The main focus of this study was to develop a numerical scheme with new expressions for interface flux approximations based on the upwind approach in the finite volume method. Our new proposed numerical scheme is unconditionally stable with second-order accuracy in both space and time. The method is based on the second-order formulation for the temporal approximation, and an upwind approach of the finite volume method is used for spatial interface approximation. Some numerical experiments have been conducted to illustrate the performance of the new numerical scheme for a convection–diffusion problem. For the phenomena of convection dominance and diffusion dominance, we developed a comparative study of this new upwind finite volume method with an existing upwind form and central difference scheme of the finite volume method. The modified numerical scheme shows highly accurate results as compared to both numerical schemes.

In this analysis, the thermal and flow properties of modified hybrid nanofluids (MNFs) have been investigated under the effects of electroosmosis and homogeneous-heterogeneous chemical reactions. Three types of nanoparticles of Cu, CuO, and Al2O3 are utilized to monitor the performance of the MNFs with water as a working liquid. The determination of the heating phenomenon is explored by incorporating the effects of NPs shape, temperature reliant viscosity, Joule heating, heat generation/absorption and viscous dissipation. In this exploration, equal diffusion factors for the auto catalyst and reactants are assumed. The model formulation contains a highly non-linear PDE system, which is converted to ODEs under physical assumptions with lubrication and Debye–Huckel. The solution treatment involves the Homotopy perturbation method for solving the governing differential equations is used. A major outcome discloses that an addition in heterogeneous reaction parameter aids in enhancing the concentration profile. In a result, the temperature curve decreases at increasing volume fraction of the NPs. Modified hybrid NFs have higher heat transfer rate as compared to base H20, or ordinary Al2O3–H20 and hybrid Cu + Al2O3–H20 NFs. Pressure gradient decreases by improving electroosmotic parameter. Further a comparison between analytically (HPM) and numerical results (NDSolve) show that both results are in good agreement.

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Based on the finite volume method (FVM), a numerical scheme is constructed to simulate the unsteady convection–diffusion transport problem. New expressions are obtained for interface approximation of the field variable, subsequently, these newly obtained interface expressions are used to develop the numerical scheme. Convection-dominant and diffusion-dominant phenomena are simulated by taking different values of convective velocity [Formula: see text] and diffusion coefficient [Formula: see text]. This newly proposed numerical scheme gives second order of convergence along space and time. Experiments are carried out to test the new proposed upwind approach. Numerical results produced by the proposed approach are compared with the conventional finite volume method, step-wise approach FVM and quadratic upwind interpolation finite volume approach. This comparative study indicates that for different cases for convection-dominant and diffusion-dominant problems, our proposed approach gives highly accurate and stable solution. The conventional finite volume method and other approaches result solution with non-physical oscillations. Our obtained numerical results are consistent and support our theoretical approach.