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Mathematical Modelling of Triple Diffusion in Natural Convection Flow in a Vertical Duct with Robin Boundary Conditions, Viscous Heating, and Chemical Reaction Effects

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Abstract

A triple diffusive convective flow (thermal diffusion and dual species diffusion) in a viscous fluid flowing within a vertical duct is investigated under the Robin boundary conditions at the duct walls. Viscous heating and homogenous chemical reaction effects are included. The mass transfer (solutal) buoyancy effects due to concentration gradients of the dispersed components are taken into account using the Boussinesq approximation. Symmetric and asymmetric wall conditions for the temperature are taken into account. The conservation equations are rendered into a dimensionless form via suitable transformations, and the emerging ordinary differential equations feature a number of dimensionless parameters. Those include the thermal Grashof number, two solutal Grashof numbers (one for each of the diffusing components, i.e., species 1 and species 2), left and right duct wall thermal Biot numbers, species 1 and species 2 chemical reaction parameters, Brinkman number, and temperature difference ratio. These coupled and nonlinear dimensionless conservation equations are solved numerically using the Runge-Kutta shooting method. The solutions obtained numerically are validated with approximate analytical solutions obtained via a regular perturbation method, which are valid for small values of the Brinkman number. The effect of selected parameters on the velocity, temperature, and dual species concentration distributions is presented graphically. Furthermore, the variation of the skin friction and the Nusselt number with these parameters is tabulated. The solutions obtained numerically and analytically are found to be equal in the absence of viscous dissipation. However, the deviation grows with large values of the Brinkman number. In the absence of chemical reaction, the results concur with the earlier computations by Zanchini (1998). Increase in the second species solutal Grashof number is observed to decelerate the flow in the left duct half space, accelerate the flow in the right duct half space, and consistently reduce the temperatures across the entire duct width. With rise in the species 1 chemical reaction parameter, the concentration magnitudes increase in the left duct half space and decrease in the right duct half space. A similar response is computed for the influence of the species 2 reaction parameter on the concentration profile. The temperatures are strongly enhanced across the duct width with increase in the Brinkman number and are symmetric about the channel centerline for the symmetric Biot number case (equal thermal Biot numbers at the left and right walls). These profiles change in the asymmetric Biot number case (equal thermal Biot numbers at the left and right walls). The temperatures descend from the left wall toward the right wall, although they are still enhanced with increase in the Brinkman number. The simulations are relevant to geochemical transport phenomena, industrial materials processing, and thermal duct design.

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Authors and Affiliations

  1. Dipartimento di Ingegneria dell’Università degli Studi della Campania “Luigi Vanvitelli”, CE, Aversa, Italy

    J. C. Umavathi

  2. Department of Aeronautical/Mechanical Engineering, University of Salford, M5 4WT, Manchester, UK

    O. Anwar Bég

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Umavathi, J.C., Anwar Bég, O. Mathematical Modelling of Triple Diffusion in Natural Convection Flow in a Vertical Duct with Robin Boundary Conditions, Viscous Heating, and Chemical Reaction Effects. J. Engin. Thermophys. 29, 348–373 (2020). https://doi.org/10.1134/S1810232820020162

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