The movement of biological fluids in the human body is a premium field of interest to overcome growing biomedical challenges. Blood behavior shows different behavior in capillaries, veins, and arteries during circulation. In this paper, a new mathematical relation for the nano-layer of biological fluids flows with the effect of TiO₂ and Ag hybrid nanoparticles was developed. Further, we explain the engineering phenomena of biological fluids and the role of hybrid nanoparticles in the blood vessel system. The improvement of drug delivery systems by using low seepage Reynolds number was associated with expansion/contraction and was discussed in detail through the rectangular domain. Using similarity transformation, the governing equations were converted into non-linear ordinary differential equations, and the mathematical problem was solved by employing the numerical shooting method. Plots of momentum, temperature, skin friction coefficient, as well as the Nusselt number on different non-dimensionless parameters are displayed via lower/upper porous walls of the channel. It was analyzed that the walls of the channel showed different results on magnetized physical parameters. Values of thermophoresis and the Brownian motion flow of the heat transfer rate gradually increased on the upper wall and decreased on the lower wall of the channel. The important thing is that the hybrid nanoparticles, rather than nano, were more useful for improving thermal conductivity, heat transfer rate, and the nano-layer.