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Implementing the Galerkin Method Associated with the Shifted Vieta-Lucas Polynomials for Studying Numerically the Bionanofluid Flow Which Is Saturated by Gyrotactic Microorganisms over a Slippery Stretching Sheet

Heat transfer is a critical function in many technical, industrial, home, and commercial structures. As a result, the purpose of this study is to investigate the effects of slip velocity and variable fluid characteristics on Casson bionanofluid flow across a stretching sheet that has been saturated by gyrotactic microorganisms. The suggested system will be converted to a computationally tractable form using the Galerkin method. The shifted Vieta-Lucas polynomials are then used as basis functions on the provided domain to solve the nonlinear system of ordinary differential equations that has been constructed (ODEs). The results are presented in the form of graphs and tables to assess the impact of the problem’s governing parameters. The estimated solutions produced by using the proposed techniques were physically acceptable and accurate. The current outcomes are confirmed by comparing them to the available literature. It appears that the temperature distribution is enhanced whereas the velocity distribution declines, caused by rising values of the magnetic parameter, slip parameter, and Casson parameter. Also, the local Nusselt number escalates with the strength of the viscosity parameter while the friction drag decays with the same parameter. In addition, the effectiveness and accuracy of the proposed method are satisfied by computing and the residual error function.