Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow
The mathematical formulation of fluid flow problems often results in coupled nonlinear partial differential equations (PDEs); hence, their solutions remain a challenging task for researchers. The present study offers a solution for the flow differential equations describing a bio-inspired flow field...
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AIMS Press
2025-01-01
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| author | Khalil Ur Rehman Nosheen Fatima Wasfi Shatanawi Nabeela Kousar |
| author_facet | Khalil Ur Rehman Nosheen Fatima Wasfi Shatanawi Nabeela Kousar |
| author_sort | Khalil Ur Rehman |
| collection | DOAJ |
| description | The mathematical formulation of fluid flow problems often results in coupled nonlinear partial differential equations (PDEs); hence, their solutions remain a challenging task for researchers. The present study offers a solution for the flow differential equations describing a bio-inspired flow field of non-Newtonian fluid with gyrotactic microorganisms. A methanol-based nanofluid with ferrous ferric oxide, copper, and silver nanoparticles was considered in a stretching permeable cylinder. The chemical reaction, activation energy, viscous dissipation, and convective boundary conditions were considered. The Casson fluid, a non-Newtonian fluid model, was used as flowing over a cylinder. The fundamental PDEs were established using boundary layer theory in a cylindrical coordinate system for concentration, mass, momentum, and microorganisms' field. These PDEs were then transformed into nonlinear ODEs by applying transforming variables. ODEs were then numerically solved in MATLAB software using the built-in solver bvp4c algorithm. We established an artificial neural network (ANN) model, incorporating Tan-Sig and Purelin transfer functions, to enhance the accuracy of predicting skin friction coefficient (SFC) values along the surface. The networks were trained using the Levenberg–Marquardt method. Quantitative results show that the ferrous ferric oxide nanofluid is superior in increasing Nusselt number, Sherwood number, velocity, and microorganism density number; silver nanofluid is superior in increasing skin friction coefficient, temperature, and concentration. Interestingly, heat transfer rate decreases with the magnetic and curvature parameters and Eckert number, whereas the skin friction coefficient increases with the magnetic parameter and Darcy–Forchheimer number. The present results are validated with the previous existing studies. |
| format | Article |
| id | doaj-art-d9d13bbbf76540218affec3b78cb9eba |
| institution | DOAJ |
| issn | 2473-6988 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | AIMS Press |
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| spelling | doaj-art-d9d13bbbf76540218affec3b78cb9eba2025-08-20T02:48:13ZengAIMS PressAIMS Mathematics2473-69882025-01-0110159863310.3934/math.2025027Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flowKhalil Ur Rehman0Nosheen Fatima1Wasfi Shatanawi2Nabeela Kousar3Department of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh, 11586, Saudi ArabiaDepartment of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, PakistanDepartment of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh, 11586, Saudi ArabiaDepartment of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, PakistanThe mathematical formulation of fluid flow problems often results in coupled nonlinear partial differential equations (PDEs); hence, their solutions remain a challenging task for researchers. The present study offers a solution for the flow differential equations describing a bio-inspired flow field of non-Newtonian fluid with gyrotactic microorganisms. A methanol-based nanofluid with ferrous ferric oxide, copper, and silver nanoparticles was considered in a stretching permeable cylinder. The chemical reaction, activation energy, viscous dissipation, and convective boundary conditions were considered. The Casson fluid, a non-Newtonian fluid model, was used as flowing over a cylinder. The fundamental PDEs were established using boundary layer theory in a cylindrical coordinate system for concentration, mass, momentum, and microorganisms' field. These PDEs were then transformed into nonlinear ODEs by applying transforming variables. ODEs were then numerically solved in MATLAB software using the built-in solver bvp4c algorithm. We established an artificial neural network (ANN) model, incorporating Tan-Sig and Purelin transfer functions, to enhance the accuracy of predicting skin friction coefficient (SFC) values along the surface. The networks were trained using the Levenberg–Marquardt method. Quantitative results show that the ferrous ferric oxide nanofluid is superior in increasing Nusselt number, Sherwood number, velocity, and microorganism density number; silver nanofluid is superior in increasing skin friction coefficient, temperature, and concentration. Interestingly, heat transfer rate decreases with the magnetic and curvature parameters and Eckert number, whereas the skin friction coefficient increases with the magnetic parameter and Darcy–Forchheimer number. The present results are validated with the previous existing studies.https://www.aimspress.com/article/doi/10.3934/math.2025027nonlinear pdesmotile gyrotactic microorganismscasson nanofluidinclined mhdartificial neural network (ann) model |
| spellingShingle | Khalil Ur Rehman Nosheen Fatima Wasfi Shatanawi Nabeela Kousar Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow AIMS Mathematics nonlinear pdes motile gyrotactic microorganisms casson nanofluid inclined mhd artificial neural network (ann) model |
| title | Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow |
| title_full | Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow |
| title_fullStr | Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow |
| title_full_unstemmed | Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow |
| title_short | Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow |
| title_sort | mathematical solutions for coupled nonlinear equations based on bioconvection in mhd casson nanofluid flow |
| topic | nonlinear pdes motile gyrotactic microorganisms casson nanofluid inclined mhd artificial neural network (ann) model |
| url | https://www.aimspress.com/article/doi/10.3934/math.2025027 |
| work_keys_str_mv | AT khalilurrehman mathematicalsolutionsforcouplednonlinearequationsbasedonbioconvectioninmhdcassonnanofluidflow AT nosheenfatima mathematicalsolutionsforcouplednonlinearequationsbasedonbioconvectioninmhdcassonnanofluidflow AT wasfishatanawi mathematicalsolutionsforcouplednonlinearequationsbasedonbioconvectioninmhdcassonnanofluidflow AT nabeelakousar mathematicalsolutionsforcouplednonlinearequationsbasedonbioconvectioninmhdcassonnanofluidflow |