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

Abstract

The triple-diffusive convective flow (thermal diffusion and dual species diffusion) in a viscous fluid flowing within a vertical duct is investigated subject to 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 dimensionless form via suitable transformations and the emerging ordinary differential equations feature a number of dimensionless parameters including 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 theRunge-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 Brinkman number.The impact of selected parameters on velocity, temperature and dual species concentration distributions is visualized graphically. Furthermore, the variation of skin friction and Nusselt number with these parameters is also tabulated. The solutions obtained numerically and analytically are found to be equal in the absence of viscous dissipation. However, the deviation is magnified with large values of Brinkman number. In the absence of chemical reaction, the results concur with the earlier computations of Zanchini (1998). Increasing second species solutal Grashof number is observed to decelerate the flow in the left duct half space, to accelerate the flow in the right duct half space and consistently reduce temperatures across the entire duct width. With increasing species 1 chemical reaction parameter the concentration magnitudes are elevated in the left duct half space whereas they are depressed in the right duct half space. A similar response is computed for the influence of species 2 reaction parameter on the concentration profile. Temperatures are strongly enhanced across the duct width with increasing Brinkman number and are symmetric in nature about the channel centerline for the symmetric Biot number case (equal thermal Biot numbers at the left and right walls). These profiles are morphed for the asymmetric Biot number case (equal thermal Biot numbers at the left and right walls). Temperatures descend from the left wall to the right wall, although they are still enhanced with increasing Brinkman number. The simulations are relevant to geochemical transport phenomena, industrial materials processing and thermal duct design.