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Neural Network Analysis for Bioconvection Flow of Casson Fluid Over a Vertically Extending Sheet

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Abstract

The aim of this article is to study the bioconvection flow of Casson fluid over a vertically stretching sheet. Applying similarity transformations, the system of non-linear ordinary differential equations is obtained from the equations governing the flow. The significance of this study is solving these nonlinear ordinary differential equations by employing ANN (Artificial Neural Networks). In the trial functions, a multi-layer perceptron with customizable parameters (the biases and weights) is used. The Adam (Adaptive Moment Estimation) optimization algorithm is adopted to determine the adjustable parameters of the trial solution. The results of the current approach are validated by utilizing the shooting method in conjunction with the Runge–Kutta 4th-order method. It is observed that using a modest number of hidden neurons might give sufficient accuracy. However, the results suggest that the efficiency of artificial neural network-based methods improves as the number of neurons in the hidden layer of the neural network increases. It is noted that the skin friction coefficient, Nusselt number, and Sherwood number decrease as bioconvection Peclet number, bioconvection Schmidth number, and dimensionless bioconvection constant increase, whereas motile microorganism density increases.

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Data available on request from the authors.

References

  1. Casson, N.: Flow equation for pigment-oil suspensions of the printing ink-type. In: Mill, C. C. (ed.) Rheology of Disperse Systems, pp. 82–104. Pergamon Press, New York (1959)

  2. Ullah, I., Shafie, S., Khan, I.: Effects of slip condition and Newtonian heating on MHD flow of Casson fluid over a nonlinearly stretching sheet saturated in a porous medium. J. King Saud Univ. Sci. 29(2), 250–259 (2017)

    Article  Google Scholar 

  3. El-Aziz, A., Afify, A.A.: Influences of slip velocity and induced magnetic field on MHD stagnation-point flow and heat transfer of Casson fluid over a stretching sheet. Math. Probl. Eng. 2018, 9402836 (2018)

    Google Scholar 

  4. Abd El-Aziz, M., Afify, A.A.: MHD Casson fluid flow over a stretching sheet with entropy generation analysis and Hall influence. Entropy 21(6), 592 (2019)

    Article  MathSciNet  Google Scholar 

  5. Hussanan, A., Salleh, M.Z., Alkasasbeh, H.T., Khan, I.: MHD flow and heat transfer in a Casson fluid over a nonlinearly stretching sheet with Newtonian heating. Heat Transf. Res. 49(12), 1185–1198 (2018)

    Article  Google Scholar 

  6. Mabood, F., Das, K.: Outlining the impact of melting on MHD Casson fluid flow past a stretching sheet in a porous medium with radiation. Heliyon 5(2), e01216 (2019)

    Article  Google Scholar 

  7. Tantry, I.A., Wani, S., Agrawal, B.: Study of MHD boundary layer flow of a casson fluid due to an exponentially stretching sheet with radiation effect. Int. J. Stat. Appl. Math. 6, 138–144 (2021)

    Google Scholar 

  8. Zhou, J.C., Abidi, A., Shi, Q.H., Khan, M.R., Rehman, A., Issakhov, A., Galal, A.M.: Unsteady radiative slip flow of MHD Casson fluid over a permeable stretched surface subject to a non-uniform heat source. Case Stud. Therm. Eng. 26, 101141 (2021)

    Article  Google Scholar 

  9. Renu, D.E., Poply, V., Mani, M.A.: Effect of aligned magnetic field and inclined outer velocity in Casson fluid flow over a stretching sheet with heat source. J. Therm. Eng. 7(4), 823–844 (2021)

    Article  Google Scholar 

  10. Pal, D., Roy, N., Vajravelu, K.: Effects of thermal radiation and Ohmic dissipation on MHD Casson nanofluid flow over a vertical non-linear stretching surface using scaling group transformation. Int. J. Mech. Sci. 114(10), 257–267 (2016)

    Article  Google Scholar 

  11. Vajravelu, K., Prasad, K.V., Vaidya, H., Basha, N.Z., Ng, C.O.: Mixed convective flow of a Casson fluid over a vertical stretching sheet. Int. J. Appl. Comput. Math. 3, 1619–1638 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kasim, A.R., Arifin, N.S., Syazwani, M.Z., Ariffin, N.A.N., Salleh, M.Z., Anwar, M.I.: Mathematical model of simultaneous flow between Casson fluid and dust particle over a vertical stretching sheet. Int. J. Integr. Eng. 12(3), 253–260 (2020)

    Google Scholar 

  13. Razzaq, R., Farooq, U., Cui, J., Muhammad, T.: Non-similar solution for magnetized flow of Maxwell nanofluid over an exponentially stretching surface. Math. Probl. Eng. 2021, 5539542 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Javed, M., Imran, N., Arooj, A., Sohail, M.: Meta-analysis on homogeneous-heterogeneous reaction effects in a sinusoidal wavy curved channel. Chem. Phys. Lett. 16(763), 138200 (2021)

    Article  Google Scholar 

  15. Nazir, U., Sohail, M., Selim, M.M., Alrabaiah, H., Kumam, P.: Finite element simulations of hybrid nano-Carreau Yasuda fluid with hall and ion slip forces over rotating heated porous cone. Sci. Rep. 11(1), 19604 (2021)

    Article  Google Scholar 

  16. Pedley, T.J., Hill, N.A., Kessler, J.O.: The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms. J. Fluid Mech. 195, 223–237 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Babu, M.J., Sandeep, N.: Effect of nonlinear thermal radiation on non-aligned bio-convective stagnation point flow of a magnetic-nanofluid over a stretching sheet. Alex. Eng. J. 55(3), 1931–1939 (2016)

    Article  Google Scholar 

  18. Pal, D., Mondal, S.K.: MHD nanofluid bioconvection over an exponentially stretching sheet in the presence of gyrotactic microorganisms and thermal radiation. Bio Nanosci. 8(1), 272–287 (2018)

    Google Scholar 

  19. Ray, A.K., Vasu, B., Bég, O.A., Gorla, R.S.R., Murthy, P.V.: Magneto-bioconvection flow of a Casson thin film with nanoparticles over an unsteady stretching sheet: HAM and GDQ computation. Int. J. Numer. Meth. Heat Fluid Flow 29(11), 4277–4309 (2019)

    Article  Google Scholar 

  20. Sabir, Z., Akhtar, R., Zhiyu, Z., Umar, M., Imran, A., Wahab, H.A., Shoaib, M., Raja, M.A.: A computational analysis of two-phase casson nanofluid passing a stretching sheet using chemical reactions and gyrotactic microorganisms. Math. Probl. Eng. 2019, 1490571 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Magagula, V.M., Shaw, S., Kairi, R.R.: Double dispersed bioconvective Casson nanofluid fluid flow over a nonlinear convective stretching sheet in suspension of gyrotactic microorganism. Heat Transf. 49(5), 2449–2471 (2020)

    Article  Google Scholar 

  22. Sankad, G.C., Maharudrapppa, I.S., Dhange, M.Y.: Bioconvection in Casson fluid flow with gyrotactic microorganisms and heat transfer over a linear stretching sheet in presence of magnetic field. Adv. Math. Sci. J. 10, 155–169 (2021)

    Article  Google Scholar 

  23. Kumaraswamy Naidu, K., Harish Babu, D., Satya Narayana, P.V.: Bioconvection in magneto hydrodynamics Casson nanoliquid (Fe3O4-sodium alginate) with gyrotactic microorganisms over an exponential stretching sheet. J. Nanofluids 10(3), 327–338 (2021)

    Article  Google Scholar 

  24. Cui, J., Razzaq, R., Farooq, U., Khan, W.A., Farooq, F.B., Muhammad, T.: Impact of non-similar modeling for forced convection analysis of nano-fluid flow over stretching sheet with chemical reaction and heat generation. Alex. Eng. J. 61(6), 4253–4261 (2022)

    Article  Google Scholar 

  25. Cui, J., Munir, S., Farooq, U., Rabie, M.E., Muhammad, T., Razzaq, R.: On numerical thermal transport analysis of three-dimensional bioconvective nanofluid flow. J. Math. 2021, 5931989 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sangeetha, E., De, P.: Activation energy and binary chemical reaction were taken into consideration Bioconvection in nanofluid flow embedded in non-darcy porous medium with viscous dissipation and ohmic heating. J. Porous Media 24(1), 15–23 (2021)

    Article  Google Scholar 

  27. De, P., Gorji, M.R.: Activation energy and binary chemical reaction on unsteady MHD Williamson nanofluid containing motile gyrotactic micro-organisms. Heat Transf. 49, 3030–3043 (2020)

    Article  Google Scholar 

  28. Sangeetha, E., De, P.: Bioconvective Casson nanofluid flow toward stagnation point in non-Darcy porous medium with buoyancy effects, chemical reaction, and thermal radiation. Heat Transf. 52, 1529–1551 (2023)

    Article  Google Scholar 

  29. Lee, H., Kang, I.S.: Neural algorithm for solving differential equations. J. Comput. Phys. 91(1), 110–131 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lagaris, I.E., Likas, A., Fotiadis, D.I.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9(5), 987–1000 (1998)

    Article  Google Scholar 

  31. Pham, D.T., Liu, X.: Neural Networks for Identification, Prediction and Control. Springer, London (1995)

    Book  Google Scholar 

  32. Yadav, N., Yadav, A., Kumar, M.: An Introduction to Neural Network Methods for Differential Equations. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  33. Chakraverty, S., Mall, S.: Artificial Neural Networks for Engineers and Scientists: Solving Ordinary Differential Equations. CRC Press (2017)

    Book  MATH  Google Scholar 

  34. Meade, A.J., Jr., Fernandez, A.A.: Solution of nonlinear ordinary differential equations by feedforward neural networks. Math. Comput. Model. 20(9), 19–44 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. Piscopo, M.L., Spannowsky, M., Waite, P.: Solving differential equations with neural networks: applications to the calculation of cosmological phase transitions. Phys. Rev. D 100(1), 016002 (2019)

    Article  MathSciNet  Google Scholar 

  36. Sahari, M.F., Nezhad, A.H.: Estimation of the flow and heat transfer in MHD flow of a power law fluid over a porous plate using artificial neural networks. Middle East J. Sci. Res. 22(9), 1422–1429 (2014)

    Google Scholar 

  37. Ziaei-Rad, M., Saeedan, M., Afshari, E.: Simulation and prediction of MHD dissipative nanofluid flow on a permeable stretching surface using artificial neural network. Appl. Therm. Eng. 99, 373–382 (2016)

    Article  Google Scholar 

  38. Reddy, P.B.A., Das, R.: Estimation of MHD boundary layer slip flow over a permeable stretching cylinder in the presence of chemical reaction through numerical and artificial neural network modeling. Eng. Sci. Technol. Int. J. 19(3), 1108–1116 (2016)

    Google Scholar 

  39. Elayarani, M., Shanmugapriya, M.: Artificial neural network modeling of MHD stagnation point flow and heat transfer towards a porous stretching sheet. AIP Conf. Proc. 2161(1), 020043 (2019)

    Article  Google Scholar 

  40. Mutuk, H.: A neural network study of Blasius equation. Neural Process. Lett. 51, 2179–2194 (2020)

    Article  Google Scholar 

  41. Raja, M.A., Shoaib, M., Hussain, S., Nisar, K.S., Islam, S.: Computational intelligence of Levenberg-Marquardt backpropagation neural networks to study thermal radiation and Hall effects on boundary layer flow past a stretching sheet. Int. Commun. Heat Mass Transfer 130, 105799 (2022)

    Article  Google Scholar 

  42. Kingma, D. P., and Ba, J.: ADAM: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. 2014 Dec 22.

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Warangal, Telangana State, 506004, India

    D. Srinivasacharya & R. Shravan Kumar

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  1. D. Srinivasacharya
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Contributions

DS conceived of the presented idea. DS and RSK developed the theory, performed the computations. RSK developed the code in PYTHON and DS verified the code by the number of data points, hidden neurons and giving different values for various parameters involved in the problem. DS encouraged RSK to present the effect of various parameters graphically and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.

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Correspondence to D. Srinivasacharya.

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Srinivasacharya, D., Shravan Kumar, R. Neural Network Analysis for Bioconvection Flow of Casson Fluid Over a Vertically Extending Sheet. Int. J. Appl. Comput. Math 9, 80 (2023). https://doi.org/10.1007/s40819-023-01556-w

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