Dufour Effect with Ramped Wall Temperature and Specie Concentration on Natural Convection Flow Through A Channel

In this paper, we have obtained an analytical solution to the problem of unsteady free convection and mass transfer flow of an incompressible fluid through a vertical channel in the presence of Dufour effect (or diffusion thermo). The bounding plates are assumed to have ramped wall temperature as well as specie concentration. The mathematical model responsible for the physical situation is presented in dimensionless form and solved analytically using the powerful Laplace Transform Technique (LTT) under relevant initial and boundary conditions. In order to cross check the accuracy of the analytical results, numerical solutions are obtained using PDEPE solver in MATLAB. The expressions for temperature, concentration, and velocity are obtained. The effects of Dufour parameter, Prandtl number (Pr), Schmidt number (Sc), and dimensionless time are described during the course of these discussions. The temperature, concentration, and velocity profiles are graphically presented for some realistic values of Pr = 0.025, 0.71, 7.0, 11.62, 100.0 and Sc = 0.22, 0.60, 1.00, 2.62, while the values of all other parameters are arbitrarily taken.


Introduction
Fluid flow, and the combined heat and mass transfer through a channel, have received less attention than the case of a single plate. This configuration is very frequently encountered in numerous applications, such as fire engineering, combustion modelling, nuclear energy, petroleum reservoir etc. Transport phenomena, involving the combined influence of thermal and concentration buoyancy, are often seen in many engineering systems. They are frequently encountered in the design of modern electric circuits, heat exchangers, solar panels, ventilation system in buildings, chemical distilleries, and thermal protection systems [1][2][3][4][5][6][7][8][9][10][11]. Most of these studies are directed towards the effects of various fluid dynamical processes and flow geometry. Moreover, most of the previous studies were carried out by either numerical or experimental methods.
Study on free convection flow, due to the effect of temperature change has drawn significant interest recently [12][13][14][15][16]. Authors in [17] formulated the problem of coupled heat and mass transfer by natural convection from a vertical, semi-infinite flat plate, embedded in a porous medium in the presence of an external magnetic field and internal heat generation or absorption effects. In this paper, the plate surface was maintained at either, constant temperature or constant heat flux, and was permeable to allow for possible fluid wall suction or blowing. Chamkha [18] considered unsteady, laminar, double-diffusive, natural convection flow inside a rectangular enclosure filled with a uniform porous medium with cooperating temperature and concentration gradients. Ismail et al. [19] investigated unsteady magnetohydrodynamics natural convection flow through a porous medium, parameter, magnetic field parameter, and the angle of inclination in both ramped and isothermal cases. The authors in [12] investigated the Dufour effect on free convective flow with heat absorption, thermal radiation, chemical reaction, and magnetic field with ramped wall temperature over a single vertical plate contained in a permeable medium.
The objective of the present work is to provide an analytical solution for the unsteady free convection and mass transfer flow, in a vertical channel, with ramped wall temperature and ramped wall specie concentration, in the presence of Dufour effect. The influences of the ramped walls are analysed with different parameters embedded in the problem. The main benefits resulting from the analytical approach is a transparent dependence of the quantity of engineering interest of the governing parameters. The accuracy of the analytical solutions is ensured by making a comparison with the numerical solutions. Furthermore, the results presented here are compared with existing literature Singh et al. [34], where the boundary conditions are constant and Dufour effect is absent. In fact, the solutions obtained by Singh et al. is a particular case of the solutions obtained in the present work. The present results will have specific applications in the design of solar energy collectors, geothermal systems, cooling of electronic equipment, and in the design of chemical processing equipment.

Governing Equations
Mathematical model responsible for transient free convection and mass transfer flow in a vertical channel in the presence Dufour effect is considered. Let -axis be along the vertical direction andaxis normal to it ( Figure 1).

Figure 1. Physical configuration.
At the initial state ′ ≤ 0, the two bounding walls as well as the fluid are at rest and considered to have the same constant temperature ′ and concentration ′ . At ′ > 0, the temperature and concentration at the wall ′ = 0 have, respectively, a temporally ramped function given by ′ + ( 0 ′ − ′ ) ′ / 0 and ′ + ( 0 ′ − ′ ) ′ / 0 at ′ ≤ 0 , and then, at ′ > 0 , the temperature and concentration are maintained at 0 ′ and 0 ′ respectively. The temperature and concentration at the other wall ′ = remain the same as the initial temperature ′ and concentration ′ giving raise to free convection currents. The system of equations shows that the temperature is influenced by specie concentration which leads to Dufour effect. We presume that the heat flux that radiates along the vertical axis is negligible relative to that along the horizontal axis. Since both the two walls are of infinite length, all the physical quantities will depend only upon ′ and ′. Within the framework of such assumptions, the equations which govern free convection and mass transfer flow in the presence of Dufour effect under Boussinesq approximation are: Subject to the following initial and boundary conditions { ′ ≤ 0: ′ = 0, ′ = ′ , ′ = ′ for 0 < ′ ≤ ′ > 0: { at ′ = 0: To change Equations (1)-(3) to non-dimensional forms, we make the following substitutions: Thus, Equations (1)

Solutions
The ramped BCs for temperature and concentration given in Equation (9) can be written in linear form using the Heaviside step function ( ) as Then applying LTT to (6)- (8) in conjunction with (9) and taking (13) into consideration we obtain where, denotes the Laplace parameter and By taking the inverse transforms of (14)- (16) and applying the second shift theorem [35][36][37], we obtain the following solutions: 4 12 4 12

Solution for the Temperature When =
It can be observed in (18)

Solution for the Velocity When
It is clear that (20)

Results and Discussions
The Dufour effect on the flow of a viscous fluid, through a vertical channel with ramped boundary conditions, is investigated. The analytical and numerical solutions to the system of governing Equations (6)-(8) with appropriate ICs and BCs given in (9) [38].
The solutions for temperature, concentration, velocity, Nuselt number, Sherwood number and wall Skin-friction are graphically reported in Figures 2-14, thereby revealing the influence of the embedded parameters on the flow.
In order to highlight the accuracy of the analytical and numerical results, comparison is made and presented in Table 1. From the table, it is clear that the analytical and numerical results are in good agreement.
From Figure 2, it can be observed that the temperature descents from the ramped state on the wall to the free stream state. As increases, the temperature also increases. Likewise, when the Schmidt number increases, the temperature rises as well for all . In Figure 3, concentration is seen to reduce as the Schmidt number increases, while it increases with increase in time.   Figure 4 shows that the isothermal wall temperature is greater than the ramped wall temperature (or they are at least the same). This happens because heating of the fluid takes place more slowly in the case of ramped boundary conditions than the case of constant boundary conditions. Similarly, it can be seen in Figure 5, that the species concentration is greater in the case of isothermal plate than in the case of ramped temperature at the plate. However, it is observed that the fluid temperature demonstrates significant change over time than the concentration.
Dufour effect with both ramped and constant boundary conditions are respectively presented in Figures 6-9. It is observed that the temperature rises when and increase. Moreover, the particular case of = 0 in Figure 8 corresponds to the work of Singh et al. [34] and it indicates that the energy flux is not caused by specie concentration gradient. Thus, we see that Dufour effect has the tendency to change the fluid's temperature. So also, it is clear that the temperature is higher in the case of constant boundary conditions compared to when ramped wall conditions are used since heating of the fluid is slower in the latter than in the former.      Figure 10 shows the influence of Dufour effect on the velocity over time. As increases, the velocity also increases and has higher value when the boundary conditions are constant, but has lower value with ramped wall conditions. This is because in the latter, the buoyancy force increases slowly as increases. Figure 11 shows that increase in causes the velocity to rise and apparently the time taken for the velocity to reach steady-state is very short compared to the work of Singh et al. [34].   , ℎ and are presented in Figures 12-14 respectively. and depend on and while ℎ depends on and . From Figure 12, it is clear that for all and with ramped boundary conditions, ascends for 0 < ≤ 1 and descends when > 1. However, decreases generally in the case of constant boundary conditions. Likewise, for all and with ramped boundary conditions, ℎ ascends for 0 < ≤ 1 and descends for > 1. Moreover, ℎ decreases generally with constant boundary conditions. demonstrates an insignificant change over time for fluids with large Prandtl number ( ). However, variations become pretty responsive to minor change in time for fluids with smaller . Moreover, is higher in the case of constant boundary condition.

Conclusions
A mathematical model, that investigates the Dufour effect on unsteady free convection, and mass transfer flow of an incompressible fluid, through a vertical channel, in the presence of Dufour effect, has been developed. Influences of the ramped walls are analysed with various parameter values. The study shows that incorporating ramped BCs makes the fluid temperature, specie concentration, flow velocity, coefficient of rate of heat transfer, coefficient of rate of mass transfer and coefficient of Skin-friction to be lower compared to the case of constant BCs. Moreover, Dufour effect changes the fluid temperature. This happens because specie concentration gradient which occur as a combined effect of a permanent process can also generate heat flux. The velocity as well is affected by the Dufour effect. However, from the settings of the governing equations, the concentration cannot be influenced by the Dufour parameter. The results could have immediate relevance in the separation of isotopes, cooling of vertical printed circuit boards, geothermal systems and in the design of chemical processing equipment.