Many biotechnology sectors that depend on fluids and their physical characteristics, including the phenomenon of bioconvection, have generated a great deal of discussion. The term “bioconvection” describes the organized movement of microorganisms, such as bacteria or algae. Microorganisms that participate in bioconvection display directed movement, frequently in the form of upward or downward streaming, which can lead to the production of distinctive patterns. The interaction between the microbes’ swimming behavior and the physical forces acting on them, such as buoyancy and fluid flow, is what drives these patterns. This work considers the laminar-mixed convection incompressible flow at the stagnation point with viscous and gyrotactic microorganisms in an unsteady electrically conducting hybrid nanofluid (Fe₃O₄-Cu/water). In addition, hybrid nanofluid flow over a horizontal porous stretched sheet, as well as external and induced magnetic field effects, can be used in biological domains, including drug delivery and microcirculatory system flow dynamics. The governing system has been reduced to a set of ordinary differential equations (ODEs) through the use of the group technique. The current research was inspired by an examination of the impacts of multiple parameters, including Prandtl number, $Pr$, magnetic diffusivity, ${\eta }_0$, shape factor$, n$, microorganism diffusion coefficient, $D_n$, Brownian motion coefficient, $D_B$, thermophoresis diffusion coefficient, $D_T$, bioconvection Peclet number, $Pe$, temperature difference, $\delta t$, and concentration difference, $\delta c$. The results show that as $Pr$ rises, temperature, heat flux, and nanoparticles all decrease. In contrast, when the ${\eta }_0$ value increases, the magnetic field and velocity decrease. Heat flow, bacterial density, and temperature decrease as the $D_B$ value rises, yet the number of nanoparticles increases. As the $D_T$ value increases, the temperature, heat flow, and concentration of nanoparticles all rise while the density of bacteria decreases. Even though temperature, heat flux, nanoparticles, and bacterial density all decrease as $\delta c$ values climb, bacterial density rises as $D_n$ values do although bacterial density falls with increasing, $\delta t$ and $Pe$ values; on the other hand, when n values increase, temperature and heat flow increase but the density of bacteria and nanoparticle decrease. The physical importance and behavior of the present parameters were illustrated graphically.

Forward scattering of (3 + 1)-dimensional Jimbo-Miwa (JM) equation is investigated by singular manifold method (SMM). The detected Lax pair is reduced through three parameters group method into a compatible couple of ordinary differential equations (ODEs). Three forms of symmetry variables have been discussed leading to integrable system of ODEs. Some new closed form solutions have been obtained. The three parameters group method, which is rarely used in solving PDEs, has emphasized its efficiency in creating symmetry solutions of evolution equations.