Synchronized effects of melting heat on aligned MHD Williamson nanofluid comprising microorganisms to the leading edge: a numerical approach

The study of microbial flow with nanofluids involves investigating the behavior and interactions of microorganisms in the presence of nanofluids. Nanofluids are colloidal suspensions containing nanoparticles dispersed in a base fluid, and they have unique properties that can significantly impact microbial flow dynamics. This research focuses on the simultaneous effects of melting heat, Joule heating, chemical reaction, and thermal radiation on an aligned magnetohydrodynamic Williamson nanofluid (WNF) with the Cattaneo–Christov heat flux model (CCHFM) comprising microorganisms near the leading edge having stagnation flow characteristics. A melting approach is employed to study the behavior of fluid flow and heat transfer in the presence of these combined phenomena. A mathematical model is developed based on the governing equations, including conservation equations for mass, momentum, energy, concentration and microbes. To analyze the system, a set of nonlinear partial differential equations is transformed into ordinary differential equations (ODEs) by using similarity measures. The solutions of these nonlinear ODEs are obtained via the RK-4th method in MATLAB, and the results are presented in graphs and tables. Boundary layer equations for microbe propagation, fluid temperature, nano-inclusion volume fraction, and fluid velocity are formulated and discussed for various influential parameters. A higher melting parameter \({M}_{b}\) $M_b$ results in thicker boundary layers for velocity and temperature profiles, accompanied by an increased concentration profile for \(\alpha =\frac{\pi }{4}\) $\alpha =\frac{\pi }{4}$ . Notably, the trend is more pronounced in \(\alpha =\frac{\pi }{4}\) $\alpha =\frac{\pi }{4}$ compared \(\alpha =\frac{\pi }{2}\) $\alpha =\frac{\pi }{2}$ . The impact of \(Nb\) $\mathit{Nb}$ and \(Nt\) $\mathit{Nt}$ on the motile profile exhibits a positive influence, leading to an increased boundary layer thickness for the gyrotactic microorganisms. The numerical analysis in Table 1 reveals that the melting heat parameter has a more pronounced impact on the motile configuration, as evidenced by the respective rates of change \((- 0.076783, - 0.337954, 0.597858, 0.893040)\) $(-0.076783,-0.337954,0.597858,0.893040)$ for key physical quantities compared to other configurations. The convergence and validation of the numerical solutions ensure the accuracy and reliability of the findings, establishing a foundation for further research in this area.