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Bivariate Spectral Local Linearisation Method (BSLLM) for Unsteady MHD Micropolar-Nanofluids with Homogeneous–Heterogeneous Chemical Reactions Over a Stretching Surface

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Abstract

A mathematical model of MHD micropolar-nanofluid flow deformed by a stretchable surface is presented with a homogeneous–heterogeneous reactions given by isothermal cubic autocatalator kinetics and first order kinetics. We assumed the existence of an induced magnetic field. The basic microrotation flow and heat mass transfer nonlinear equations are solved using the bivariate spectral local linearisation method. An analysis of the accuracy of the method is given using residual errors, and the influence of certain variables on the fluid properties are discussed. The results show, that the concentration distribution is reduced by an increase in the homogeneous reaction parameter while it increases with the Schmidt number. The rate of heat transfer is enhanced by larger values of the Prandtl number and the thermophoresis parameter.

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Abbreviations

\(C_a, C_b\) :

Concentration of homogeneous–heterogeneous reactions species

\(C_{fx }\), \(Nu_{x}\), \(Sh_{x}\) :

Local skin friction, Nusselt and Sherwood number

\(C_{\infty }\) :

Species concentration far away from the wall

\(T_{\infty }\) :

Temperature of the fluid far away from the wall

\(C_{p}\) :

Specific heat at constant pressure

\(D_B\) :

Mass diffusivity

f :

Dimensionless stream function

Ha :

Hartmann number

K :

Material parameter

\(K_s\) :

Heterogeneous reaction parameter

\(k_{1}^{*}\) :

Thermal conductivity of the fluid

N :

Angular velocity

\(Q_{0}\) :

Heat generation coefficient

Pr :

Prandtl number

\(D_{B}\) :

Brownian diffusion coefficient

\(D_{T}\) :

Thermophoretic diffusion coefficient

\(N_b\) :

Brownian motion parameter

\(N_t\) :

Thermophoresis parameter

\(Sc_A\) :

Scidth number

u, v :

Velocity component

\(\psi \) :

Stream function

\(\lambda \) :

Homogeneous reaction rate parameter

\(\rho \) :

Density of the fluid

\(\mu \) :

Dynamic viscosity of the fluid

\(\nu \) :

Kinematic viscosity

\(\xi , \eta \) :

Transformed variables

\(\epsilon \) :

Ratio of the diffusion coefficient

C :

Concentration

T :

Temperature

w :

Conditions at the wall

\(\infty \) :

Free stream condition

References

  1. Kumari, M., Slaouti, A., Takhar, H.S., Nakamura, S., Nath, S.: Unsteady free convection flow over a continuous moving vertical surface. Acta Mech. 116, 75–82 (1996)

    Article  Google Scholar 

  2. Pal, D., Chatterjee, S.: Heat and mass transfer in MHD non-Darcian flow of a micropolar fluid over a stretching sheet embedded in a porous media with non-uniform heat source and thermal radiation. Commun. Nonlinear Sci. Numer. Simulat. 15, 1843–1857 (2010)

    Article  Google Scholar 

  3. Sithole, H., Mondal, H., Sibanda, P.: Entropy generation in a second grade magnetohydrodynamic nanofluid flow over a convectively heated stretching sheet with nonlinear thermal radiation and viscous dissipation. Result Phys. 9, 1077–1085 (2018)

    Article  Google Scholar 

  4. Ishak, A., Nazar, R., Pop, I.: Heat transfer over a stretching surface with variable surface heat flux in micropolar fluids. Phys. Lett. A 372, 559–561 (2008)

    Article  Google Scholar 

  5. Muthuraj, R., Srinivas, S.: Fully developed MHD flow of a micropolar and viscous fluids in a vertical porous space using HAM. Int. J. Appl. Math. Mech. 6(11), 55–78 (2010)

    Google Scholar 

  6. Shercliff, J.A.: A Text Book of Magnetohydrodynamics. Pergamon Press Inc., New York (1965)

    Google Scholar 

  7. Pal, D., Mondal, H.: Soret and Dufour effects on MHD non-Darcian mixed convection heat and mass transfer over a stretching sheet with non-uniform heat source/sink. Phys. B 407, 642–651 (2012)

    Article  Google Scholar 

  8. Sarkar, A., Kundu, P.K.: Outcomes of non-uniform heat source/sink on micropolar nanofluid flow in presence of slip boundary conditions. Int. J. Appl. Comput. Math. 3, 801–812 (2017)

    Article  MathSciNet  Google Scholar 

  9. Kameswaran, P.K., Shaw, S., Sibanda, P., Murthy, P.V.S.N.: Homogeneous–heterogeneous reactions in a nanofluid flow due to a porous stretching sheet. Int. J. Heat Mass Transf. 57(2), 465–472 (2013)

    Article  Google Scholar 

  10. Ravikiran, G., Radhakrishnamacharya, G.: Effect of homogeneous and heterogeneous chemical reactions on peristaltic transport of a Jeffrey fluid through a porous medium with slip condition. J. Appl. Fluid Mech. 8(3), 521–528 (2015)

    Article  Google Scholar 

  11. RamReddy, C., Pradeepa, T.: Influence of convective boundary condition on nonlinear thermal convection flow of a micropolar fluid saturated porous medium with homogenous–heterogenous reactions. Front. Heat Mass Transf. 8(6), 1–10 (2017)

    Google Scholar 

  12. Shaw, S., Kameswaran, P.K., Sibanda, P.: Homogeneous–heterogeneous reactions in micropolar fluid flow from a permeable stretching or shrinking sheet in a porous medium. Bound. Value Prob. 2013, 77 (2013)

    Article  MathSciNet  Google Scholar 

  13. Magagula, V.M.: Bivariate Pseudospectral Collocation Algorithms for Nonlinear Partial Differential Equations. University of KwaZulu-Natal, Ph.D Thesis (2016)

  14. Ghadikolaei, S.S., Hosseinzadeh, K., Ganji, D.D., Jafari, B.: Nonlinear thermal radiation effect on magneto Casson nanofluid flow with Joule heating effect over an inclined porous stretching sheet. Case Stud. Therm. Eng. 12, 176–187 (2018)

    Article  Google Scholar 

  15. Afridi, M.I., M, Q., Khan, I., Tlili, I.: Entropy generation in MHD mixed convection stagnation-point flow in the presence of joule and frictional heating. Case Stud. Therm. Eng. 12, 292–300 (2018)

    Article  Google Scholar 

  16. Nadeem, S., Masood, S., Mehmood, R., Sadiq, M.A.: Optimal and numerical solutions for an MHD micropolar nanofluid between rotating horizontal parallel plates. PLoS ONE. (2015). https://doi.org/10.1371/journal.pone.0124016

  17. Khan, W.A., Alshomrani, A.S., Khan, M.: Assessment on characteristics of heterogeneous–homogenous processes in three-dimensional flow of Burgers fluid. Results Phys. 6, 772–779 (2016)

    Article  Google Scholar 

  18. Hayat, T., Sajjad, R., Ellahi, R., Alsaedi, A., Muhammad, T.: Homogeneous–heterogeneous reactions in MHD flow of micropolar fluid by a curved stretching surface. J. Mol. Liq. 240, 209–220 (2017)

    Article  Google Scholar 

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Acknowledgements

This work was supported by a Claude Leon Foundation Postdoctoral Fellowship, and the University of KwaZulu-Natal, South Africa.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsvile, Pietermaritzburg, 3209, South Africa

    Hloniphile Sithole, Hiranmoy Mondal, Precious Sibanda & S. Motsa

  2. Department of Mathematics, Faculty of Science and Engineering, University of Swaziland, Private Bag 4, Matsapha, Swaziland

    Vusi Mpendulo Magagula & S. Motsa

Authors
  1. Hloniphile Sithole
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  2. Hiranmoy Mondal
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  3. Vusi Mpendulo Magagula
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  4. Precious Sibanda
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Correspondence to Hiranmoy Mondal.

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Sithole, H., Mondal, H., Magagula, V.M. et al. Bivariate Spectral Local Linearisation Method (BSLLM) for Unsteady MHD Micropolar-Nanofluids with Homogeneous–Heterogeneous Chemical Reactions Over a Stretching Surface. Int. J. Appl. Comput. Math 5, 12 (2019). https://doi.org/10.1007/s40819-018-0593-8

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