The heat and mass transfer within non-Newtonian fluid flow results in complex mathematical equations and solution in this regard remains a challenging task for researchers. The present paper offers a numerical solution for the non-Newtonian flow field by using Artificial neural networking (ANN) model with the Levenberg Marquardt training technique. To be more specific, we considered thermally magnetized non-Newtonian flow headed for inclined heated surfaces. The flow is carried with viscous dissipation, stagnation point, heat generation, mixed convection, and thermal radiation effects. The concentration aspects are entertained by the owing concentration equation. The shooting method is used to solve the mathematical flow equations. The quantity of interest includes the temperature and heat transfer coefficient. Two different artificial neural networking models have been built. The training of networks is done by use of the Levenberg Marquardt technique. The values of the coefficient of determination suggest artificial neural networks as the best method for predicting the Nusselt number at both surfaces. The thermal radiation parameter and Prandtl number admit a direct relationship to the Nusselt number while the differing is the case for variable thermal conductivity and Casson parameters. Further, by using Nusselt number (NN)-ANN models, we found that for cylindrical surface, the strength of the NN is greater than the flat surface.
Citation: Khalil Ur Rehman, Wasfi Shatanawi, Zeeshan Asghar, Haitham M. S. Bahaidarah. Neural networking analysis for MHD mixed convection Casson flow past a multiple surfaces: A numerical solution[J]. AIMS Mathematics, 2023, 8(7): 15805-15823. doi: 10.3934/math.2023807
The heat and mass transfer within non-Newtonian fluid flow results in complex mathematical equations and solution in this regard remains a challenging task for researchers. The present paper offers a numerical solution for the non-Newtonian flow field by using Artificial neural networking (ANN) model with the Levenberg Marquardt training technique. To be more specific, we considered thermally magnetized non-Newtonian flow headed for inclined heated surfaces. The flow is carried with viscous dissipation, stagnation point, heat generation, mixed convection, and thermal radiation effects. The concentration aspects are entertained by the owing concentration equation. The shooting method is used to solve the mathematical flow equations. The quantity of interest includes the temperature and heat transfer coefficient. Two different artificial neural networking models have been built. The training of networks is done by use of the Levenberg Marquardt technique. The values of the coefficient of determination suggest artificial neural networks as the best method for predicting the Nusselt number at both surfaces. The thermal radiation parameter and Prandtl number admit a direct relationship to the Nusselt number while the differing is the case for variable thermal conductivity and Casson parameters. Further, by using Nusselt number (NN)-ANN models, we found that for cylindrical surface, the strength of the NN is greater than the flat surface.
[1] | M. A. Dweib, C. M. ÓBrádaigh, Extensional and shearing flow of a glass-mat-reinforced thermoplastics (GMT) material as a non-Newtonian viscous fluid, Compos. Sci. Technol., 59 (1999), 1399–1410. https://doi.org/10.1016/S0266-3538(98)00182-1 doi: 10.1016/S0266-3538(98)00182-1 |
[2] | K. A. Fisher, R. J. Wakeman, T. W. Chiu, O. F. J. Meuric. Numerical modelling of cake formation and fluid loss from non-Newtonian muds during drilling using eccentric/concentric drill strings with/without rotation, Chem. Eng. Res. Design, 78 (2000), 707–714. https://doi.org/10.1205/026387600527888 doi: 10.1205/026387600527888 |
[3] | Y. A. Berezin, V. A. Chugunov, K. Hutter, Hydraulic jumps on shallow layers of non-Newtonian fluids, J. Non-newtonian Fluid Mech., 101 (2001), 139–148. https://doi.org/10.1016/S0377-0257(01)00154-9 doi: 10.1016/S0377-0257(01)00154-9 |
[4] | H. Z. Li, Y. Mouline, N. Midoux, Modelling the bubble formation dynamics in non-Newtonian fluids, Chem. Eng. Sci., 57 (2002), 339–346. https://doi.org/10.1016/S0009-2509(01)00394-3 doi: 10.1016/S0009-2509(01)00394-3 |
[5] | M. Anand, K. R. Rajagopal, A shear-thinning viscoelastic fluid model for describing the flow of blood, Int. J. Cardiovascular Medicine Sci., 4 (2004), 59–68. http://www.cs.cmu.edu/afs/cs.cmu.edu/project/taos-10/publications/MAKRR2004.pdf |
[6] | Z. Yu, A. Wachs, Y. Peysson, Numerical simulation of particle sedimentation in shear-thinning fluids with a fictitious domain method, J. Non-Newtonian Fluid Mech., 136 (2006), 126–139. https://doi.org/10.1016/j.jnnfm.2006.03.015 doi: 10.1016/j.jnnfm.2006.03.015 |
[7] | J. Marn, P. Ternik, Laminar flow of a shear-thickening fluid in a 90 pipe bend, Fluid Dyn. Res.., 38 (2006), 295. https://doi.org/10.1016/j.fluiddyn.2006.01.003 doi: 10.1016/j.fluiddyn.2006.01.003 |
[8] | S. Guillou, R. Makhloufi, Effect of a shear-thickening rheological behaviour on the friction coefficient in a plane channel flow: A study by direct numerical simulation, J. Non-Newtonian Fluid Mech., 144 (2007), 73–86. https://doi.org/10.1016/j.jnnfm.2007.03.008 doi: 10.1016/j.jnnfm.2007.03.008 |
[9] | S. U. Siddiqui, S. Mishra, A study of modified Casson's fluid in modelled normal and stenotic capillary-tissue diffusion phenomena, Appl. Math. Comp., 189 (2007), 1048–1057. https://doi.org/10.1016/j.amc.2006.11.151 doi: 10.1016/j.amc.2006.11.151 |
[10] | M. H. Abolbashari, N. Freidoonimehr, F. Nazari, M. M. Rashidi, Analytical modeling of entropy generation for Casson nano-fluid flow induced by a stretching surface, Adv. Powder Tech., 26 (2015), 542–552. https://doi.org/10.1016/j.apt.2015.01.003 doi: 10.1016/j.apt.2015.01.003 |
[11] | K. Bhattacharyya, M. S. Uddin, G. C. Layek, Exact solution for thermal boundary layer in Casson fluid flow over permeable shrinking sheet with variable wall temperature and thermal radiation, Alex. Eng. J., 55 (2016), 1703–1712. https://doi.org/10.1016/j.aej.2016.03.010 doi: 10.1016/j.aej.2016.03.010 |
[12] | M. Abd El-Aziz, A. S. Yahya, Perturbation analysis of unsteady boundary layer slip flow and heat transfer of Casson fluid past a vertical permeable plate with Hall current, Appl. Math. Comput., 307 (2017), 146–164. https://doi.org/10.1016/j.amc.2017.02.034 doi: 10.1016/j.amc.2017.02.034 |
[13] | M. Nawaz, R. Naz, M. Awais, Magnetohydrodynamic axisymmetric flow of Casson fluid with variable thermal conductivity and free stream, Alex. Eng. J., 57 (2018), 2043–2050 https://doi.org/10.1016/j.aej.2017.05.016 doi: 10.1016/j.aej.2017.05.016 |
[14] | M. Usman, F. A. Soomro, R. Ul Haq, W. Wang, O. Defterli, Thermal and velocity slip effects on Casson nanofluid flow over an inclined permeable stretching cylinder via collocation method, Inter. J. Heat Mass Trans., 122 (2018), 1255–1263 https://doi.org/10.1016/j.ijheatmasstransfer.2018.02.045 doi: 10.1016/j.ijheatmasstransfer.2018.02.045 |
[15] | A. Tassaddiq, I. Khan, K. S. Nisar, J. Singh, MHD flow of a generalized Casson fluid with Newtonian heating: A fractional model with Mittag–Leffler memory, Alex. Eng. J., 59 (2020), 3049–3059. https://doi.org/10.1016/j.aej.2020.05.033 doi: 10.1016/j.aej.2020.05.033 |
[16] | F. Hussain, M. Nazeer, M. Altanji, A. Saleem, M. M. Ghafar, Thermal analysis of Casson rheological fluid with gold nanoparticles under the impact of gravitational and magnetic forces, Case Stud. Thermal Eng., 28 (2021), 101433. https://doi.org/10.1016/j.aej.2020.05.033 doi: 10.1016/j.aej.2020.05.033 |
[17] | M. T. Akolade, Y. O. Tijani, A comparative study of three dimensional flow of Casson–Williamson nanofluids past a riga plate: Spectral quasi-linearization approach, Part. Diff. Eqs. Appl. Math., 4 (2021), 100108. https://doi.org/10.1016/j.padiff.2021.100108 doi: 10.1016/j.padiff.2021.100108 |
[18] | B. K. Siddiqui, S. Batool, M. Y. Malik, Q. Mahmood ul Hassan, Ali S. Alqahtani, Darcy Forchheimer bioconvection flow of Casson nanofluid due to a rotating and stretching disk together with thermal radiation and entropy generation, Case Studies Thermal Eng., 27 (2021), 101201. https://doi.org/10.1016/j.csite.2021.101201 doi: 10.1016/j.csite.2021.101201 |
[19] | S. G. Bejawada, Y. D. Reddy, W. Jamshed, K. Sooppy Nisar, A. N. Alharbi, R. Chouikh. Radiation effect on MHD Casson fluid flow over an inclined non-linear surface with chemical reaction in a Forchheimer porous medium, Alex. Eng. J., 61 (2022), 8207–8220. https://doi.org/10.1016/j.aej.2022.01.043 doi: 10.1016/j.aej.2022.01.043 |
[20] | M. R. Khan, A. S. Al-Johani, A. MA Elsiddieg, T. Saeed, A. Mousa Abd Allah, The computational study of heat transfer and friction drag in an unsteady MHD radiated Casson fluid flow across a stretching/shrinking surface, Int. Comm. Heat Mass Trans., 130 (2022), 105832. https://doi.org/10.1016/j.icheatmasstransfer.2021.105832 doi: 10.1016/j.icheatmasstransfer.2021.105832 |
[21] | A. C. Venkata Ramudu, K. Anantha Kumar, V. Sugunamma, N. Sandeep, Impact of Soret and Dufour on MHD Casson fluid flow past a stretching surface with convective–diffusive conditions, J. Therm. Anal. Calorim., 147 (2022), 1–11. https://doi.org/10.1007/s10973-021-10569-w doi: 10.1007/s10973-021-10569-w |
[22] | T. Hayat, S. A. Khan, S. Momani, Finite difference analysis for entropy optimized flow of Casson fluid with thermo diffusion and diffusion-thermo effects, Inter. J. Hydrogen Ener., 47 (2022), 8048–8059. https://doi.org/10.1016/j.ijhydene.2021.12.093 doi: 10.1016/j.ijhydene.2021.12.093 |
[23] | M. Afrand, N. Sina, H. Teimouri, A. Mazaheri, M. R. Safaei, M. H. Esfe, et al., Effect of magnetic field on free convection in inclined cylindrical annulus containing molten potassium, Inter. J. Appl. Mech., 7 (2015), 1550052. https://doi.org/10.1142/S1758825115500520 doi: 10.1142/S1758825115500520 |
[24] | M. S. Dehghani, D. Toghraie, B. Mehmandoust, Effect of MHD on the flow and heat transfer characteristics of nanofluid in a grooved channel with internal heat generation, Inter. J. Num. Methods. Heat Fluid Flow., 29 (2018), 1403–1431. https://doi.org/10.1108/HFF-05-2018-0235 doi: 10.1108/HFF-05-2018-0235 |
[25] | K. Ur, Rehman, M. Awais, A. Hussain, N. Kousar, M. Y. Malik, Mathematical analysis on MHD Prandtl‐Eyring nanofluid new mass flux conditions, Math. Method. Appl. Sci., 42 (2019), 24–38. https://doi.org/10.1002/mma.5319 doi: 10.1002/mma.5319 |
[26] | K. Ur. Rehman, W. Shatanawi, S. Yaseen, A comparative numerical study of heat and mass transfer individualities in Casson stagnation point fluid flow past a flat and cylindrical surfaces, Mathematics., 11 (2023), 470. https://doi.org/10.3390/math11020470 doi: 10.3390/math11020470 |
[27] | K. Ur. Rehman, W. Shatanawi, U. Firdous, A comparative thermal case study on thermophysical aspects in thermally magnetized flow regime with variable thermal conductivity, Case Stud. Therm. Eng., 44 (2023), 102839. https://doi.org/10.1016/j.csite.2023.102839 doi: 10.1016/j.csite.2023.102839 |
[28] | G. Tunc, Y. Bayazitoglu, Heat transfer in microtubes with viscous dissipation, Inter. J. Heat Mass Trans., 44 (2001), 2395–2403. https://doi.org/10.1016/S0017-9310(00)00298-2 doi: 10.1016/S0017-9310(00)00298-2 |
[29] | K. Ur. Rehman, Q. M. Al-Mdallal, M. Y. Malik, Symmetry analysis on thermally magnetized fluid flow regime with heat source/sink, Case Stud. Therm. Eng., 14 (2019), 100452. https://doi.org/10.1016/j.csite.2019.100452 doi: 10.1016/j.csite.2019.100452 |
[30] | P. Barnoon, D. Toghraie, R. B. Dehkordi, H. Abed, MHD mixed convection and entropy generation in a lid-driven cavity with rotating cylinders filled by a nanofluid using two phase mixture model, J. Magnet. Magnet. Material, 483 (2019), 224–248. https://doi.org/10.1016/j.jmmm.2019.03.108 doi: 10.1016/j.jmmm.2019.03.108 |
[31] | D. Toghraie, Numerical simulation on MHD mixed convection of Cu-water nanofluid in a trapezoidal lid-driven cavity, Inter. J. Appl. Electrom., 62 (2020), 683–710. https://doi.org/10.3233/JAE-190123 doi: 10.3233/JAE-190123 |
[32] | H. Sadaf, Z. Asghar, N. Iftikhar, Cilia-driven flow analysis of cross fluid model in a horizontal channel, Comp. Part. Mech., (2022), 1–8. https://doi.org/10.1007/s40571-022-00539-w doi: 10.1007/s40571-022-00539-w |
[33] | A. Aabid, S. Afghan Khan, M. Baig, Numerical analysis of a microjet-based method for active flow control in convergent-divergent nozzles with a sudden expansion duct, Fluid Dynam. Mater. Process., 18 (2022), 1–24. https://doi.org/10.32604/fdmp.2022.021860 doi: 10.32604/fdmp.2022.021860 |
[34] | I. S. Hussain, D. Prakash, B. Abdalla, M. Muthtamilselvan, Analysis of Arrhenius activation energy and chemical reaction in nanofluid flow and heat transfer over a thin moving needle, Current Nanosci., 19 (2023), 39–48. https://doi.org/10.2174/1573413717666211117150656 doi: 10.2174/1573413717666211117150656 |
[35] | K. Ur Rehman, A. Batur Çolak, W. Shatanawi, Artificial neural networking (ANN) model for drag coefficient optimization for various obstacles, Mathematics, 10 (2022), 2450. https://doi.org/10.3390/math10142450 doi: 10.3390/math10142450 |
[36] | A. B. Çolak, An experimental study on the comparative analysis of the effect of the number of data on the error rates of artificial neural networks, Int. J. Ener. Resear., 45 (2021), 478–500. https://doi.org/10.1002/er.5680 doi: 10.1002/er.5680 |
[37] | A. Shafiq, A. Batur Çolak, T. Naz 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. Num. Methods Fluids, 93 (2021), 3384–3404. https://doi.org/10.1002/fld.5038 doi: 10.1002/fld.5038 |
[38] | A. Shafiq, A. Batur Çolak, T. Naz Sindhu, Q. M. Al-Mdallal, T. Abdeljawad, Estimation of unsteady hydromagnetic Williamson fluid flow in a radiative surface through numerical and artificial neural network modeling, Sci. Rep., 11 (2021), 14509. https://doi.org/10.1038/s41598-021-93790-9 doi: 10.1038/s41598-021-93790-9 |
[39] | A. B. Colak, Experimental study for thermal conductivity of water‐based zirconium oxide nanofluid: developing optimal artificial neural network and proposing new correlation, Int. J. Energy Res., 45 (2021), 2912–2930. https://doi.org/10.1002/er.5988 doi: 10.1002/er.5988 |
[40] | M. Adamu, A. Batur Çolak, Y. E. Ibrahim, S. I. Haruna, M. F. Hamza, Prediction of mechanical properties of rubberized concrete incorporating fly ash and nano silica by artificial neural network technique, Axiom., 12 (2023), 81. https://doi.org/10.3390/axioms12010081 doi: 10.3390/axioms12010081 |