An Optimistic Solver for the Mathematical Model of the Flow of Johnson Segalman Fluid on the Surface of an Infinitely Long Vertical Cylinder

Naveed Ahmad Khan; Fahad Sameer Alshammari; Carlos Andrés Tavera Romero; Muhammad Sulaiman; Seyedali Mirjalili

doi:10.3390/ma14247798

An Optimistic Solver for the Mathematical Model of the Flow of Johnson Segalman Fluid on the Surface of an Infinitely Long Vertical Cylinder

Materials (Basel). 2021 Dec 16;14(24):7798. doi: 10.3390/ma14247798.

Authors

Naveed Ahmad Khan¹, Fahad Sameer Alshammari², Carlos Andrés Tavera Romero³, Muhammad Sulaiman¹, Seyedali Mirjalili^{4

5}

Affiliations

¹ Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan.
² Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia.
³ COMBA R&D Laboratory, Faculty of Engineering, Universidad Santiago de Cali, Cali 76001, Colombia.
⁴ Centre for Artificial Intelligence Research and Optimisation, Torrens University Australia, Fortitude Valley, Brisbane, QLD 4006, Australia.
⁵ Yonsei Frontier Lab, Yonsei University, Seoul 03722, Korea.

Abstract

In this paper, a novel soft computing technique is designed to analyze the mathematical model of the steady thin film flow of Johnson-Segalman fluid on the surface of an infinitely long vertical cylinder used in the drainage system by using artificial neural networks (ANNs). The approximate series solutions are constructed by Legendre polynomials and a Legendre polynomial-based artificial neural networks architecture (LNN) to approximate solutions for drainage problems. The training of designed neurons in an LNN structure is carried out by a hybridizing generalized normal distribution optimization (GNDO) algorithm and sequential quadratic programming (SQP). To investigate the capabilities of the proposed LNN-GNDO-SQP algorithm, the effect of variations in various non-Newtonian parameters like Stokes number (St), Weissenberg number (We), slip parameters (a), and the ratio of viscosities (ϕ) on velocity profiles of the of steady thin film flow of non-Newtonian Johnson-Segalman fluid are investigated. The results establish that the velocity profile is directly affected by increasing Stokes and Weissenberg numbers while the ratio of viscosities and slip parameter inversely affects the fluid's velocity profile. To validate the proposed technique's efficiency, solutions and absolute errors are compared with reference solutions calculated by RK-4 (ode45) and the Genetic algorithm-Active set algorithm (GA-ASA). To study the stability, efficiency and accuracy of the LNN-GNDO-SQP algorithm, extensive graphical and statistical analyses are conducted based on absolute errors, mean, median, standard deviation, mean absolute deviation, Theil's inequality coefficient (TIC), and error in Nash Sutcliffe efficiency (ENSE). Statistics of the performance indicators are approaching zero, which dictates the proposed algorithm's worth and reliability.

Keywords: Johnson Segalman model; computational fluid dynamics; drainage problems; generalized normal distribution optimization; hybrid soft computing; sequential quadratic programming; weighted legendre neural networks.