Mathematical modeling of fully developed mixed convection of Newtonian fluid in a vertical channel

Abstract

In many industrial and engineering applications, the understanding of flow patterns and heat transfer processes is vital. Nowadays, the major concern amongst the scientists and engineers is to achieve an optimal heat transfer process in advanced energy systems by increasing or decreasing the heat transfer rate. To solve the issue, the occurrences of flow reversal as well as the effects of nanoliquids and a conductive wall were studied in this research. A fully developed, mixed convection in differentially heated vertical channel was selected as the geometrical model. The flow in the mathematical models were considered to be Newtonian, two-dimensional, incompressible and laminar. The Boussinesq approximation was used for the density variations. The shape of the nanoparticles was assumed to be spherical. As for the porous medium, the Brinkman- Forchheimer extended Darcy model was adopted. In the problems, a fully-developed flow was assumed and the governing equations were transformed to a system of nonlinear ordinary differential equations (ODEs). The fourth-order Runge-Kutta (RK4) method incorporating a shooting technique with Newton’s method was used to solve the ODEs. An auxiliary parameter was proposed in the shooting method to handle the high nonlinearity in the ODEs. All numerical schemes were programmed in MATLAB, except for the final problem, where a built-in routine in MAPLE, dsolve, was used instead. Validation studies to the previously published problems were carried out to verify the accuracy of the present computations. The numerical results of the flow and temperature variables as well as the heat transfer rates were presented graphically. First, the occurrence of flow reversal on a fully developed vertical channel were studied. It was revealed that flow reversal occurs only for a sufficiently large mixed convection parameter and internal heat parameter but with low values of local heating exponent. Meanwhile, for the case involving water based nanoliquid with boundary conditions on temperature, it was concluded that the presence of nanoparticles in the base liquid causes a noticeable decrease in the velocity and the temperature profile and reflow occur with higher value of nanoparticle volume fraction. The obtained result also shows that the viscosity gives greater impact than buoyancy and Biot numbers on the heat transfer rate as nanoparticle volume fraction increases. Next, the problem involving non-Darcy flow with boundary conditions on temperature concluded that Robin boundary condition gives a more satisfactory and realistic result, in comparison with the Dirichlet condition and Neumann condition. Further, for the problem involving nanoliquids combined with porous medium, it was concluded that the critical values of mixed convection parameter for the occurrence of reversed flow decreases with increasing temperature ratio and increases with increasing nanoparticles mass flux. Finally, the problem involving unsteady oscillatory flow with non-uniform internal heating concluded that the oscillatory heat transfer for both walls generally have the highest amplitude at relatively low oscillating frequency.