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Due to dynamical applications of nanoparticles in various engineering and biomedical applications, our work focuses on examining and exploring the numerical study of bioconvected unsteady Williamson fluid flow at a heated permeable stretched sheet in a porous medium, along with the existence of particular Cattaneo–Christov heat and mass flux, viscous dissipation, activation energy, Brownian and thermophoresis motion. A mathematical model has been developed by considering the effects of the aforementioned terminologies on the flow field which are expected to be similar to the physical behavior and already published work. By using suitable similarity transformations, the system of the flow field is transformed from partial differential equations to ordinary differential equations. The developed differential system is then solved numerically through bvp4c in the computational software MATLAB. The tabular and graphical analysis have been presented against velocity, temperature, concentration, and density profiles that imply the impact of physical parameters. The skin friction coefficient, rate of heat, and mass transfer are also examined. The results coincide very well to previous published work for limiting cases that authenticate the validity of the current work. It was observed that upon increasing magnetic field (M) and porosity (Kp) parameters, the velocity profile decelerates but the opposite behavior was seen for motile density, concentration, and temperature profiles. The temperature and concentration of Williamson nanofluid reduced by enhancing the significant thermal and mass stratification. The concentration profile declines for rising values of Schmidt number (Sc), the chemical reaction rate (σ), and temperature difference (δ) parameters but accelerates for higher values of Activation energy (Ea). Also, by increasing Peclet number (Pe), bioconvection Lewis number (Lb), and microorganism concentration difference parameter Ω motile density profile decreases. Analyze the impacts of bioconvection and activation energy on Williamson nanofluid. We numerically imitate the outcomes of viscous dissipation, bioconvection, and activation energy at a heated permeable stretched sheet. We examine the effect of A, M, We, Kp, and S on − 1 2 C f (R e) 1 2 . We examine the effect of M, Kp, and We on S h x (R e x) − 1 2 . We examine the effect of the local density number with M, Kp, We, A, and injection and suction parameters. [ABSTRACT FROM AUTHOR]

Numerical outcomes for the steady and unsteady flow situations are investigated for the mixed convection flow of Casson nanofluid caused by the motion of a nonlinear stretching sheet in porous media with chemical reaction and magnetic effects. In the present study, the impact of convective boundary conditions on the thermal and concentration profiles as well as in the presence of Brownian motion and thermophoresis effects on the flow field are also examined. Highly non-linear and coupled partial differential equations (PDEs) are transformed into a system of ordinary differential equations (ODEs) by using suitable similarity transformations and subsequently portray numerical and graphical results with the help of bvp4c tool using MATLAB. The validity of numerical results are compared and checked with previously published results in the literature. For deliberating the physical nature of the investigation, graphical and tabulated results are provided. The Casson parameter helps to reduce the intensity of the skin friction, while Casson fluid has a converse impact on velocity and temperature field, but parallel results are seen in the case of concentration and temperature profile. The impacts for the magnetic parameter on velocity and temperature is revealed more effective than nanoparticle concentration. The impact of thermal and concentration buoyancy parameter on velocity, temperature and concentration profiles are observed and found that velocity profile increases with slight increase in thermal buoyancy parameter but decreases the other two profiles. [ABSTRACT FROM AUTHOR]