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Effect of Magneto Convection Nanofluid Flow in a Vertical Channel

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Abstract

An analytical study of the effect of the magneto-convective flow of immiscible fluids through a vertical channel has been investigated in the presence of a chemical reaction. One region is saturated by electrically conducting incompressible fluid, and the other is saturated by nanofluid in a vertical channel with constant transport properties. The coupled nonlinear governing equations are solved by the regular perturbation method, with the Brinkman number as a perturbation parameter since its value is always less than unity. The results are discussed in detail using plots to analyze the flow phenomena. The increase in thermal and mass Grashof numbers enhances the fluid velocity and temperature profile, whereas Hartman number, solid volume fraction, and chemical reaction parameters exhibit the opposite effect. The effect of an increase in the nanoparticle volume fraction opposes the fluid flow and diminishes the temperature distribution due to the enhanced viscosity of the nanofluid.

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Data Availability

Enquiries about data availability should be directed to the authors.

Abbreviations

\(E_{o}\) :

Applied electric field (\({\text{V/m}}\))

\(\sigma_{e}\) :

Electrical conductivity (\({\text{S/m}}\))

\(E\) :

Electric field load parameter

\(B_{o}\) :

Magnetic field (Tesla)

\(\mu_{nf}\) :

Viscosity of nanofluid (\({\text{Pa}}\;{\text{s}}\))

\(U_{0}\) :

Average velocity (\({\text{m/s}}\))

\(\rho_{nf}\) :

Density of nanofluid \(({\text{kg/m}}^{3} )\)

\(k_{f}\) :

Thermal conductivity (\({\text{W}}\;{\text{m}}^{ - 1} \;{\text{k}}^{ - 1}\))

\(k_{nf}\) :

Thermal conductivity of nanofluid

\(Br\) :

Brinkman number

\(C_{w1} ,C_{w2}\) :

Concentrations (\({\text{mol/m}}^{3}\))

\(\overline{{C_{1} ,}} \overline{{C_{2} }}\) :

Reference concentrations (\({\text{mol/m}}^{3}\))

\(C_{P}\) :

Specific heat at constant pressure (\({\text{J/kg}}\;{\text{K}}\))

\(D_{nf} \;{\text{and}}\;D_{f}\) :

Diffusion coefficient of nanofluid and base fluid (\({\text{m}}^{2} {\text{/s}}\))

g:

Acceleration due to gravity (\({\text{m}}^{2} {\text{/s}}\))

\(Gr\) :

Grashof number

\(Gc\) :

Modified Grashof number

\(M\) :

Hartman number

\(h\) :

Width of regions

\(T_{w1} ,\,T_{w2}\) :

Temperature of the boundaries (\({\text{K}}\))

\(X,Y\) :

Space co-ordinates (m)

\(\mu_{nf} ,\mu_{f}\) :

Viscosities of the nanofluid and fluid (\({\text{Pa}}\;{\text{s}}\))

\(\beta_{tf,} \;\beta_{tnf} \;{\text{and}}\;\beta_{cf,} \;\beta_{cnf}\) :

Thermal and concentration expansion coefficient of fluid and nanofluid

\(\varepsilon\) :

Solid volume fraction

\(\phi_{1} ,\phi_{2}\) :

Non-dimensional concentrations (\({\text{mol/m}}^{3}\))

\(\theta_{1} ,\theta_{2}\) :

Non-dimensional temperatures \(({\text{K}})\)

\(\alpha_{1} ,\alpha_{2}\) :

Chemical reaction parameters

References

  1. Brinkman, H.C.: The viscosity of concentrated suspensions and solution. J. Chem. Phys. 20, 571–581 (1952)

    Article  Google Scholar 

  2. Xuan, Y., Roetzel, W.: Conceptions for heat transfer correlation of nanofluids. Int. J. Heat Mass Transf. 43, 3701–3707 (2000)

    Article  Google Scholar 

  3. Hamilton, R.L., Crosser, O.K.: Thermal conductivity of heterogeneous two-component systems. I&EC Fundam. 1(3), 187–191 (1962)

    Article  Google Scholar 

  4. Choi, S.U.S., Eastman, J.A.: Enhancing thermal conductivity of fluids with nanoparticles. No. ANL/MSD/CP-84938. CONF-951135-29. Argonne National Lab.(ANL). Argonne, IL (United States), (1995)

  5. Buongiorno, J., Hu, W.: Nanofluid coolants for advanced nuclear power plants. Proc. ICAPP 5, 15–19 (2005)

    Google Scholar 

  6. Chen, H., Yulong, D., Yurong, H., Chunqing, T.: Rheological behaviour of ethylene glycol based titaniananofluids. Chem. Phys. Lett. 4–6, 333–337 (2007)

    Article  Google Scholar 

  7. Duangthongsuk, W., Wongwises, S.: Effect of thermophysical properties models on the predicting of the convective heat transfer coefficient for low concentration nanofluid. Int. Commun. Heat Mass Transf. 35(10), 1320–1326 (2008)

    Article  Google Scholar 

  8. Turkyilmazoglu, M.: Buongiorno model in a nanofluid filled asymmetric channel fulfilling zero net particle flux at the walls. Int. J. Heat Mass Transf. 126, 974–979 (2018)

    Article  Google Scholar 

  9. Turkyilmazoglu, M.: On the transparent effects of Buongiornonanofluid model on heat and mass transfer. Eur. Phys. J. Plus 4, 1–15 (2021)

    Google Scholar 

  10. Mabood, F., Khan, W.A., Ismail, A.M.: MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: a numerical study. J. Magn. Magn. Mater. 374, 569–576 (2015)

    Article  Google Scholar 

  11. Ghadikolaei, S.S., Hosseinzadeh, K., Hatami, M., Ganji, D.D.: MHD boundary layer analysis for micropolar dusty fluid containing Hybrid nanoparticles (Cu Al2O3) over a porous medium. J. Mol. Liq. 268, 813–823 (2018)

    Article  Google Scholar 

  12. Khan, M.I., Qayyum, S., Hayat, T., Waqas, M., Khan, M.I., Alsaedi, A.: Entropy generation minimization and binary chemical reaction with Arrhenius activation energy in MHD radiative flow of nanomaterial. J. Mol. Liq. 259, 274–283 (2018)

    Article  Google Scholar 

  13. Abbas, S.Z., Khan, M.I., Kadry, S., Khan, W.A., Israr-Ur Rehman, M., Waqas, M.: Fully developed entropy optimized second order velocity slip MHD nanofluid flow with activation energy. Comput. Methods Progr. Biomed. 190, 105362 (2019). https://doi.org/10.1016/j.cmpb.2020.105362

    Article  Google Scholar 

  14. Jamshed, W.: Numerical investigation of MHD impact on Maxwell nanofluid. Int. Commun. Heat Mass Transf. 120, 104973 (2021)

    Article  Google Scholar 

  15. Waqas, H., Farooq, U., Naseem, R., Hussain, S., Alghamdi, M.: Impact of MHD radiative flow of hybrid nanofluid over a rotating disk. Case Stud. Therm. Eng. 26, 101015 (2021). https://doi.org/10.1016/j.csite.2021.101015

    Article  Google Scholar 

  16. Mustafa, M., Khan, J.A., Hayat, T., Alsaedi, A.: Buoyancy effects on the MHD nanofluid flow past a vertical surface with chemical reaction and activation energy. Int. J. Heat Mass Transf. 108, 1340–1346 (2017)

    Article  Google Scholar 

  17. Hamid, A., Khan, M.: Impacts of binary chemical reaction with activation energy on unsteady flow of magneto-Williamson nanofluid. J. Mol. Liq. 262, 435–442 (2018)

    Article  Google Scholar 

  18. Ramesh, G.K., Madhukesh, J.K., Prasannakumara, B.C., Roopa, G.S.: Significance of aluminium alloys particle flow through a parallel plates with activation energy and chemical reaction. J. Therm. Anal. Calorim. 147(12), 6971–6981 (2022)

    Article  Google Scholar 

  19. Li, Y.X., Mishra, S.R., Pattnaik, P.K., Baag, S., Li, Y.M., Khan, M.I., Khan, N.B., Alaoui, M.K., Khan, S.U.: Numerical treatment of time dependent magnetohydrodynamicnanofluid flow of mass and heat transport subject to chemical reaction and heat source. Alex. Eng. J. 61(3), 2484–2491 (2022)

    Article  Google Scholar 

  20. Ahmed, M.A., Shuaib, N.H., Yusoff, M.Z.: Numerical investigations on the heat transfer enhancement in a wavy channel using nanofluid. Int. J. Heat Mass Transf. 55(21–22), 5891–5898 (2012)

    Article  Google Scholar 

  21. Karimipour, A., D’Orazio, A., Shadloo, M.: The effects of different nano particles of Al2O3 and Ag on the MHD nano fluid flow and heat transfer in a microchannel including slip velocity and temperature jump. Phys. E Low Dimens. Syst. Nanostruct. 86, 146–153 (2017)

    Article  Google Scholar 

  22. Ibanez, G., Lopez, A., Lopez, I., Pantoja, J., Moreira, J., Lastres, O.: Optimization of MHD nanofluid flow in a vertical microchannel with a porous medium, nonlinear radiation heat flux, slip flow and convective–radiative boundary conditions. J. Therm. Anal. Calorim. 135, 3401–3420 (2019)

    Article  Google Scholar 

  23. Izadi, M., Pour, H., Karimdoost, S.M.R., Yasuri, A., Chamkha, A.J.: Mixed convection of a nanofluid in a three-dimensional channel. J. Therm. Anal. Calorim. 136(6), 2461–2475 (2019)

    Article  Google Scholar 

  24. Tu, J., Qi, C., Tang, Z., Tian, Z., Chen, L.: Experimental study on the influence of bionic channel structure and nanofluids on power generation characteristics of waste heat utilisation equipment. Appl. Therm. Eng. 202, 117893 (2022)

    Article  Google Scholar 

  25. Sarchami, A., Najafi, M., Imam, A., Houshfar, E.: Experimental study of thermal management system for cylindrical Li-ion battery pack based on nanofluid cooling and copper sheath. Int. J. Therm. Sci. 171, 107244 (2022)

    Article  Google Scholar 

  26. Javid, K., Ellahi, M., Al-Khaled, K., Raza, M., Khan, S.U., Khan, M.I., El-Zahar, E.R., Gouadria, S., Afzaal, M., Khan, M.I.: EMHD creeping rheology of nanofluid through a micro-channel via ciliated propulsion under porosity and thermal effects. Case Stud. Therm. Eng. 30, 101746 (2022)

    Article  Google Scholar 

  27. Siddabasappa, C., Siddheshwar, P.G., Makinde, O.D.: A study on entropy generation and heat transfer in a magnetohydrodynamic flow of a couple-stress fluid through a thermal nonequilibrium vertical porous channel. Heat Transf. 50(6), 6377–6400 (2021)

    Article  Google Scholar 

  28. Siddabasappa, C., Kalpana, G., Babitha.: A numerical study on heat transfer and MHD flow of a Newtonian through a porous channel—local thermal nonequilibrium model. Heat Transf. 52, 949–967 (2023)

  29. Siddabasappa, C.: Unsteady magneto-hydrodynamic flow through saturated porous medium with thermal non-equilibrium and radiation effects. Int. J. Appl. Comput. Math. 6, 1–23 (2020)

    Article  MathSciNet  Google Scholar 

  30. Prathapkumar, J., Umavathi, J.C., Shreedevi, K.: Free convective flow of electrically conducting and viscous immiscible fluid flow in a vertical channel in the presence of first-order chemical reaction. Heat Transf. Asian Res. 44(7), 657–680 (2015)

    Article  Google Scholar 

  31. Umavathi, J.C., Hemavathi, K.: Flow and heat transfer of composite porous medium saturatured with nanofluid. Propuls. Power Res. 8(2), 173–181 (2019)

    Article  Google Scholar 

  32. Maxwell, J.: Treatise a on electricity and magnetisum, 2nd edn. Oxford University Press, Cambridge, UK (1904)

  33. Shekhar, M., Umavathi, J.C.: Influence of viscous dissipation on non-Darcy mixed convection flow of nanofluid. Heat Transf. Asian Res. 46, 176–199 (2017)

    Article  Google Scholar 

  34. Malshetty, M.S., Umavathi, J.C., Kumar, J.P.: Magnetoconvection of two immiscible fluids in vertical enclosure. Heat Mass Transf. 42, 977–993 (2006)

    Article  Google Scholar 

  35. Umavathi, J.C., Liu, I.C., Sheremet, M.A.: Convective heat transfer in a vertical rectangular duct filled with nanofluid. HTAR 45, 661–679 (2016)

    Google Scholar 

  36. Umavathi, J.C., Shermet, M.A.: Influence of temperature dependent conductivity of a nanofluid in a vertical rectangular duct. Int. J. Nonlinear Mech. 78, 17–28 (2016)

    Article  Google Scholar 

  37. Umavathi, J.C., Ojjela, O., Vajravelu, K.: Numerical analysis of natural convective flow and heat transfer of nanofluids in a vertical rectangular duct using Darcy-Forchheimer–Brinkman model. IJTS 11, 511–524 (2017)

    Google Scholar 

  38. Venkateswarlu, M., Ramana Reddy, G.V., Lakshmi, D.V.: Radiation effects on MHD boundary layer flow of liquid metal over a porous stretching surface in porous medium with heat generation. J. Korean Soc. Ind. Appl. Math. 1983, 102 (2015)

    MathSciNet  Google Scholar 

  39. Turkyimazoglu, M., Pop, I.: Soret and heat source effects on the unsteady radiative MHD free convection flow from an impulsively started infinite vertical plate. Int. J. Heat Mass Transf. 55, 7635–7644 (2012)

    Article  Google Scholar 

  40. Venkateswarlu, M., Ramana Reddy, G.V., Lakshmi, D.V.: Thermal diffusion and radiation effects on unsteady MHD free convection heat and mass transfer flow past a linearly accelerated vertical porous plate with variable temperature and mass diffusion. J. Korean Soc. Ind. Appl. Math. 18, 257–268 (2014)

    MathSciNet  Google Scholar 

  41. Attar, M.A., Roshani, M., Hosseinzadeh, Kh., Ganji, D.D.: Analytical solution of fractional differential equations by Akbari–Ganji’s method. Partial Differ. Equ. Appl. Math. 6, 100450 (2022). https://doi.org/10.1016/j.padiff.2022.100450

    Article  Google Scholar 

  42. Fallah Najafabadi, M., Talebi Rostami, H., Hosseinzadeh, K., Ganji, D.D.: Hydrothermal study of nanofluid flow in channel by RBF method with exponential boundary conditions. Proc. Inst. Mech. Eng. Part E J. Process Mech. Eng. 237(6), 2268–2277 (2023). https://doi.org/10.1177/09544089221133909

    Article  Google Scholar 

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Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sharnbasava University, Kalaburgi, Karnataka, 585103, India

    K. Shreedevi & G. Yamanappa

  2. Department of Sciences and Humanities, CHRIST (Deemed to be University), Bengaluru, Karnataka, 560 074, India

    C. Siddabasappa

  3. Department of Mathematics, Manipal Institute of Technology, Bengaluru Manipal Academy of Higher Education, Manipal, India

    S. Sindhu

Authors
  1. K. Shreedevi
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  2. G. Yamanappa
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  3. C. Siddabasappa
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  4. S. Sindhu
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Contributions

K. Shreedevi have done the Formulation of the problem, G. Yamanappa has done the programming part, solution, and graphs, C. Siddabasappa and S. Sindhu has wrote this manuscript, and approved the final manuscript.

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Correspondence to K. Shreedevi.

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Appendix

Appendix

$$\begin{aligned} & B_{1} = \frac{{hsinh(\alpha_{2} )}}{{d\alpha cosh(\alpha_{2} )sinh(\alpha_{1} ) + hcosh(\alpha_{1} )sinh(\alpha_{2} )}}, \\&\quad \;B_{2} = \frac{{d\alpha cosh(\alpha_{2} )}}{{d\alpha cosh(\alpha_{2} )sinh(\alpha_{1} ) + hcosh(\alpha_{1} )sinh(\alpha_{2} )}}, \end{aligned}$$
$$\begin{aligned}& B_{3} = \frac{{hsinh(\alpha_{2} )}}{{d\alpha cosh(\alpha_{2} )sinh(\alpha_{1} ) + hcosh(\alpha_{1} )sinh(\alpha_{2} )}}, \\&\quad \;B_{4} = \frac{{hcosh(\alpha_{2} )}}{{d\alpha cosh(\alpha_{2} )sinh(\alpha_{1} ) + hcosh(\alpha_{1} )sinh(\alpha_{2} )}}, \end{aligned}$$
$$ c_{1} = - \frac{1}{1 + hk},\;c_{2} = \frac{hk}{{1 + hk}},\;c_{3} = - \frac{hk}{{1 + hk}},\;c_{4} = \frac{hk}{{1 + hk}},k = k_{nf} {/}k_{f} ,d = D_{nf} {/}D_{f} , $$
$$ r_{1} = \frac{{ - \left( {M^{2} E - GR_{T} c_{2} + p} \right)}}{{M^{2} }},\;r_{2} = \frac{{GR_{T} c_{1} }}{{M^{2} }},\;r_{3} = \frac{{ - GR_{C} B_{1} }}{{\alpha_{1}^{2} - M^{2} }},\;r_{4} = \frac{{ - GR_{C} B_{2} }}{{\alpha_{1}^{2} - M^{2} }}, $$
$$ r_{5} = \frac{{GR_{T} a_{1} c_{4} + a_{2} }}{{\sigma^{2} }},\;r_{6} = \frac{{GR_{T} a_{1} c_{3} }}{{\sigma^{2} }},\;r_{7} = \frac{{ - GR_{C} a_{1} B_{3} }}{{\alpha_{2}^{2} - \sigma^{2} }},\;r_{8} = \frac{{ - GR_{C} a_{1} B_{4} }}{{\alpha_{2}^{2} - \sigma^{2} }}, $$
$$\begin{aligned} s_{1} & = \frac{{p_{1} }}{2},\;s_{2} = \frac{{p_{2} }}{6},\;s_{3} = \frac{{p_{3} }}{12},\;s_{4} = \frac{{p_{4} \alpha_{1} - 2p_{15} }}{{\alpha_{1}^{3} }}, \\ \;s_{5} &= \frac{{p_{5} \alpha_{1} - 2p_{14} }}{{\alpha_{1}^{3} }},\;s_{6} = \frac{{p_{6} }}{{4\alpha_{1}^{2} }},\;s_{7} = \frac{{p_{7} }}{{4\alpha_{1}^{2} }},\;s_{8} = \frac{{p_{8} }}{{4M^{2} }},\end{aligned} $$
$$\begin{aligned} s_{9} &= \frac{{p_{9} }}{{4M^{2} }},\;s_{10} = \frac{{p_{10} M - 2p_{13} }}{{M^{3} }},\;s_{11} = \frac{{p_{11} M - 2p_{12} }}{{M^{3} }}, \\ \;s_{12} &= \frac{{p_{12} }}{{M^{2} }},\;s_{13} = \frac{{p_{13} }}{{M^{2} }},\;s_{14} = \frac{{p_{14} }}{{\alpha_{1}^{2} }},\;s_{15} = \frac{{p_{15} }}{{\alpha_{1}^{2} }},\end{aligned} $$
$$ s_{16} = \frac{{p_{16} }}{{\left( {\alpha_{1} + M} \right)^{2} }},\;s_{17} = \frac{{p_{17} }}{{\left( {\alpha_{1} - M} \right)^{2} }},\;s_{18} = \frac{{p_{18} }}{{\left( {\alpha_{1} + M} \right)^{2} }},\;s_{19} = \frac{{p_{19} }}{{\left( {\alpha_{1} - M} \right)^{2} }}, $$
$$\begin{aligned} p_{1} &= \frac{{ - \left( {2r_{2}^{2} + 2r_{1}^{2} M^{2} + 2E^{2} M^{2} + \left( {r_{3}^{2} - r_{4}^{2} } \right)\left( {M^{2} - \alpha_{1}^{2} } \right) + 4Er_{1} M^{2} } \right)}}{2}, \\ \;p_{2} &= - \left( {2r_{1} r_{2} M^{2} + 2Er_{2} M^{2} } \right),\end{aligned} $$
$$ p_{3} = - r_{2}^{2} M^{2} ,\;p_{4} = - \left( {2r_{2} r_{4} \alpha_{1} + 2r_{1} r_{3} M^{2} + 2Er_{3} M^{2} } \right),\;p_{5} = - \left( {2r_{2} r_{3} \alpha_{1} + 2r_{1} r_{4} M^{2} + 2Er_{4} M^{2} } \right), $$
$$ \begin{aligned} p_{6}& = \frac{{ - \left( {r_{3}^{2} + r_{4}^{2} } \right)\left( {\alpha_{1}^{2} + M^{2} } \right)}}{2},\;p_{7} = - r_{3} r_{4} \left( {\alpha_{1}^{2} + M^{2} } \right), \\ \;p_{8} &= - \left( {A_{1}^{2} M^{2} + A_{2}^{2} M^{2} } \right),\;p_{9} = - 2A_{1} A_{2} M^{2} ,\end{aligned} $$
$$\begin{aligned} p_{10} &= - \left( {2A_{2} r_{2} M + 2A_{1} r_{1} M^{2} + 2A_{1} EM^{2} } \right), \\ \;p_{11}& = - \left( {2A_{1} r_{2} M + 2A_{2} r_{1} M^{2} + 2A_{2} EM^{2} } \right),\;p_{12} = - 2A_{1} r_{2} M^{2} ,\end{aligned} $$
$$ \begin{aligned} p_{13} &= - 2A_{2} r_{2} M^{2} ,\;p_{14} = - 2r_{2} r_{3} M^{2} ,\;p_{15} = - 2r_{2} r_{4} M^{2} , \\ \;p_{16} &= - \left( {A_{1} r_{3} \alpha_{1} M + r_{4} A_{2} M^{2} + r_{4} A_{2} M\alpha_{1} + r_{3} A_{1} M^{2} } \right),\end{aligned} $$
$$ \begin{aligned} p_{17}& = - \left( {A_{2} r_{4} \alpha_{1} M + r_{3} A_{1} M^{2} - r_{3} A_{1} M\alpha_{1} - r_{4} A_{2} M^{2} } \right), \\ \;p_{18} &= - \left( {A_{2} r_{3} \alpha_{1} M + r_{4} A_{1} M^{2} + r_{4} A_{1} M\alpha_{1} + r_{3} A_{2} M^{2} } \right),\end{aligned} $$
$$ p_{19} = - \left( {A_{2} r_{3} \alpha_{1} M + r_{4} A_{1} M^{2} - r_{4} A_{1} M\alpha_{1} - r_{3} A_{2} M^{2} } \right), $$
$$ H_{1} = \frac{{GR_{T} \left( {M^{4} A_{6} + 2s_{1} M^{2} + 24s_{3} } \right)}}{{M^{6} }},\;H_{2} = \frac{{GR_{T} \left( {M^{2} A_{5} + 6s_{2} } \right)}}{{M^{4} }},\;H_{3} = \frac{{GR_{T} \left( {M^{2} s_{1} + 12s_{3} } \right)}}{{M^{4} }},\;H_{4} = \frac{{GR_{T} s_{2} }}{{M^{2} }}, $$
$$ H_{5} = \frac{{GR_{T} s_{3} }}{{M^{2} }},\;H_{6} = \frac{{2\alpha_{1} GR_{T} s_{15} - GR_{T} s_{4} \left( {\alpha_{1}^{2} - M^{2} } \right)}}{{\left( {\alpha_{1}^{2} - M^{2} } \right)^{2} }},\;H_{7} = \frac{{2\alpha_{1} GR_{T} s_{14} - GR_{T} s_{5} \left( {\alpha_{1}^{2} - M^{2} } \right)}}{{\left( {\alpha_{1}^{2} - M^{2} } \right)^{2} }}, $$
$$ H_{8} = \frac{{ - GR_{T} s_{6} }}{{4\alpha_{1}^{2} - M^{2} }},\;H_{9} = \frac{{ - GR_{T} s_{7} }}{{4\alpha_{1}^{2} - M^{2} }},\;H_{10} = \frac{{ - GR_{T} s_{8} }}{{3M^{2} }}, $$
$$ H_{11} = \frac{{ - GR_{T} s_{9} }}{{3M^{2} }},\;H_{12} = \frac{{GR_{T} s_{12} - 2GR_{T} s_{11} M}}{{4M^{2} }},\;H_{13} = \frac{{GR_{T} s_{13} - 2GR_{T} s_{10} M}}{{4M^{2} }}, $$
$$ H_{14} = \frac{{ - GR_{T} s_{14} }}{{\alpha_{1}^{2} - M^{2} }},\;H_{15} = \frac{{ - GR_{T} s_{15} }}{{\alpha_{1}^{2} - M^{2} }},\;H_{16} = \frac{{ - GR_{T} s_{16} }}{{\left( {\alpha_{1} + M} \right)^{2} - M^{2} }},\;H_{17} = \frac{{ - GR_{T} s_{17} }}{{\left( {\alpha_{1} - M} \right)^{2} - M^{2} }}, $$
$$ H_{18} = \frac{{ - GR_{T} s_{18} }}{{\left( {\alpha_{1} + M} \right)^{2} - M^{2} }},\;H_{19} = \frac{{ - GR_{T} s_{19} }}{{\left( {\alpha_{1} - M} \right)^{2} - M^{2} }},\;H_{20} = \frac{{ - GR_{T} s_{13} }}{4M},\;H_{21} = \frac{{ - GR_{T} s_{12} }}{4M}, $$
$$\begin{aligned} s_{20} &= \frac{{12r_{7} r_{8} \alpha_{2} a_{3} }}{24}, \\ \;s_{21} &= \frac{{ - 12r_{6}^{2} a_{3} }}{24} - \frac{{12r_{5}^{2} a_{3} \sigma^{2} }}{24} - \frac{{6r_{7}^{2} a_{3} \sigma^{2} }}{24} + \frac{{6r_{8}^{2} a_{3} \sigma^{2} }}{24} + \frac{{6r_{7}^{2} a_{3} \alpha_{2}^{2} }}{24} - \frac{{6r_{8}^{2} a_{3} \alpha_{2}^{2} }}{24},\end{aligned} $$
$$\begin{aligned} s_{22} &= \frac{{ - 8r_{5} r_{6} a_{3} \sigma^{2} }}{24},\;s_{23} = \frac{{ - 2r_{6}^{2} a_{3} \sigma^{2} }}{24}, \\ \;s_{24} &= \frac{{ - 48A_{3} r_{5} a_{3} \sigma }}{24\sigma } + \frac{{48A_{4} r_{6} a_{3} }}{24\sigma },\;s_{25} = \frac{{ - 48A_{4} r_{5} a_{3} }}{24} + \frac{{48A_{3} r_{6} a_{3} }}{24\sigma },\end{aligned} $$
$$ s_{26} = \frac{{ - 48A_{3} r_{6} a_{3} }}{24},\;s_{27} = \frac{{ - 48A_{4} r_{6} a_{3} }}{24},\;s_{28} = \frac{{ - 6A_{3}^{2} a_{3} }}{24} - \frac{{ - 6A_{4}^{2} a_{3} }}{24},\;s_{29} = \frac{{ - 12A_{3} A_{4} a_{3} }}{24}, $$
$$\begin{aligned} s_{30} &= \frac{{ - 48r_{5} r_{7} A_{4} a_{3} \sigma^{2} }}{{24\alpha_{2}^{2} }} - \frac{{48r_{6} r_{8} a_{3} }}{{24\alpha_{2} }} + \frac{{96r_{6} r_{8} a_{3} \sigma^{2} }}{{24\alpha_{2}^{3} }}, \\ \;s_{31} &= \frac{{96r_{6} r_{7} a_{3} \sigma^{2} }}{{24\alpha_{2}^{3} }} - \frac{{48r_{5} r_{8} a_{3} \sigma^{2} }}{{24\alpha_{2}^{2} }} - \frac{{48r_{6} r_{7} a_{3} }}{{24\alpha_{2} }},\end{aligned} $$
$$\begin{aligned} s_{32} &= \frac{{ - 3r_{7}^{2} a_{3} }}{24} - \frac{{3r_{8}^{2} a_{3} }}{24} - \frac{{3r_{7}^{2} a_{3} \sigma^{2} }}{{24\alpha_{2}^{2} }} - \frac{{3\sigma^{2} r_{8}^{2} a_{3} }}{{24\alpha_{2}^{2} }}, \\ \;s_{33} &= \frac{{ - 6r_{7} r_{8} a_{3} }}{24} - \frac{{6r_{7} r_{8} a_{3} \sigma^{2} }}{{24\alpha_{2}^{2} }},\;s_{34} = - \frac{{48r_{6} r_{7} a_{3} \sigma^{2} }}{{24\alpha_{2}^{2} }},\end{aligned} $$
$$ s_{35} = - \frac{{48r_{6} r_{8} a_{3} \sigma^{2} }}{{24\alpha_{2}^{2} }},\;s_{36} = - \frac{{24A_{3} r_{7} a_{3} \sigma }}{{24\left( {\sigma - \alpha_{2} } \right)}} + \frac{{24A_{4} r_{8} a_{3} \sigma }}{{24\left( {\sigma - \alpha_{2} } \right)}},\;s_{37} = - \frac{{24A_{4} r_{7} a_{3} \sigma }}{{24\left( {\sigma - \alpha_{2} } \right)}} + \frac{{24A_{3} r_{8} a_{3} \sigma }}{{24\left( {\sigma - \alpha_{2} } \right)}}, $$
$$\begin{aligned} s_{38} &= - \frac{{24A_{4} r_{7} a_{3} \sigma }}{{24\left( {\sigma + \alpha_{2} } \right)}} - \frac{{24A_{3} r_{8} a_{3} \sigma }}{{24\left( {\sigma + \alpha_{2} } \right)}}, \\ \;s_{39} &= - \frac{{24A_{3} r_{7} a_{3} \sigma }}{{24\left( {\sigma + \alpha_{2} } \right)}} - \frac{{24A_{4} r_{8} a_{3} \sigma }}{{24\left( {\sigma + \alpha_{2} } \right)}},\;s_{40} = - \frac{{3r_{7}^{2} a_{3} \sigma^{2} }}{{24\alpha_{2}^{2} }} - \frac{{3r_{8}^{2} a_{3} \sigma^{2} }}{{24\alpha_{2}^{2} }},\end{aligned} $$
$$\begin{aligned} H_{22} &= a_{1} \left( {\frac{{\sigma^{4} A_{10} + \sigma^{4} s_{40} + \sigma^{2} 2s_{21} + 24s_{23} }}{{\sigma^{6} }}} \right), \\ \;H_{23} &= a_{1} \left( {\frac{{\sigma^{2} A_{9} + \sigma^{2} s_{20} + 6s_{22} }}{{\sigma^{4} }}} \right),\;H_{24} = a_{1} \left( {\frac{{\sigma^{2} s_{21} + 12s_{23} }}{{\sigma^{4} }}} \right),\end{aligned} $$
$$\begin{aligned} H_{25} &= \frac{{a_{1} s_{22} }}{{\sigma^{2} }},\;H_{26} = \frac{{a_{1} s_{23} }}{{\sigma^{2} }},\;H_{27} = - a_{1} \left( {\frac{{s_{24} }}{2\sigma } - \frac{{s_{27} }}{{4\sigma^{2} }}} \right), \\ \;H_{28}& = - a_{1} \left( {\frac{{s_{25} }}{2\sigma } - \frac{{s_{26} }}{{4\sigma^{2} }}} \right),\;H_{29} = \frac{{ - a_{1} s_{26} }}{4\sigma },\;H_{30} = \frac{{ - a_{1} s_{27} }}{4\sigma },\end{aligned} $$
$$ \begin{aligned} H_{31} &= \frac{{ - a_{1} s_{28} }}{{3\sigma^{2} }},\;H_{32} = \frac{{ - a_{1} s_{29} }}{{3\sigma^{2} }},\;H_{33} = \frac{{ - a_{1} s_{30} }}{{\alpha_{2}^{2} - \sigma^{2} }}, \\ \;H_{34} &= \frac{{ - a_{1} s_{31} }}{{\alpha_{2}^{2} - \sigma^{2} }},\;H_{35} = \frac{{ - a_{1} s_{32} }}{{4\alpha_{2}^{2} - \sigma^{2} }},\;H_{36} = \frac{{ - a_{1} s_{33} }}{{4\alpha_{2}^{2} - \sigma^{2} }},\end{aligned} $$
$$ \begin{aligned} H_{37}& = - a_{1} \left( {\frac{{s_{34} y\left( {\alpha_{2}^{2} - \sigma^{2} } \right) - 2\alpha_{2} s_{35} }}{{\left( {\alpha_{2}^{2} - \sigma^{2} } \right)^{2} }}} \right), \\ \;H_{38} &= - a_{1} \left( {\frac{{s_{35} y\left( {\alpha_{2}^{2} - \sigma^{2} } \right) - 2\alpha_{2} s_{34} }}{{\left( {\alpha_{2}^{2} - \sigma^{2} } \right)^{2} }}} \right),\;H_{39} = \frac{{ - a_{1} s_{36} }}{{\left( {\sigma - \alpha_{2} } \right)^{2} - \sigma^{2} }},\end{aligned} $$
$$ H_{40} = \frac{{ - a_{1} s_{37} }}{{\left( {\sigma - \alpha_{2} } \right)^{2} - \sigma^{2} }},\;H_{41} = \frac{{ - a_{1} s_{38} }}{{\left( {\sigma + \alpha_{2} } \right)^{2} - \sigma^{2} }},\;H_{42} = \frac{{ - a_{1} s_{39} }}{{\left( {\sigma + \alpha_{2} } \right)^{2} - \sigma^{2} }}. $$

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Shreedevi, K., Yamanappa, G., Siddabasappa, C. et al. Effect of Magneto Convection Nanofluid Flow in a Vertical Channel. Int. J. Appl. Comput. Math 10, 82 (2024). https://doi.org/10.1007/s40819-024-01709-5

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