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The current analysis deals with the physiological flow of hybrid nanofluid within an expanding and contracting vertical annulus embedding with a permeable medium. Additionally, in the free convective viscous flow, the impact of thermal radiation on the heat transport phenomenon is vital. The significance of this research lies in the utilization of Hamilton–Crosser thermal conductivity to investigate the behavior of nanoparticles with different shapes. In this study, both oxide nanoparticles like Al2O3 and metal nanoparticles like Ag are incorporated into the blood, which serves as the base fluid. This incorporation aims to enhance the performance of hybrid nanofluids in influencing flow phenomena. The proposed mathematical model designed for the above-mentioned constraints is solved analytically following the transformation into their corresponding non-dimensional form. The statistical behavior of the diversified parameters is presented graphically and the numerical results for the rate coefficients are deployed via table. Further, the important characteristics of the results are depicted as follows: the combined effect of the particle concentration used in hybrid nanofluid favors in reducing the fluid velocity greatly than that of nanofluid and pure fluid. [ABSTRACT FROM AUTHOR]