MHD Casson Nanofluid in Darcy-Forchheimer Porous Medium in the Presence of Heat Source and Arrhenious Activation Energy: Applications of Neural Networks

ABSTRACT

The present study investigates the significance of activation energy during chemical reactions, thermal radiations and temperature gradient of 3-D steady Magnetohydrodynamic Casson nanofluid flow (MHDCNFM) in Darcy−Forchheimer medium over an oscillating disk by using an intelligent numerical-based computing solver through the Levenberg-Marquardt backpropagation neural network scheme (LMBNNS). The adjusted Buongiorno model is utilized to develop the system of partial differential equations (PDEs) for MHDCNFM, and further, by invoking Von Karman similarity transformation, the system of governing PDEs is turned out into ordinary differential equations (ODEs). First, a data set for the magnetohydrodynamic Casson nanofluid flow model (MHDNNFM) is generated for a range of sundry parameters for the radial velocity, tangential velocity, heat distribution and concentration profile by the variations of Casson parameter, magnetic parameter, Brownian motion parameter, thermophoresis parameter, Forchheimer parameter, activation energy parameter, chemical reaction parameter, stretching parameter, porosity parameter and Schmidt number through the Lobatto IIIA technique, and after this, by applying an intelligent computing algorithm through nftool training, testing and validation steps are taken into account to find out the approximate solution for various cases. The designed solver LMBNNS is used to solve the MHDCNFM through mean squared error (MSE), regression, gradient analysis and histogram studies.

KEYWORDS:

• Arrhenius activation energy
• magnetohydrodynamic
• Casson nanofluid
• Darcy−Forchheimer porous medium
• heat source
• neural networks
• Levenberg−Marquardt backpropagation

Nomenclatures

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Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Muhammad Shoaib

Muhammad Shoaib is the resident of village Basal of district Attock, Punjab Pakistan. He has completed his master's degree in Mathematics, in 2012 Punjab University Lahore, Pakistan and Ph.D. Mathematics in 2016 from Riphah International University Islamabad Pakistan He is currently working as Assistant Professor in COMSATS University Islamabad Attock Campus from 2017. He has about 150 research publications in international Impact Factor Journals with impact factor approximately 450 plus. His area of interest includes mathematical modeling, neural networks, artificial intelligence and implementation of computational techniques based on traditional as well as the heuristic paradigms.

Shafaq Naz

Shafaq Naz was born in historical place Gujrat , Pakistan. She was incredible student during her education period. She obtained her Master’s degree from university of Gujrat and got Bronze’s Medalt. She has completed her MPhil degree in 2009 from Quaid-I Azam University, Islamabad, Pakistan. After that she continued her education and completed her PhD degree in 2015 in the field of Algebra. Now since 2014, she served University of Gujrat, Pakistan as Assistant Professor. She is a multitasking researcher. She has supervised several MPhil students and PhD. She is very devoting to her research activities. Her area of research is fuzzy algebra, mathematical modeling, artificial intelligence and neural networking.

Kottakkaran Sooppy Nisar

Prof (Dr). Kottakkaran Sooppy Nisar is a full professor of Applied Mathematics at Prince Sattam bin Abdulaziz University, Saudi Arabia. His current research interests are Special functions, Fractional Calculus, Fluid dynamics, Bio Mathematics, Machine Learning, SAC-OCDMA and Multidisciplinary applications of Mathematics. He is an editorial board member of many reputed journals and has published more than 800 research papers in Scopus/WoS indexed journals. He is the winner of a number of awards like PSAU research excellent awards (Best researcher of the University) in 2019 and 2020, Riemann Young researcher award 2020, Indo Asian-Niels Henrik Distinguished innovative Scientist Award 2020 and Obada Prize 2021 for Young distinguished researcher.

Muhammad Asif Zahoor Raja

Muhammad Asif Zahoor Raja has done his M.Sc. Mathematics, M.Sc. Nuclear Engineering, from Ph.D. Electronic Engineering from prestigious universities of Pakistan in 1996, 1999 and 2011, respectively. He is involved in research and development assignment of Engineering and Scientific Commission of Pakistan from 1999 to 2012. From 2012 to 2020, he works as an Assistant Professor in department of Electrical Engineering, COMSATS University Islamabad, Since Oct 2020 to date, he is working as Professor in Future Technology Research Center, National Yunlin University of Science and Technology, Yunlin, Taiwan. Dr. Raja has developed the Fractional least mean square algorithm, fractional neural networks and others related computational platform for solving fractional/integer order differential equation using artificial intelligence techniques. Dr. Raja has been author of more than 384+ WoS publications in the reputed journal publications, with H-index 47 and citations of 7504. Dr. Raja acts as a resource person and gives invited talks on many workshops and conferences held at the national and international level. His areas of interest are solving linear and nonlinear differential equation of arbitrary order, computation fluid dynamics, nanotechnology, active noise control system, fractional adaptive signal processing, nonlinear system identification, direction of arrival estimation and Bioinformatics problems.

Sana Aslam

Sana Aslam is currently competed her MPhil degree from department of Mathematics, University of Gujrat, Pakistan. She has completed her MSc-(Mathematics) degree from University of Gujrat (UOG). She is potential and hardworking student during her research period. She has research interest in Mathematical Modeling, computational mathematics, artificial intelligence.

Iftikhar Ahmad

Iftikhar Ahmad was born in village Shahaya, Tehsil: Hasssan Abdal District: Atttock, Pakistan. Ahmad has obtained MSc Mathematics degree from Govt. Asghar Mall College Rawalpindi, Pakistan in 1993, MPhil in Mathematics from Quaid-i-Azam, University, Islamabad, Pakistan in 1997 and PhD in field of Inflationary Cosmology from Chinese Academy of Sciences (CAS) Beijing, China, in 2009. He has served Punjab Higher Education Department as lecturer in 1997 to 2009. After that he has served as Assistant Professor in 2009 to 2017 and associate professor in 2017 to 2021, Department of Mathematics, University of Gujrat, Gujrat, Pakistan. Currently Ahmad is working as Professor/ Chairperson Department of Mathematics University of Gujrat, Pakistan. Ahmad has been author of more than 50 impact factor publications and have active research collaboration with renewed professors from USA, Spain, Italy, Malaysia, Turkey and Chinese Academy of Sciences. Dr. Ahmad is working in multidiscipline inflationary cosmology, artificial intelligent, supervised learning, deep learning, computational mathematics, mathematical modeling of differential equations and partial differential equations, wave propagation.