FLOW REVERSAL OF FULLY DEVELOPED MIXED CONVECTION OF NANOFLUIDS IN A VERTICAL CHANNEL FILLED WITH POROUS MEDIUM

The present analysis is concerning the criteria for the onset of flow reversal of the fully developed flow mixed convection of nanofluids in a vertical channel filled with porous medium. The governing equations and the critical values of the buoyancy force are solved and calculated numerically via dsolve package in MAPLE. It was found that the critical values of mixed convection parameter for the occurrence of reversed flow decreases with increasing temperature difference ratio and increases with increasing nanoparticles mass flux. AMS Subject Classification: 76S99, 12D05

of mixed convection flow in a heated vertical parallel-plate channel filled with porous medium has received considerable attention because of its wide range of applications such as geothermal system, cooling of nuclear reactors, thermal insulation, energy storage and conservation, chemical, food and metallurgical industries petroleum reservoirs and operations of electronic devices. One of the earliest studies on mixed convection in a vertical channel with asymmetric wall heating had been studied by [9]. They have modeled the porous layer using non-Darcy. [3] included the heat generation, absorption and hydromagnetics effects. [14] studied the effect of inertial forces by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the BrinkmanForchheimer-extended Darcy equations. Later [13] studied the problem with boundary conditions of third kind.
The volcanic eruption, material fabrication, cooking and transportation utilizing internal combustion and jet engines, unintentionally release nanoparticles into the atmosphere. These nanoparticles dispersed in the pure fluid such as water known as nanofluids. In recent years a new type of nanoparticles source coming form nanotechnology has been introduced, and followed by an engineered nanofluids, product of [6]. Nanofluids in vertical porous media with viscous heating had been studied by [12]. [10] and [11] studied nanofluids in a vertical channel filled with highly porous medium. They found that flow reversal at the cold wall of the channel becomes smaller for nanofluids compared to regular fluids and the flow reversal extents by mixed convection parameter. However, they overlooked the effects of nanofluids combined with the porous medium properties in determining the criteria for the onset of flow reversal. Therefore, in this article, we present the conditions for the onset of flow reversal.
The high buoyancy force due to a differentially heated wall combined with an upward flow leads to a high fluid flow adjacent to the walls that can precipitate a downward flow (i.e. reversal flow) emanating from the open top of the channel in order to augment the increased upward flow. Theory on the appearance of flow reversal was conducted by [1] and [5] for mixed convection in vertical channel filled with regular fluids. Criteria for the occurrence of reversal flow, adjacent to the colder wall under effects of concentration, magnetic field including internal heating, micropolar as well as thermophoretic were presented by [7], [2], [4] and [8], respectively. aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa Nanofluids Figure 1: Schematic representation of the model.

Mathematical Formulation
Consider the steady flow of a viscous and incompressible nanofluids between two vertical and parallel plane walls. The distance between the walls, i.e., the channel width, is ℓ. A coordinate system is chosen such that the x-axis is parallel to the gravitational acceleration vector g, but with the opposite direction. The y-axis is orthogonal to the channel walls, and the origin of the axes is such that the positions of the channel walls are y = −ℓ/2 and y = ℓ/2, respectively. A sketch of the system and of the coordinate axes is reported in Fig. 1. The walls at y = −ℓ/2 and y = ℓ/2 are isothermal at given temperatures T 1 and T 2 , where we assume that T 1 ≥ T 2 . The nanofluids has a uniform vertical upward stream wise velocity distribution U 0 at the channel entrance. The Boussinesq approximation is employed and homogeneity and local thermal equilibrium in the porous medium are considered. The nanofluids is assumed to saturate the solid matrix and both are in thermodynamic equilibrium. The Brinkman-Forchheimer extended Darcy equation was adopted for the porous layer. Based on these assumptions and taking into account the effect of viscous dissipation,the following relations apply: in which v is the velocity component in the y−direction and P is the pressure. This will lead us to the governing equations for fully developed flow: where T 0 is the reference temperature and the coefficient v eff = v 1 / isthe effective viscosity of fluid in the porous region. Meanwhile, ζ and Φ are respectively a constant and the viscous heating due to viscous dissipation [10]. They are defined as: The walls of the channel are assumed isothermal with no-slip conditions: With reference to equation (2), the first and second derivatives of T with respect to y is substituted into the thermal energy equation, i.e. equation (3).Hence, we obtain: The following dimensionless parameters were applied to transform these equations into dimensionless forms: Using these dimensionless parameters, the dimensionless governing equations for the problem are: subject to the boundary conditions: Here, Br is the Brinkman number, R T is the temperature difference ratio, G R = Gr/Re, where Gr is the Grashof number and Re is the Reynolds number.

Solutions Method
System (11) and (12) subject to the boundary conditions (14)-(16) is a kind of nonlinear boundary value problem (BVP) for which no available exact solutions have been reported. Therefore, the BVP will be solved numerically by dsolve from Maple. The nonlinearities was handled by applying continuation, optional in dsolve. The continuation is utilized to maintain Newton method (inside package dsolve) to reach the convergence. Custom user interfaces in Maple could be created. Maple is a commercial computer algebra system developed and sold commercially by Maplesoft, a software company also based in Waterloo, Ontario, Canada. We use Maple version 16, which is an improvement from its predecessor Maple 15. tion. Therefore, an increase in the buoyancy force increases the velocity in the upward direction. The effects of various levels of Darcy number on dimensionless velocity is presented in Figure 4. Obviously, increasing Da increases the velocity. The maximum velocity location moves to the middle channel as the Darcy number get higher. Flow reversal is found near the colder wall for Da < 0.1. Generally, decreasing the Darcy number enhances the flow reversal and suppress the maximum velocity near the hotter wall. It is observed at Da = 0.001 that the flow reversal is slightly smaller compared to the flow reversal at Da = 0.005. This means that a sufficiently low Darcy number weakens the velocity at each position including flow reversal near the colder wall. Figure 5 presents effects of various levels of Brinkman number on dimensionless velocity. Obviously, increasing viscous dissipation increases the velocity and decrease the flow reversal. This fact is because of a greater energy generated by viscous dissipation yields a greater fluid temperature that leads to increasing the buoyancy force at each position and as a consequence it tends to contrast the flow reversal. Flow reversal occurs only for a large mixed convection parameter combined with low nanoparticles mass flux parameter, low Darcy number and small Brinkman number. Parameter zones for the occurrence of reversed flow would be presented next. Figure 6 shows the critical value G Rc for the onset of a flow reversal with N for different Br at R T = 1, Da = 0.01 and F = 1. It was observed that increasing the nanoparticles mass flux increases G Rc . It notes that N = 0 refers to regular fluids. The effect of variation Br on G Rc nanofluids (N = 2.0) is more profound than G Rc regular fluid. Brinkman number (Br) represents the effect of heat dissipation in the medium such that large values of Br show that more heat dissipates in the medium. Heat dissipation can act as a heat source in the medium and raises the nanofluids temperature. Figure 7 shows the critical value G Rc for the onset of a flow reversal with N for different Da at R T = 1, Br = 0.05 and F = 1. Increasing the nanoparticles mass flux increases G Rc for the considered Da. The G Rc increases by increasing Da at fixed nanoparticles mass flux parameter. This is due to the fluid velocity increases with an increase in the porous medium permeability (i.e., Da increases). In fact, with an increase in the permeability of the porous medium at a fixed pressure gradient, more fluid can pass through the porous medium, thus, the velocity of the nanofluids increases.  Figure 8 shows the critical value G Rc for the onset of a flow reversal with N for different F at R T = 1, Br = 0.05 and Da = 0.01. Increasing the nanoparticles mass flux increases G Rc significantly for the considered Forchheimer number. The G Rc decreases as Forchheimer number increases at fixed N . The Forchheimer number represents the inertial effects in the porous media. The presence of porous medium in the channel increases the flow resistance. In addition, the inertial effect by increasing Forchheimer number assist this resistance, which further decreases the buoyancy force. Figure 9 shows the critical value GR c for the onset of a flow reversal with R T for different N at F = 1, Br = 0.05 and Da = 0.01. We limited the R T range from 0.6 to 1 because of G Rc exists only in this range for N = 1. It notes that symmetrical wall temperatures situation when R T = 0 and at this condition, the flow reversal is impossible to occur in the channel. We observed from Fig. 9 that increasing the temperature difference ratio decreases G Rc for all N . In particular, the G Rc dives in 0.6 < R T < 0.625. This fact due to lower degree of the walls temperature variation leading to prevent the occurrence of down flow.  Figure 10 shows the critical value GR c for the onset of a flow reversal with R T for different F at N = 1, Br = 0.05 and Da = 0.01. Increasing the R T decrease G Rc for all F . G Rc decreases significantly in 0.6 < R T < 0.625 for F = 1 while G Rc decreases significantly in 0.625 < R T < 0.65 for F = 4, 8, 16. Variation G Rc by adjusting the Forchheimer number is insignificant for R T ≥ 0.6. Therefore, neglecting the inertial effect (Forchheimer number) in high temperature difference ratio could not cause significant errors.

Conclusions
The present numerical simulation study the effects of nanofluids combined with the porous medium properties in determining the criteria for the onset of flow reversal. The dimensionless forms of the governing equations are solved using MAPLE. The main conclusions of the present analysis are as follows: 1. Flow reversal occurs only for a sufficiently large mixed convection parameter combined with low nanoparticles mass flux parameter, low Darcy