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After applying a magnetic field, the behavior of the partly ionized liquids is completely different from that of the ordinary fluids. In this study, we concentrated on the Cattaneo–Christov heat flux model-based three-dimensional partly ionized bio-convective flow of a second-order fluid on a bidirectional permeable stretching surface. The development of the thermal and solutal flow models takes into account the impacts of non-uniform sources and sinks, Ohmic viscous dissipation, and chemical reactions. In addition, the surface boundary effects of electron and ion collisions with convective boundary conditions are seen. The mathematical flow model is transformed appropriately to create an ordinary differential equations, which is then numerically solved with MATLAB’s BVP4C approach. To demonstrate the physical relevance of the flow field along various developing parameters, graphical and tabular results are created. It is noteworthy to note that while fluid temperature decreases with stronger values of the second-order fluid parameter, fluid velocity improves in both directions. In addition, it is shown that raising the thermal and concentration relaxation parameters, respectively, causes a drop in the fluid temperature and nanoparticle concentration.Graphical AbstractThe graphical description of the second-grade fluid parameter along velocity field. It is seen from the figure that stronger values of second-grade fluid parameter improve the fluid velocity consequently, as a result momentum boundary layer thickness boosts up.

The persistent challenge of optimizing heat transfer and energy storage in industrial and engineering applications has spurred the development of advanced heat transfer fluids, such as nanofluids and hybrid nanofluids. A third-generation heat transfer fluid, known as modified hybrid nanofluids (MHNs), has recently emerged, created through the synthesis of ternary nanomaterials and the base fluid. This new class of fluids demonstrates enhanced effectiveness over simple nanofluids in a variety of engineering and industrial applications. Thus, the current analysis concentrates on a thorough investigation that employs modified versions of Fick’s and Fourier’s laws to explore a variety of scientific ideas in a synergistic manner, including magnetohydrodynamics, micropolar trihybrid nanofluids, second-grade fluids, and stagnation point flow. Additionally, the effects of thermal radiation, activation energy, viscous dissipation, and double stratification are considered. The study goes on to use the modified versions of Fourier’s and Fick’s principles, which offer a thorough foundation for comprehending heat-mass communication in such complex systems. By using suitable similarity variables, the flow model equations are converted into their non-dimensional forms. These non-dimensional equations are then numerically solved through the BVP4C technique implemented in MATLAB. Results are produced both numerically and graphically, and they are analyzed for different emergent dimensionless parameters. For both unary and trihybrid nanofluid flow, the thermal and concentration relaxation time parameters decrease the temperature and concentration profile. Furthermore, when ternary hybrid nanofluid and unary nanofluid are present, the fluid velocity and temperature flow rapidly increase with bigger values of the second-grade fluid parameters.

This paper deals with an unsteady magnetohydrodynamic two-dimensional second-grade fluid flow towards a permeable exponentially stretching surface with heterogeneous–homogeneous reactions. The nonuniform heat source/sink, thermal slip, and thermal radiation effect are also considered to analyze the thermal attributes. The modeled equations of motions are converted into nonlinear ordinary differential equations (ODEs) by suitable transformations. A MATLAB Bvp4c approach is employed for the numerical solution of ODEs. The outcomes of various parameters are scrutinized by graphs. The quantities of interests such as Nusselt number and the skin friction are presented and discussed. The resistance effects take place due to higher estimations of second-grade parameter, as a result, the velocity field declines. The temperature field raises with the increment of radiation parameter. The concentration of nanoparticles decaying when heterogeneous-homogeneous reactions become larger. Moreover, from the tabulated data, it is noticed the growing estimations of Kand Mboosts the coefficient of skin friction.

The main aim of current research work is to examine the impacts of homogeneous/heterogeneous reactions on the three-dimensional second-grade micropolar fluid flow caused by an exponentially stretching Riga plate. The thermal and solutal energy transportation characteristics are observed in the existence of Cattaneo–Christov heat flux, non-uniform heat source/sink and joule heating. Moreover, the features of thermal slip and porous medium are also incorporated in the mathematical model. The dimensionless process is adopted for the conversion of the PDEs into the self-similar form of ordinary differential equations (ODEs). The numerical approach Bvp4c is employed to solve ODEs and a comprehensive discussion is presented of arising physical parameters in this research work. The velocity profile rises as boosting the values of the microrotation parameter. Further, microrotation profiles along with x- and y-axes show decaying behaviour for the higher estimations of microrotation parameters. The homogeneous reaction profile has rising behaviour for higher values of Schmidt number and diminishing for higher estimations of strength homogeneous reaction parameter, respectively.