The Succession of Heat and Mass Driven Natural Convection Regimes Along a Vertical Impermeable Wall

This paper presents the analysis of the natural convection process that takes place near a vertical plane wall embedded in a constant temperature and linearly mass stratified fluid (the Prandtl number and the Smith number are smaller than 1.0, while the Lewis number is greater than 1.0). The wall has a constant temperature, while the flux of a certain constituent is constant at this boundary. The scale analysis and the finite differences method are used as techniques of work. The scale analysis proves the existence, at equilibrium, of heat and/or mass driven convection regimes along the wall. The finite differences method is used solve the governing equations and to verify the scale analysis results using two particular parameters sets.


Introduction
The analysis of the natural convection process along a vertical plane wall is a classical problem that was taken into consideration along the past decades in applications of a specific or general character ( Figure 1 presents the vertical plane wall and the x-y system of co-ordinates associated to it in dimensional (Fig. 1a) and in dimensionless (Fig. 1b) representations.

Mathematical Formulation
The temperature at the wall is Tw, while the environment has a constant temperature, T∞. The mass flux of a certain constituent, mw, is constant at the wall, while the environment registers a concentration of the constituent of (1) (2) ( (4) and the boundary conditions (Neagu, 2018): define the point of start of the scale analysis.

Scale Analysis
This analysis follows the same pattern used before in the scientific literature (Bejan 1995;Neagu, 2018Neagu, , 2021) for a better correlation, comprehension of the research characteristics and following the evolution of the natural convection regime that develops along the boundary: the initial state (section 3.1), the heat driven convection (HDC) regime (section 3.2) and the mass driven convection (MDC) regime (section 3.3).

The Initial State
Because this section is similar to the results of previous analysis, only the most significant results will be mentioned here (Neagu, 2018): the boundary layer thickness of the temperature field: (10) the boundary layer thickness of the concentration field: (11) 12 the heat driven convection regimes that exists at the beginning at each point along the wall will be replaced by a mass driven convection regime only if the equilibrium time of the regime is bigger than the transition time, ttrz: (12) the inequality , that is valid at the beginning at each point of the wall, ceases to be valid if the equilibrium time of the regime is bigger than ts: (13) (14)

The Heat Driven Convection Regime
Invoking the equilibrium between the horizontal diffusion and the vertical convection of heat, the temperature equilibrium time, boundary layer thickness and vertical velocity orders of magnitude are (Neagu, 2018): , then the equilibrium between the horizontal diffusion of the constituent and the vertical convection of it gives us the equilibrium time of the concentration field: (18) Further, using equation (11), the boundary layer thickness order of magnitude becomes: The equilibrium time of the concentration field, ( ) T C , ech t , is bigger than the transition time, trz t , if: , then the concentration equilibrium time and boundary layer thickness are: 1.
The analysis of the results presented above reveals that there are only two possibilities: 2. a HDC regime along the wall if

The Mass Driven Convection Regime
The vertical velocity order of magnitude was derived by Lin, Armfield and Patterson (2008): (26)

MDCSc Regime.
In this case, the equilibrium between the horizontal diffusion and the vertical convection of the constituent requires: Replacing v from equation (26), the concentration equilibrium time and the boundary layer thickness are: At equilibrium, the vertical velocity order of magnitude scales as: The inequality ( defines the X co-ordinate that separates the MDCC and the MDCSc regimes in the figure 2(b).
Scale analysis of the temperature field in the MDCSc regime. The equilibrium state requires: . Using the equation (10) and the equation (29), the equilibrium temperature boundary layer thickness order of magnitude is: (31)

MDCC Regime
The equilibrium between the horizontal diffusion and the vertical convection requires . Replacing the equations (11) and (29), the equilibrium time and the concentration boundary layer thickness become: (32) ( ) while the vertical velocity scales as: Scale analysis of the temperature field in the MDCC regime.
The equilibrium between the thermal horizontal diffusion and the vertical thermal convection requires Further, the scale analysis results of section 3 will be verified using the finite differences method for two particular parameters sets.

Numerical Modeling
The stream function formulation of the velocity field, , and the vorticity definition, (37) The boundary conditions take the following form: (42) 0 X X X X (43) The conservation equations (36)−(39) with the boundary conditions (40)−(43) were solved using the finite differences method. A software was built by Neagu (2018) using a higher order hybrid scheme (Tennehill, Anderson & Pletcher, 1997) through an iterative process: at each time step, equation (36) was solved iteratively till the relative error of  , at each point of the grid, became less than 10 -6 . The iterative process stopped when the relative errors of  ,  and  became less than 10 -6 at each grid point.

Results and Discussions
Two particular parameter sets are used to run the program explained above: The 5000 Ra = The results for this particular parameters set are present by Fig. 3. It shows the temperature (Figure 3(a)), concentration (Figure 3(b)), stream function (Figure 3(c)) and X / C   (Figure 3(d)) fields. The concentration field (Figure 3(b)) is not greater than  In the HDCC region, three sections present the temperature (Figure 4(a)), the concentration (Figure 4(b)) and the vertical velocity (Figure 4(c)) plots for thre abscissa: 0.5; 1.0 and 1.5. The scaled plots (Fig. 4(d)-Fig. 4(f)

Conclusions
The natural convection process developed in the boundary layer of a vertical plane wall continues to be the subject of new discoveries. If the temperature and the heat flux of a certain constituent are constant at the wall, then a constant temperature and a linearly mass stratified environment ( , (Neagu, 2018).
These results open the gate for further scientific paths: a closer analysis of the way in which the non-dimensionalisation process is realised for these particular cases.
a closer analysis of the implications that these results could have on the stability management of the natural convection process.