magnetic field.

Keywords

#### References

References:
1.    Carreau, P.J., “Rheological equations from molecular network theories”, Transactions of the Society of Rheology, 16(1), pp.99-127 (1972).
2.    Carreau, P.J., “An analysis of the viscous behavior of polymer solutions”, Can. J. Chem. Eng., Vol. 57, pp. 135-140 (1979).
3.    Griffiths, P. T. “Flow of a generalised Newtonian fluid due to a rotating disk”, Journal of Non-Newtonian Fluid Mechanics, Vol. 221, pp. 9-17 (2015).
4.    Machireddy, G. R. and Naramgari, S. “Heat and mass transfer in radiative MHD Carreau fluid with cross diffusion”, Ain Shams Engineering Journal, (2016).
5.    Kumar, K. G., Gireesha, B. J., Rudraswamy, N. G., et al. “Radiative heat transfers of Carreau fluid flow over a stretching sheet with fluid particle suspension and temperature jump”, Results in physics, Vol. 7, pp. 3976-3983 (2017).
6.    Irfan, M., Khan, M. and Khan, W. A. “Numerical analysis of unsteady 3D flow of Carreau nanofluid with variable thermal conductivity and heat source/sink”, Results in physics, Vol. 7, pp. 3315-3324 (2017).
7.    Kefayati, G. R. and Tang, H. “MHD thermosolutal natural convection and entropy generation of Carreau fluid in a heated enclosure with two inner circular cold cylinders”, using LBM. International Journal of Heat and Mass Transfer, vol. 126, pp. 508-530 (2018).
8.    Irfan, M., Khan, W. A., Khan, M. and M. M. Gulzar, “Influence of Arrhenius activation energy in chemically reactive radiative flow of 3D Carreau nanofluid with nonlinear mixed convection”, Journal of Physics and Chemistry of Solids, Vol. 125, pp. 141-152 (2019).
9.    Choi, S. U. S., and Eastman, J. A., “Enhancing thermal conductivity of fluids with nanoparticles, Presented at ASME International Mechanical Engineering Congress and Exposition”, (1995).
10.    Sohail, M., Naz, R. and Raza, R. "Application of double diffusion theories to Maxwell nanofluid under the appliance of thermal radiation and gyrotactic microorganism", Multidiscipline Modeling in Materials and Structures, (2019).
11.    Bilal, S., Sohail, M., Naz, R. and Malik, M. Y. “Dynamical and Optimal Procedure to Analyse the Exhibition of Physical Attribute Imparted by Sutterby Magneto Nano Fluid in Darcy medium yield by Axially Stretched Cylinder”, Canadian J. Phys., https://doi.org/10.1139/cjp-2018-0581, (2019).
12.    Sohail, M. and Naz, R., “Modified heat and mass transmission models in the magnetohydrodynamic flow of Sutterby nanofluid in stretching cylinder”, Physica A: Statistical Mechanics and its Applications, p.124088, (2020).
13.    Sohail, M. and Raza, R. "Analysis of radiative magneto nano pseudo-plastic material over three dimensional nonlinear stretched surface with passive control of mass flux and chemically responsive species", Multidiscipline Modeling in Materials and Structures, (2020).
14.    Khan, H., Haneef, M., Shah, Z., Islam, S., Khan, W. and Muhammad, S., “The combined magneto hydrodynamic and electric field effect on an unsteady Maxwell nanofluid flow over a stretching surface under the influence of variable heat and thermal radiation”, Applied Sciences, 8(2), p.160, 2018.
15.    Sohail, M., Naz, R. and Bilal, S., “Thermal performance of an MHD radiative Oldroyd-B nanofluid by utilizing generalized models for heat and mass fluxes in the presence of bioconvective gyrotactic microorganisms and variable thermal conductivity”, Heat Transfer-Asian Research, 48(7), pp.2659-2675, (2019).
16.    Bhatti, M. M., Zeeshan, A., Ellahi, R. and Shit, G. C. “Mathematical modeling of heat and mass transfer effects on MHD peristaltic propulsion of two-phase flow through a Darcy-Brinkman-Forchheimer porous medium”, Advanced Powder Technology, Vol. 29 (5), pp. 1189-1197 (2018).
17.    Dogonchi, S., Alizadeh, M., and Ganji, D. D. “Investigation of MHD Go-water nanofluid flow and heat transfer in a porous channel in the presence of thermal radiation effect”, Advanced Powder Technology, Vol. 28 (7), pp. 1815-1825, (2017).
18.    Prasad, P. D., Kumar, R. V. M. S. S. K., and Varma, S. V. K. “Heat and mass transfer analysis for the MHD flow of nanofluid with radiation absorption”, Ain Shams Engineering Journal, (2016).
19.    Hamid, M., Usman, M., Khan, Z.H., Haq, R.U. and Wang, W., “Numerical study of unsteady MHD flow of Williamson nanofluid in a permeable channel with heat source/sink and thermal radiation”, The European Physical Journal Plus, 133(12), p.527, (2018).
20.    Zhao, G., Wang, Z. and Jian, Y. “Heat transfer of the MHD nanofluid in porous microtubes under the electrokinetic effects”, International Journal of Heat and Mass Transfer, Vol. 130, pp. 821-830 (2019).
21.    Cengel, Y. A., and Ghanjar, A. J. “Heat and mass transfer, Fundamentals and applications, 5th edition”, pp. 1208.
22.    Vasilyeva, M., Babaei, M. E., Chung, T. and Spiridonov, D. “Multiscale modeling of heat and mass transfer in fractured media for enhanced geothermal systems applications”, Applied Mathematical Modelling, Vol. 67, pp. 159-178 (2019).
23.    Vandewalle, L. A., Vijver, R. V. D., Geem, K. M. V., and Marin, G. B. “The role of mass and heat transfer in the design of novel reactors for oxidative coupling of methane”, Chemical Engineering Science, Vol. 198, pp. 268-289 (2019).
24.    Huminic, G. and Huminic, A. “Heat transfer capability of the hybrid nanofluids for heat transfer applications”, Journal of Molecular Liquids, Vol. 272, pp. 857-870 (2018).
25.    Karman, T.V. “Uber laminare and turbulente Reibung”, Z. Angew. Math. Mech., Vol. 1, pp. 233-252 (1921).
26.    Boujo, E. and Cadot, O. “Stochastic modeling of a freely rotating disk facing a uniform flow”, Journal of Fluids and Structures, Vol. 86, pp. 34-43 (2019).
27.    Usman, M., Hamid, M., Haq, R.U. and Wang, W., “Heat and fluid flow of water and ethylene-glycol based Cu-nanoparticles between two parallel squeezing porous disks: LSGM approach”, International Journal of Heat and Mass Transfer, 123, pp.888-895, (2018).
28.    Lok, Y. Y., Merkin, J. H. and Pop, I. “Axisymmetric rotational stagnation-point flow impinging on a permeable stretching/shrinking rotating disk”, European Journal of Mechanics-B/Fluids, Vol. 72, pp. 275-292, (2018).
29.    Platt, J. R. “Bioconvection patterns in cultures of free-swimming organisms”, Science, Vol. 133, pp. 1766-1767 (1961).
30.    Chakraborty, T., Das, K., and Kundu, P. K. “Framing the impact of external magnetic field on bioconvection of a nanofluid flow containing gyrotactic microorganisms with convective boundary conditions”, Alexandria engineering journal, Vol. 57(1), pp. 61-71 (2018).
31.    Khan, M., Irfan, M. and Khan, W. A. “Impact of nonlinear thermal radiation and gyrotactic microorganisms on the Magneto-Burgers nanofluid”, International Journal of Mechanical Sciences, Vol. 130, pp. 375-382 (2017).
32.    Usman, M., Hamid, M. and Rashidi, M.M. “Gegenbauer wavelets collocation-based scheme to explore the solution of free bio-convection of nanofluid in 3D nearby stagnation point”, Neural Computing and Applications, pp.1-17, 2018.
33.    Khan, W. A. and Makinde, O. D. “MHD nanofluid bioconvection due to gyrotactic microorganisms over a convectively heat stretching sheet”, International Journal of Thermal Sciences, Vol. 81, pp. 118-124 (2014).
34.    Zhao, M., Xiao, Y., and Wang, S. “Linear stability of thermal-bioconvection in a suspension of gyrotactic micro-organisms”, International Journal of Heat and Mass Transfer, Vol. 126, pp. 95-102 (2018).
35.    Marinca, Vasile, Herişanu, N., and Nemeş, I. “Optimal homotopy asymptotic method with application to thin film flow”, Open Physics, Vol. 6.3, pp. 648-653 (2008).
36.    Ali, L., Islam, S., Gul, T., et al. “New version of Optimal Homotopy Asymptotic Method for the solution of nonlinear boundary value problems in finite and infinite intervals”, Alexandria Engineering Journal, vol. 55(3), pp. 2811- 2819 (2016).
37.    Bilal, S., Sohail, M., Naz, R., and Malik, M. Y. “Upshot of ohmically dissipated Darcy-Forchheimer slip flow of magnetohydrodynamic Sutterby fluid over radiating linearly stretched surface in view of Cash and Carp method”, Appl. Math. Mech.-Engl. Ed. https://doi.org/10.1007/s10483-019-2486-9, (2019).
38.    Naz, R., Sohail, M. and Hayat, T. "Numerical exploration of heat and mass transport for the flow of nanofluid subject to Hall and ion slip effects", Multidiscipline Modeling in Materials and Structures, (2020).
39.    Sohail, M. and Tariq, S., "Dynamical and optimal procedure to analyze the attributes of yield exhibiting material with double diffusion theories", Multidiscipline Modeling in Materials and Structures, (2019).
40.    Makinde, O., & Animasaun, I. “Bioconvection in MHD nanofluid flow with nonlinear thermal radiation and quartic autocatalysis chemical reaction past an upper surface of a paraboloid of revolution”, International Journal of Thermal Sciences, 109, 159-171, (2016).
41.    Makinde, D., and Animasaun, I. L. “Thermophoresis and Brownian motion effects on MHD bioconvection of nanofluid with nonlinear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution”, Journal of Molecular Liquids, Vol. 221, 733-743, (2016).
42.    Mutuku, W. N., and Makinde, O. D. “Hydromagnetic bioconvection of nanofluid over a permeable vertical plate due to gyrotactic microorganisms”, Computers & Fluids, 95, 88-97, (2014).
43.    Khan, W., Makinde, O., and Khan, Z. “MHD boundary layer flow of a nanofluid containing gyrotactic microorganisms past a vertical plate with Navier slip”, International Journal of Heat and Mass Transfer, 74, 285-291, (2014).
44.    Makinde, O. D., Kumara, B. P., Ramesh, G., and Gireesha, B. J. “Simultaneous Convection of Carreau Fluid with Radiation Past a Convectively Heated Moving Plate”, Defect and Diffusion Forum, 389, 60-70, (2018).
45.    Atif, S., Hussain, S. and Sagheer, M. “Effect of thermal radiation on MHD micropolar Carreau nanofluid with viscous dissipation, Joule heating, and internal heating”, Scientia Iranica, 26(6), pp.3875-3888, (2019).
46.    Iqbal, M., Ghaffari, A. and Mustafa, I. “Investigation into thermophoresis and Brownian motion effects of nanoparticles on radiative heat transfer in Hiemenz flow using spectral method”, Scientia Iranica, 26(6), pp. 3905-3916, (2019).
47.    Avinash, K., Sandeep, N., Makinde, O.D. and Animasaun, I.L. “Aligned magnetic field effect on radiative bioconvection flow past a vertical plate with thermophoresis and Brownian motion”, In Defect and Diffusion Forum (Vol. 377, pp. 127-140). Trans Tech Publications Ltd, (2017).