Fully Developed Free Convection Flow in a Vertical Slot with Open to Capped Ends: D’Alembert’s Approach

The fully developed free convection flow in a vertical slot with open to capped ends discussed by Weidman [5] and Magyari [6] is scrutinized in this present work. Exact solution of momentum and energy equations under relevant boundary conditions as discussed in [5, 6] is obtained using the D’Alembert’s method. Numerical comparison of this present work is made with previous result of [6] and the results were justified using the well-known implicit finite difference method (IFDM); this gives an excellent comparison. During the course of numerical investigation, it is found that D’Alembert’s approach is a simpler, reliable and accurate tool for solving coupled equations.


Introduction
Laminar flow between two differentially-heated vertical plates is a classical problem in free convection flow due to its applications in industrial and technological world.
A lot of researchers have worked on fluid flow when both ends are capped as well as when both ends are opened. Batchelor [1] gives a detailed analysis of the conduction and convection regime flow whereas the analytical studies and experimental values of τ were carried out by Elder [2]. Daniel [3] discussed a transition from the conductive to convective regime for closed cavities for flow with large Prandtl numbers. Bu hler [4] reported a significant result of stable laminar convection in a tall slender cavity. For conduction regime flow, he proposed a generalized case by assuming the ends of the channel to be porous.
Weidman [5] presented a convection regime flow in a vertical slot: continuum of solutions from capped to open ends and found that identical viscous and thermal boundary layer exist at the opposing walls when the cavity is capped. In addition, he concluded that as the flow evolves to one with open ends, there is intensification (attenuation) of the boundary layers near the hot (cold) walls. Furthermore, Magyari [6] revisited the work of Weidman [5] and presented another approach "normal modes" of solution to the coupled energy and momentum equations. The conduction regime is seen to correspond exactly with Weidman [5]. Other research articles related to the present student can be found in [7][8][9] The aim of this present work is to present an alternative approach (D'Alembert's method) to derive exact solutions for the mathematical model presented in [5,6]. The governing momentum and energy equations are solved exactly and the impact of all pertinent parameters are discussed with the aid of line graph.

Mathematical Analysis
This work is related to the recent works of Bu hler [4], Weidman [5] and Magyari [6] in which they investigated the conduction [4] and the convection [5] regime of the free convection flow in a differentially heated tall vertical slot with open to capped ends. For the quasi-static transition from the open to the capped end situation, a continuum of solutions has been obtained for both cases.
This present work makes use of D'Alembert's approach used by Jha [10] to solve the same problem discussed in [5,6]. Following the model and assumptions presented by Weidman and Magyari [5,6], we have Along with the boundary conditions [5,6] where E and G are the Elder and Grashof numbers respectively defined by [5] as In order to solve equations (1) and (2), we choose the D'Alembert's method as discussed by Jha and Apere [10]. We multiply equation (1) by 0 and add it to equation (2) The general solution of equation (5) with boundary conditions (6) is From equation (7), we have The roots of equation (9) are Replacing A with A * and A > one after the other in equation where δ * = ,!0 * and δ > = ,!0 > (13) Solving equations (11) and (12) Substituting equation (17) into (1) and applying boundary conditions (3), we have: Equations (17) and (18) above give the conduction regime solution which is exactly the same as presented by Weidman and Magyari [5,6] respectively.

Results and Discussion
In order to verify the accuracy of the present method, we obtained numerical values of equations (14) and (15), compare it with the solutions of Magyari [6] and then use the well-known implicit finite difference method (IFDM) to justify the results. This study has been performed over the reasonable range of 0 ≤ ≤ 1 2 ⁄ . The selected reference values of = 0 corresponds to the open slot while = 1 2 ⁄ to capped slot, the aspect ratio \ = 20 at ! = 1.0 × 10 W and _ = 0.5 to determine from equation (4) for the present analysis as given in [5,6]. Tables 1 and 2 compare velocity and temperature profiles of the present work with those of Magyari [6] and IFDM for atmospheric air ( = 0.72) and water ( = 8.1) respectively. It is observed that the numerical values of both methods are exactly the same for both velocity and temperature profiles while the IFDM validates the results. For (c > 0), the velocity is seen to decrease with increase in but the reverse result is noticed for (c ≤ 0). The reversed result is observed for temperature profiles of atmospheric air ( = 0.72) . On the other hand, for water ( = 8.1) , velocity as well as temperature is seen to increase with increase in for (c ≠ 0).  Figure 1 presents temperature distributions at different values of reference temperature ( ) for fixed value of = 0.72. It is observed that temperature decrease along both walls with increase in variation of reference temperature. Also, it is evident that the temperature of the open slot is seen to be higher than capped slot. Similar result is also noticed in Figure 3 when Prandtl number is fixed at = 8.1. In addition, the center of the channel in Figures 3 and 4 for temperature and velocity of water respectively is seen to be independent on the value of reference temperature ( ).

Conclusions
Fully developed free convection flow in a vertical slot with open to capped ends discussed by Weidman and Magyari [5,6] has been studied in detail in this paper. Closed-form solution for velocity and temperature profiles is obtained with the use of D'Alembert approach. Based on the numerical values obtained from the exact solutions, we draw the following conclusions: 1. D'Alembert method is a reliable, simple and accurate method of solution. This is based on the exactness of the numerical comparison in table 1 and 2 with Magyari [6]. The graphs plotted are also seen to be exact with figures 2 and 3 of Weidman [5].
2. As an accuracy check on the proposed method of solution, the conductive regime solution of equation (17) and (18) is seen to be exactly the same with Weidman [5] and Magyari [6]. Also, the numerical value of implicit finite difference method (IFDM) result is seen to be close to the exact solution.