Research article
Special Issues
Convective boundary layer flow of MHD tangent hyperbolic nanofluid over stratified sheet with chemical reaction
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We investigated the combined impact of convective boundary conditions, thermal conductivity, and magnetohydrodynamic on the flow of a tangent hyperbolic nanofluid across the stratified surface. Furthermore, the ramifications of Brownian motion, thermophoresis, and activation energy were considered. Heat generation, chemical reactions, mixed convection, thermal conductivity, and other elements were considered when analyzing heat transfer phenomena. The governing equations were converted via similarity transformations into non-dimensional ordinary differential equations in order to analyze the system. Using the shooting method, the problem's solution was determined. We showed the mathematical significance of the temperature, concentration profiles, and velocity of each fluid parameter. These profiles were thoroughly described and shown graphically. The findings demonstrated that as the Weissenberg number and magnetic number increased, the fluid velocity profile decreased. Higher heat generation and thermophoresis parameters resulted in an increase in the temperature profile. Higher Brownian motion and Schmidt parameter values resulted in a drop in the concentration profile. Tables were used to discuss the numerical values of skin friction ($ {C}_{fx} $), Nusselt number ($ {Nu}_{x} $), and Sherwood number ($ S{h}_{x} $). For the greater values of Weissenberg number and mixed convection parameters, skin friction numerical values fell while Nusselt numbers rose.
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Keywords:
- Buongiorno nanofluid model,
- tangent hyperbolic nanofluid,
- heat generation,
- mixed convection,
- activation energy,
- stratified sheet
Citation: Reem K. Alhefthi, Irum Shahzadi, Husna A. Khan, Nargis Khan, M. S. Hashmi, Mustafa Inc. Convective boundary layer flow of MHD tangent hyperbolic nanofluid over stratified sheet with chemical reaction[J]. AIMS Mathematics, 2024, 9(7): 16901-16923. doi: 10.3934/math.2024821
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Abstract
We investigated the combined impact of convective boundary conditions, thermal conductivity, and magnetohydrodynamic on the flow of a tangent hyperbolic nanofluid across the stratified surface. Furthermore, the ramifications of Brownian motion, thermophoresis, and activation energy were considered. Heat generation, chemical reactions, mixed convection, thermal conductivity, and other elements were considered when analyzing heat transfer phenomena. The governing equations were converted via similarity transformations into non-dimensional ordinary differential equations in order to analyze the system. Using the shooting method, the problem's solution was determined. We showed the mathematical significance of the temperature, concentration profiles, and velocity of each fluid parameter. These profiles were thoroughly described and shown graphically. The findings demonstrated that as the Weissenberg number and magnetic number increased, the fluid velocity profile decreased. Higher heat generation and thermophoresis parameters resulted in an increase in the temperature profile. Higher Brownian motion and Schmidt parameter values resulted in a drop in the concentration profile. Tables were used to discuss the numerical values of skin friction ($ {C}_{fx} $), Nusselt number ($ {Nu}_{x} $), and Sherwood number ($ S{h}_{x} $). For the greater values of Weissenberg number and mixed convection parameters, skin friction numerical values fell while Nusselt numbers rose.
References
[1] S. U. Choi, J. A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, Illinois: Argonne National Laboratory, 1995. [2] Z. Haddad, H. F. Oztop, E. Abu-Nada, A. Mataoui, A review on natural convective heat transfer of nanofluids, Renew. Sust. Energy Rev., 16 (2012), 5363–5378. http://doi.org/10.1016/j.rser.2012.04.003 doi: 10.1016/j.rser.2012.04.003 
[3] M. Kalteh, K. Javaherdeh, T. Azarbarzin, Numerical solution of nanofluid mixed convection heat transfer in a lid-driven square cavity with a triangular heat source, Powder Technol., 253 (2014), 780–788. http://doi.org/10.1016/j.powtec.2013.12.039 doi: 10.1016/j.powtec.2013.12.039
[4] W. F. Sun, The dynamic effect on mechanical contacts between nanoparticles, Nanoscale, 5 (2013), 12658–12669. http://doi.org/10.1039/c3nr04354a doi: 10.1039/c3nr04354a
[5] E. Shahsavani, M. Afrand, R. Kalbasi, Using experimental data to estimate the heat transfer and pressure drop of non-Newtonian nanofluid flow through a circular tube: applicable for use in heat exchangers, Appl. Therm. Eng., 129 (2018), 1573–1581. https://doi.org/10.1016/j.applthermaleng.2017.10.140 doi: 10.1016/j.applthermaleng.2017.10.140
[6] A. Shafiq, A. B. Çolak, T. N. Sindhu, Designing artificial neural network of nanoparticle diameter and solid–fluid interfacial layer on single‐walled carbon nanotubes/ethylene glycol nanofluid flow on thin slendering needles, Int. J. Numer. Meth. Fl., 93 (2021), 3384–3404. https://doi.org/10.1002/fld.5038 doi: 10.1002/fld.5038
[7] A. Shafiq, A. B. Çolak, S. A. Lone, T. N. Sindhu, T. Muhammad, Reliability modeling and analysis of mixture of exponential distributions using artificial neural network, Math. Method. Appl. Sci., 47 (2022), 3308–3328. https://doi.org/10.1002/mma.8178 doi: 10.1002/mma.8178
[8] A. Shafiq, A. B. Çolak, T. N. Sindhu, Modeling of Soret and Dufour's convective heat transfer in nanofluid flow through a moving needle with artificial neural network, Arab. J. Sci. Eng., 48 (2023), 2807–2820. https://doi.org/10.1007/s13369-022-06945-9 doi: 10.1007/s13369-022-06945-9
[9] A. Shafq, S. A. Lone, T. N. Sindhu, Q. M. Al-Mdallal, G. Rasool, Statistical modeling for bioconvective tangent hyperbolic nanofuid towards stretching surface with zero mass flux condition, Sci. Rep., 11 (2021), 13869. https://doi.org/10.1038/s41598-021-93329-y doi: 10.1038/s41598-021-93329-y
[10] A. Shafiq, T. N. Sindhu, C. M. Khalique, Numerical investigation and sensitivity analysis on bioconvective tangent hyperbolic nanofluid flow towards stretching surface by response surface methodology, Alex. Eng. J., 59 (2020), 4533–4548. https://doi.org/10.1016/j.aej.2020.08.007 doi: 10.1016/j.aej.2020.08.007
[11] A. B. Çolak, A. Shafiq, T. N. Sindhu, Modeling of Darcy-Forchheimer bioconvective Powell Eyring nanofluid with artificial neural network, Chinese J. Phys., 77 (2022), 2435–2453. https://doi.org/10.1016/j.cjph.2022.04.004 doi: 10.1016/j.cjph.2022.04.004
[12] A. Shafiq, F. Mebarek-Oudina, T. N. Sindhu, A. Abidi, A study of dual stratification on stagnation point Walters' B nanofluid flow via radiative Riga plate a statistical approach, Eur. Phys. J. Plus, 136 (2021), 407. https://doi.org/10.1140/epjp/s13360-021-01394-z doi: 10.1140/epjp/s13360-021-01394-z
[13] A. Mishra, M. Kumar, Influence of viscous dissipation and heat generation/absorption on Ag-water nanofluid flow over a Riga plate with suction, Int. J. Fluid Mech. Res., 46 (2019), 113–125. https://doi.org/10.1615/InterJFluidMechRes.2018025291 doi: 10.1615/InterJFluidMechRes.2018025291
[14] A. Mishra, M. Kumar, Numerical analysis of MHD nanofluid flow over a wedge, including effects of viscous dissipation and heat generation/absorption, using Buongiorno model, Heat Transf., 50 (2021), 8453–8474. https://doi.org/10.1002/htj.22284 doi: 10.1002/htj.22284
[15] H. A. Khan, G. Nazeer, S. A. Shehzad, Darcy-Forchheimer tangent hyperbolic nanofluid flow through a vertical cone with non-uniform heat generation, J. Porous Media, 26 (2023), 1–14. https://doi.org/10.1615/JPorMedia.2022045225 doi: 10.1615/JPorMedia.2022045225
[16] T. Hayat, A. Shafiq, M. Nawaz, A. Alsaedi, MHD axisymmetric flow of third grade fluid between porous disks with heat transfer, Appl. Math. Mech., 33 (2012), 749–764. https://doi.org/10.1007/s10483-012-1584-9 doi: 10.1007/s10483-012-1584-9
[17] T. Hayat, A. Shafiq, A. Alsaedi, M. Awais, MHD axisymmetric flow of third grade fluid between stretching sheets with heat transfer, Comput. Fluids, 86 (2013), 103–108. https://doi.org/10.1016/J.compfluid.2013.07.003 doi: 10.1016/J.compfluid.2013.07.003
[18] T. Hayat, A. Shafiq, A. Alsaedi, MHD axisymmetric flow of third grade fluid by a stretching cylinder, Alex. Eng. J., 54 (2015), 205–212. https://doi.org/10.1016/j.aej.2015.03.013 doi: 10.1016/j.aej.2015.03.013
[19] A. Shafiq, Z. Hammouch, T. N. Sindhu, Bioconvective MHD flow of tangent hyperbolic nanofluid with newtonian heating, Int. J. Mech. Sci., 133 (2017), 759–766. https://doi.org/10.1016/j.ijmecsci.2017.07.048 doi: 10.1016/j.ijmecsci.2017.07.048
[20] F. Naseem, A. Shafiq, L. F. Zhao, A. Naseem, MHD biconvective flow of Powell Eyring nanofluid over stretched surface, AIP Adv., 7 (2017), 065013. https://doi.org/10.1063/1.4983014 doi: 10.1063/1.4983014
[21] S. Gupta, D. Kumar, J. Singh, MHD mixed convective stagnation point flow and heat transfer of an incompressible nanofluid over an inclined stretching sheet with chemical reaction and radiation, Int. J. Heat Mass Tran., 118 (2018), 378–387. https://doi.org/10.1016/j.ijheatmasstransfer.2017.11.007 doi: 10.1016/j.ijheatmasstransfer.2017.11.007
[22] G. Rasool, T. Zhang, A. J. Chamkha, A. Shafiq, I. Tlili, G. Shahzadi, Entropy generation and consequences of binary chemical reaction on MHD Darcy-Forchheimer Williamson nanofluid flow over non-linearly stretching surface, Entropy., 22 (2020), 18. https://doi.org/10.3390/e22010018 doi: 10.3390/e22010018
[23] A. Mishra, M. Kumar, Ohmic-viscous dissipation and heat generation/absorption effects on MHD nanofluid flow over a stretching cylinder with suction/injection, In: Advanced computing and communication technologie, Singapore: Springer, 2019, 45–55. https://doi.org/10.1007/978-981-13-0680-8_5 [24] A. Mishra, M. Kumar, Thermal performance of MHD nanofluid flow over a stretching sheet due to viscous dissipation, Joule heating and thermal radiation, Int. J. Appl. Comput. Math., 6 (2020), 123. https://doi.org/10.1007/s40819-020-00869-4 doi: 10.1007/s40819-020-00869-4
[25] A. Mishra, M. Kumar, Velocity and thermal slip effects on MHD nanofluid flow past a stretching cylinder with viscous dissipation and Joule heating, SN Appl. Sci., 2 (2020), 1350. https://doi.org/10.1007/s42452-020-3156-7 doi: 10.1007/s42452-020-3156-7
[26] G. Rasool, A. Shafiq, I. Khan, D. Baleanu, K. S. Nisar, G. Shahzadi, Entropy generation and consequences of MHD in Darcy-Forchheimer nanofluid flow bounded by non-linearly stretching surface, Symmetry, 12 (2020), 652. https://doi.org/10.3390/sym12040652 doi: 10.3390/sym12040652
[27] A. Shafiq, I. Zari, G. Rasool, I. Tlili, T. S. Khan, On the MHD Casson axisymmetric Marangoni forced convective flow of nanofluids, Mathematics., 7 (2019), 1087. https://doi.org/10.3390/math7111087 doi: 10.3390/math7111087
[28] M. Abbas, N. Khan, M. S. Hashmi, M. Inc, Numerical analysis of Marangoni convective flow of gyrotactic microorganisms in dusty Jeffrey hybrid nanofluid over a Riga plate with Soret and Dufour effects, J. Therm. Anal. Calorim., 148 (2023), 12609–12627. https://doi.org/10.1007/s10973-023-12549-8 doi: 10.1007/s10973-023-12549-8
[29] M. Abbas, N. Khan, M. S. Hashmi, M. Inc, Numerical simulation of magneto thermal Marangoni convective flow of dusty Sutterby hybrid nanofluid with variable thermal conductivity, ZAMM-Z. Angew. Math. Me., 104 (2024), e202300408. https://doi.org/10.1002/zamm.202300408 doi: 10.1002/zamm.202300408
[30] M. Abbas, N. Khan, M. S. Hashmi, H. Alotaibi, H. A. Khan, S. Rezapour, M. Inc, Importance of thermophoretic particles deposition in ternary hybrid nanofluid with local thermal non-equilibrium conditions Hamilton-Crosser and Yamada-Ota models, Case Stud. Therm. Eng., 56 (2024), 104229. https://doi.org/10.1016/j.csite.2024.104229 doi: 10.1016/j.csite.2024.104229
[31] S. Mukhopadhyay, I. C. Mandal, Magnetohydrodynamic (MHD) mixed convection slip flow and heat transfer over a vertical porous plate, Eng. Sci. Technol., 18 (2015), 98–105. https://doi.org/10.1016/j.jestch.2014.10.001 doi: 10.1016/j.jestch.2014.10.001
[32] M. Imtiaz, T. Hayat, A. Alsaedi, Mixed convection flow of Casson nanofluid over a stretching cylinder with convective boundary conditions, Adv. Powder Technol., 27 (2016), 2245–2256. https://doi.org/10.1016/j.apt.2016.08.011 doi: 10.1016/j.apt.2016.08.011
[33] R. U. Haq, Z. Hamouch, S. T. Hussain, T. Mekkaoui, MHD mixed convection flow along a vertically heated sheet, Int. J. Hydrogen Energ., 42 (2017), 15925–15932. https://doi.org/10.1016/j.ijhydene.2017.04.225 doi: 10.1016/j.ijhydene.2017.04.225
[34] M. Manzur, M. Khan, M. Rahman, Mixed convection heat transfer to cross fluid with thermal radiation: effects of buoyancy assisting and opposing flows, Int. J. Mech. Sci., 138 (2018), 515–523. https://doi.org/10.1016/j.ijmecsci.2018.02.010 doi: 10.1016/j.ijmecsci.2018.02.010
[35] K. Hsiao, L. Heat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipation, Commun. Nonlinear Sci., 15 (2010), 1803–1812. http://doi.org/10.1016/j.cnsns.2009.07.006 doi: 10.1016/j.cnsns.2009.07.006
[36] M. Waqas, M. Farooq, M. I. Khan, A. Alsaedi, T. Hayat, T. Yasmeen, Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition, Int. J. Heat Mass Tran., 102 (2016), 766–772. https://doi.org/10.1016/j.ijheatmasstransfer.2016.05.142 doi: 10.1016/j.ijheatmasstransfer.2016.05.142
[37] L. Ali, P. Kumar, Z. Iqbal, S. E. Alhazmi, S. Areekara, M. M. Alqarni, et al., The optimization of heat transfer in thermally convective micropolar-based nanofluid flow by the influence of nanoparticle's diameter and nanolayer via stretching sheet: sensitivity analysis approach, J. Non-Equil. Thermody., 48 (2023), 313–330. https://doi.org/10.1515/jnet-2022-0064 doi: 10.1515/jnet-2022-0064
[38] Y. I. Seini, O. D. Makinde, MHD boundary layer flow due to exponential stretching surface with radiation and chemical reaction, Math. Probl. Eng., 2013 (2013), 163614. https://doi.org/10.1155/2013/163614 doi: 10.1155/2013/163614
[39] S. Sinha, Effect of chemical reaction on an unsteady MHD free convective flow past a porous plate with ramped temperature, Proceedings of International Conference on Frontier in Mathematics, 2015,204–210. [40] L. Ali, B. Ali, T. Iqbal, Finite element analysis of the impact of particles aggregation on the thermal conductivity of nanofluid under chemical reaction, Wave Random and Complex Media, 2023 (2023), 2172962. https://doi.org/10.1080/17455030.2023.2172962 doi: 10.1080/17455030.2023.2172962
[41] L. Ali, A. Manan, B. Ali, Maxwell nanofluids: FEM simulation of the effects of suction/injection on the dynamics of rotatory fluid subjected to bioconvection, Lorentz, and Coriolis forces, Nanomaterials, 12 (2022), 3453. https://doi.org/10.3390/nano12193453 doi: 10.3390/nano12193453
[42] L. Ali, Y. J. Wu, B. Ali, S. Abdal, S. Hussain, The crucial features of aggregation in TiO2-water nanofluid aligned of chemically comprising microorganisms: a FEM approach, Comput. Math. Appl., 123 (2022), 241–251. https://doi.org/10.1016//j.camwa.2022.08.028 doi: 10.1016//j.camwa.2022.08.028
[43] L. Ali, B. Ali, M. B. Ghori, Melting effect on Cattaneo-Christov and thermal radiation features for aligned MHD nanofluid flow comprising microorganisms to leading edge: FEM approach, Comput. Math. Appl., 109 (2022), 260–269. https://doi.org/10.1016//j.camwa.2022.01.009 doi: 10.1016//j.camwa.2022.01.009
[44] M. Abbas, N. Khan, M. S. Hashmi, M. Inc, Aspects of chemical reaction and mixed convection in ternary hybrid nanofluid with Marangoni convection and heat source, Mod. Phys. Lett. B, 2023 (2023), 2450161. https://doi.org/10.1142/S0217984924501616 doi: 10.1142/S0217984924501616
[45] M. Abbas, N. Khan, S. A. Shehzad, Numerical analysis of Marangoni convected dusty second-grade nanofluid flow in a suspension of chemically reactive microorganisms, P. I. Mech. Eng. C-J. Mec., 238 (2024), 4400–4417. https://doi.org/10.1177//09544062231209828 doi: 10.1177//09544062231209828
[46] A. R. Bestman, Natural convection boundary layer with suction and mass transfer in a porous medium, Int. J. Energ. Res., 14 (1990), 389–396. https://doi.org/10.1002/er.4440140403 doi: 10.1002/er.4440140403
[47] O. D. Makinde, P. O. Olanrewaju, W. M. Charles, Unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture, Afr. Mat., 22 (2011), 65–78. https://doi.org/10.1007/s13370-011-0008-z doi: 10.1007/s13370-011-0008-z
[48] F. E. Alsaadi, I. Ullah, T. Hayat, F. E. Alsaadi, Entropy generation in nonlinear mixed convective flow of nanofluid in porous space influenced by Arrhenius activation energy and thermal radiation, J. Therm. Anal. Calorim., 140 (2020), 799–809. https://doi.org/10.1007/s10973-019-08648-0 doi: 10.1007/s10973-019-08648-0
[49] M. Abbas, N. Khan, A. Alshomrani, M. S. Hashmi, M. Inc, Performance-based comparison of Xue and Yamada–Ota models of ternary hybrid nanofluid flow over a slendering stretching sheet with activation energy and melting phenomena, Case Stud. Therm. Eng., 50 (2023), 103427. https://doi.org/10.1016//j.csite.2023.103427 doi: 10.1016//j.csite.2023.103427
[50] M. Abbas, N. Khan, S. A. Shehzad, Analytical simulation of magneto-marangoni convective flow of Walter-B fluid with activation energy and Soret-Dufour effects, Adv. Mech. Eng., 15 (2023), 1199049. https://doi.org/10.1177/16878132231199049 doi: 10.1177/16878132231199049
[51] G. K. Ramesh, A. J. Chamkha, B. J. Gireesha, MHD mixed convection flow of a viscoelastic fluid over a inclined surface with nonuniform heat source/sink, Can. J. Phys., 91, (2013), 1074–1080. https://doi.org/10.1139/cjp-2013-0173 doi: 10.1139/cjp-2013-0173
[52] K. L. Hsiao, viscoelastic fluid over a stretching sheet with electromagnetic effects and nonuniform heat source, Math. Probl. Eng., 2010 (2010), 1024–123. https://doi.org/10.1155/2010/740943 doi: 10.1155/2010/740943
[53] B. Ramandevi, J. V. R. Reddy, V. Sugunamma, N. Sandeep, Combined influence of viscous dissipation and non-uniform heat source on MHD non-Newtonian fluid flow with Cattaneo-Christov heat flux, Alex. Eng. J., 57 (2018), 1009–1018. https://doi.org/10.1016/j.aej.2017.01.026 doi: 10.1016/j.aej.2017.01.026
[54] M. Abbas, N. Khan, M. S. Hashmi, M. Inc, Scrutinization of marangoni convective flow of dusty hybrid nanofluid with gyrotactic microorganisms and thermophoretic particle deposition, J. Therm. Anal. Calorim., 149 (2024), 1443–1463. https://doi.org/10.1007/s10973-023-12750-9 doi: 10.1007/s10973-023-12750-9
[55] M. Bilal, S. Ashbar, Flow and heat transfer analysis of Eyring-Powell fluid over stratified sheet with mixed convection, J. Egypt. Math. Soc., 28 (2020), 40. https://doi.org/10.1186/s42787-020-00103-6 doi: 10.1186/s42787-020-00103-6
[56] K. Muhammad, S. A. M. Abdelmohsen, A. M. M.Abdelbacki, B. Ahmed, Darcy-Forchheimer flow of hybrid nanofluid subject to melting heat: A comparative numerical study via shooting method, Int. Commun. Heat Mass Tran., 135 (2022), 106160. https://doi.org/10.1016/j.icheatmasstransfer.2022.106160 doi: 10.1016/j.icheatmasstransfer.2022.106160
[57] H. U. Rasheed, Zeeshan, S. Islam, B. Ali, Q. Shah, R. Ali, Implementation of shooting technique for Buongiorno nanofluid model driven by a continuous permeable surface, Heat Transf., 52 (2023), 3119–3134. https://doi.org/10.1002/htj.22819 doi: 10.1002/htj.22819
[58] M. Jawad, A. H. Majeed, K. S. Nisar, M. B. B. Hamida, A. Alasiri, A. M. Hassan, et al., Numerical simulation of chemically reacting Darcy-Forchheimer flow of Buongiorno Maxwell fluid with Arrhenius energy in the appearance of nanoparticles, Case Stud. Therm. Eng., 50 (2023), 103413. https://doi.org/10.1016//j.csite.2023.103413 doi: 10.1016//j.csite.2023.103413
[59] G. K. Ramesh, S. A. Shehzad, T. Hayat, A. Alsaedi, Activation energy and chemical reaction in Maxwell magneto-nanoliquid with passive control of nanoparticle volume fraction, J. Braz. Soc. Mech. Sci. Eng., 40 (2018), 422. https://doi.org/10.1007/s40430-018-1353-8 doi: 10.1007/s40430-018-1353-8
[60] T. Hayat, S. B. Qayyum, B. Ahmad, M. Waqas, Radiative flow of a tangent hyperbolic fluid with convective conditions and chemical reaction, Eur. Phys. J. Plus, 131 (2016), 422. https://doi.org/10.1140/epjp/i2016-16422-x doi: 10.1140/epjp/i2016-16422-x
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Reem K. Alhefthi, Irum Shahzadi, Husna A. Khan, Nargis Khan, M. S. Hashmi, Mustafa Inc. Convective boundary layer flow of MHD tangent hyperbolic nanofluid over stratified sheet with chemical reaction[J]. AIMS Mathematics, 2024, 9(7): 16901-16923. doi: 10.3934/math.2024821
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- Figure 1. Physical geometry of flow
- Figure 2. (a) Impact of $ \tilde{n} $ on $ {f}^{'}\left(\eta \right) $, (b) impact of $ We $ on $ {f}^{'}\left(\eta \right) $
- Figure 3. (a) Impact of $ M $ on $ {f}^{'}\left(\eta \right) $, and (b) $ \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t} $of $ \lambda $ on $ {f}^{'}\left(\eta \right) $
- Figure 4. (a) Impact of $ N $ on $ {f}^{\text{'}\left(\xi \right)} $, and (b) Influence of Pr on θ(ξ)
- Figure 5. (a) Influence of Nb on θ(ξ), (b) Influence of $ Nt $ on $ \theta \left(\xi \right) $
- Figure 6. (a) Influence of $ S $ on $ \theta \left(\xi \right) $, (b) Influence of S$ c $ on $ \phi \left(\xi \right) $
- Figure 7. (a) Influence of $ \gamma $ on $ \phi \left(\xi \right) $, and (b) influence of $ Nt $ on $ \phi \left(\xi \right) $
- Figure 8. (a) Influence of $ Nb $ on $ \phi \left(\xi \right) $, and (b) influence of $ \delta $ on $ \phi \left(\xi \right) $
- Figure 9. (a) Influence of $ E $ on $ \phi \left(\xi \right) $, and (b) influence of $ n $ on $ \phi \left(\xi \right) $
- Figure 10. (a) Streamlines when $ M = 0.3 $, and (b) streamlines when $ M = 3.34 $
- Figure 11. (a) Isothermal graph when $ Pr = 0.3 $, and (b) isothermal graph when $ Pr = 0.6 $