Mixed convection flow in a vertical channel in the presence of wall conduction, variable thermal conductivity and viscosity

Nomenclature

k₁: Thermal conductivity of plate p₁ (Wm⁻¹s⁻¹)

k₂: Thermal conductivity of plate p₂ (Wm⁻¹s⁻¹)

μ^*: Dimensional viscosity (m²s⁻¹)

k^*: Dimensional thermal conductivity of the fluid

(Wm⁻¹s⁻¹)

λ: Variable viscosity of the fluid (m²s⁻¹)

ε: Variable thermal conductivity of the fluid (Wm⁻¹s⁻¹)

u^*: Dimensional velocity (ms⁻¹)

y^*: Dimensional distance between the plates (m)

ρ: Fluid density (kgm⁻³)

g: Acceleration due to gravity (ms⁻²)

β: Thermal expansion coefficient (K⁻¹)

T^*_f : Dimensional temperature of the fluid (K)

c_p: Specific heat at constant pressure (Jkg⁻¹K⁻¹)

Q₀: Heat absorption coefficient

T₀: Temperature of the cold plate (K)

T_w: Temperature of the heated plate (K)

T_p1 : Dimensional temperature of plate p₁ (K)

T_p2 : Dimensional temperature of plate p₂ (K)

T_p1 : Temperature of plate p₁ (K)

T_p2 : Temperature of plate p₂ (K)

h: Channel width (m)

U: Velocity of the heated plate (ms⁻¹)

x: Distance along y-direction (m)

y: Distance along x-direction (m)

k0k1:thermal conductivity ratio of fluid and plate p₁

k0k2: thermal conductivity ratio of fluid and plate p₂

P^*: Dimensional pressure (Pa)

d^*: Dimensional plate thickness (m)

d: Plate thickness (m)

1 Introduction

Heat transfer and fluid flow have been matters of research for many years due to their applications in modern industries. Fluid flows are captured in many mathematical geometries. Such flows could be laminar or turbulent. Fluid dynamics has so many applications in lubrication industries, gas turbine power plant, polymer technology, food processing industries, plasma power plant, thermal and insulating engineering, MHD power generators etc. These fluid flows are driven either by natural, forced or mixed convection.

Ajay and Feffrey [1] studied combined forced and natural convection heat transfer wherethey showed that the velocity profile reduces near the channel plate (y = 0) with increasing mixed convection while the reverse case was observed at the other plate (y = 1). Jha et al. [2] examined mixed convection flow in a vertical tube filled with porous material. They found that the velocity decreases near the inner surface with increasing mixed convection while it increases on the surface of the outer cylinder. Mohamed [3] studied transient mixed convection heat transfer along a vertical stretching surface in the presence of variable viscosity and viscous dissipation. He concluded that the effect of the mixed convection on the velocity and temperature in the steady flow is more prominent than the transient flow. He further discovered that the flow and thermal distributions are enhanced as the strength of the mixed convection increases. Jha et al. [4] studied mixed convection flow in an inclined channel filled with porous material and concluded that the flow is reversal type, since the velocity profile increased by enhancing the value of mixed convection on the lower inclined channel wall while the trend increased with increase in negative value of the mixed convection on the heated wall. Finite element analysis of heat and mass transfer by MHD mixed convection flow of a non-Newtonian power-lawnanofluid was investigated by Machal and Naikoti [5]. They concluded that, the velocity profile increases with increase in mixed convection while the reverse case was observed on the fluid temperature. Recently, Jha and Aina [6] studied effect of heat generation/absorption on MHD mixed convection flow in a vertical tube. They concluded that, heat generation increases the rate of heat transfer, while the reverse trend was observed in the case of heat absorption. Entropy generation and irreversibility analysis on a steady mixed convection flow in a vertical porous channel was investigated by Ajibade and Onoja [7]. They revealed that the velocity profile increases with increase in mixed convection while a reverse case was observed for temperature distribution. Jha and Aina [8] analysed the effect of induced magnetic field on MHD mixed convection flow in vertical microchannel.

In lubrication industries, viscous dissipation effects cannot be overlooked since internal energy is generated as a result of fluid particles’ interaction which affects the fluid flow and thermal behaviour in a medium. Pantokratoras [9] studied effects of viscous dissipation in natural convection flow along a heated vertical plate. They concluded that the viscous dissipation has a strong effect on velocity profile as it helps the upward and opposes the downward flow. Jha and Ajibade [10] studied effect of viscous dissipation on natural convection flow between vertical parallel plates. They showed that the viscous dissipation heating within the channel increases the fluid temperature above the plate temperature when the Prandtl number of the working fluid is relatively small. Kumar and Sanchayan [11] investigated effects of viscous dissipation on the limiting value of Nusselt numbers for a shear driven flow between two asymmetrically heated plates. They concluded that the temperature profile and the driven temperature difference increase with an increase in viscous dissipation. In a related work,Fransisca et al. [12] examined heat transfer with viscous dissipation in Couette-Poiseuille flow and found that viscous dissipation has significant influence on the fluid temperature. Recently, Kabir et al. [13] investigated effects of viscous dissipation on MHD natural convectionflowwith heat generation and found that the velocity and temperature profiles increase with increasing viscous dissipation. Mohamed [14] examined mixed convection flow on a micropolar fluid over an unsteady stretching surface in the presence of viscous dissipation. He concluded that the velocity and the temperature profiles increase with increase in Eckert number. Prabhakar [15] examined effects of thermal diffusion and viscous dissipation on an unsteady Magnetohydrodynamics free convection flow past a vertical porous plate in the presence of heat sink. Their work concluded that the velocity and temperature profiles increase with increase in viscous dissipation. RaihanulHaque et al. [16] examined effects of viscous dissipation on natural convection flow over a sphere with temperature dependent thermal conductivity in the presence of heat generation. They found that the velocity and temperature profiles increase with increase in viscous dissipation. All the reviewed articles mentioned above made a common assumption and that is to assume that the boundary plates in each of their investigation is of negligible thickness or rather, heat conduction through the boundary plates plays an insignificant role in the hydrodynamics and thermal behaviour within the channel.

The boundary plate thickness of a material has significant effects on the heat flux and fluid flow through a channel with thick-plate boundaries. This is in the conclusion reached in an earlier investigation of Ajibade and Umar [17] in which it was discovered that the rate of heat transfer from the plate to the fluid decreases with growing thickness in the boundary plates. In the study of effects of wall heat conduction on the reforming process of methane in microreactor, Michael and Dimos [18] concluded that heat transport through the solid wall depends on the cross-sectional area of that solid. Hassab et al. [19] analysed effect of axial wall conduction on heat transfer between two parallel plates. Their result showed that, increase in wall thickness serves as thermal resistance to the heat transfer. Ates et al. [20] analysed a transient conjugated heat transfer on a thick walled pipes and found that more heat penetrated backwardly through the upstream region by axial conduction in the thick wall. Mei et al. [21] examined criteria of axial wall heat conduction under two classical thermal boundary conditions and showed that the temperature gradient number of the thin-wall tube is higher than that of thick-wall tube. The effects of wall conduction on a mixed convection heat transfer in an externally finned pipes were studied by Moukalled et al. [22]. They observed that the axial wall conduction in the pipe wall is found to have strong effect on the flow and thermal fields. Other works that investigated the effect of boundary plates’ thickness include the work of Kevin and Barbaros [23].

Convective flow in nature and engineering phenomena requires that viscosity and thermal conductivity of fluids vary with temperature. For instance, the viscosity of dry air at 100⁰C is 21.94 × 10⁻6kg/ms while at 200⁰C, it is 26.94 × 10⁻6kg/ms . The thermal conductivity of a dry air at 100⁰C is 31.39 × 10⁻3W/mK while at 200⁰C, it is 37.95 × 10⁻3W/mK ( Adrian et al. [24]). Tarakaramu et al. [25] investigated the effects of variable thermal conductivity on 3D flow of nanofluid over a stretching sheet using numerical simulation while Dulal and Hiranmony [26] analysed the effects of temperature-dependent viscosity and variable thermal conductivity on mixed convective diffusion flow. They found that velocity profile increases while temperature decreases with increase in mixed convection. Subhas and Mehesha [27] studied heat transfer on a magnetohydrodynamics viscoelastic fluid flow over a stretching sheet with variable thermal conductivity and radiation and found that the variable thermal conductivity has an influence in enhancing the temperature and fluid flow. The works of Rahman et al. [28] and Naseem [29] have shown that the velocity and temperature profiles increase with increase in thermal conductivity. Vajravelu et al. [30], Singh and Shweta [31] and Isaac and Auselm [32] showed that, velocity distributions decrease with increase in viscosity while the temperature profiles increase with increase in variable viscosity. MHD effects on heat transfer over a stretching sheet embedded on a porous medium with variable viscosity, viscous dissipation and heat source/sink were studied by Hunegnaw and Naikoti [33].Uwanta and Murtala [34] studied heat and mass transfer flow past an infinite vertical plate with variable thermal conductivity, heat source and chemical reaction. They concluded that, velocity and temperature profiles increase with increase in thermal conductivity. Sumayya et al. [35] investigated efficiency analysis of a straight fin with variable heat transfer and thermal conductivity. They concluded that, temperature profile increases with increase in thermal conductivity.

Devi and Prakash [36] examined temperature-dependent viscosity and thermal conductivity effects on hydromagnetic flow over a slendering stretching sheet. They concluded that, increase in the viscosity decreases the velocity profile. The work of Animasaun [37] concluded that fluid velocity decreases while temperature increases with increases due to increasing viscosity. The works of Gopal and Jadav [38], Jadav and Hazarika [39], Gopal and Bandita [40], Bandita [41] and Kareem and Salawu [42] have shown that the velocity profiles decrease with increasing viscosity while the temperature profiles increase with increase in viscosity. Recently, Sreenivasulu et al. [43] studied variable thermal conductivity influence on hydromagnetic flow past a stretching cylinder in a thermally stratified medium with heat source/sink. They concluded that the velocity profile increases with increasing mixed convection and the temperature profile increases with increase in thermal conductivity. Muthtamilselvan et al. [44] investigated an internal heat generation of a dusty fluid passes through porous media of a stretching sheet using Range-Kutta fourth order method.

The aim of this work is to carry out a theoretical investigation on the effects of viscous dissipation, wall conduction, boundary plate thickness, variable viscosity and thermal conductivity on a steady mixed convection flow. Due to the wide application of viscous dissipation, conduction at the channel wall, boundary plate thickness in lubrication industries, cooling of nuclear reactors, food processing industries, cooling of electric appliances, plasma physics, aerodynamics etc., it is inappropriate to neglect these important parameters in natural and forced convection as results obtained with constant viscosity and thermal conductivity as well as neglecting the plate thickness and international heat generation as a result of fluid-particle interactions are either under-determined or over-determined as the case may be. Therefore, it is important to investigate the role of boundary plate thickness, wall conduction, viscous dissipation, variable viscosity and thermal conductivity on mixed convection flow of viscous incompressible heat generating/absorbing fluid.

Obtaining a closed form solution of this model is a very difficult task due to the coupling and nonlinearity nature of the governing equations. Various solution methods have been derived for such problems ranging from numerical solutions (Doh et al. [45], Kairi et al. [46], Kumar et al. [47], Rostami et al. [48], Hosseinzadeh et al. [49], Mukundan and Awasthi [50] etc.), perturbation methods and several other approximate solution techniques (Travis and Erturk [51], Singh et al. [52] etc.), thus, this problem has adoptedthe Homotopy Perturbation Method to carry out its investigations.

2 Mathematical analysis

We consider a two-dimensional steady mixed flow of an incompressible viscous fluid passes through vertical parallel plates of some thickness d^*. One of the plates moves in the direction of the fluid flow while the other is stationary (see figure 1). The plate at y^* = −d^* which moves uniformly with velocity U is heated to temperature T_w while the other at y^* = h + d^* is kept at ambient temperature T₀. In addition, an external pressure is applied to the flow which gave rise to mixed convection and also the presence of viscous dissipation was considered. The flow is assumed to be along x^* direction and y^* is normal to the plates. Internal energy generation is considered as a result of fluid particles interaction and the thermophysical and hydrodynamics properties are assumed to be constant except for viscosity and thermal conductivity that both vary with temperature. The physical model describing the situation, following the works of (Ajibade and Umar [17], Ajibade and Umar [53]) is given as:

Figure 1

Schematic diagram of the problem

The first, second and third terms of the momentum equation (1) are the viscosity, thermal buoyancy, and pressure gradient effects while the first, second and third term of the energy equation (3) of the fluid are the thermal conductivity, heat generation/absorption and viscous dissipation effects respectively. Equations (2) and (4) are the energy equations of the bounding slabs p₁ and p₂ respectively. Since the flow is steady, we assume that the appropriate boundary conditions of the model are:

The viscosity of the fluid and its thermal conductivity are both assumed to vary linearly with temperature so that

where μ₀ and k₀ represent the viscosity and thermal conductivity of fluid temperature T₀, a and b are the thermal factors that control the change in viscosity and thermal conductivity respectively with increase in temperature. In order to harmonize the various dimensions that are involved in the model, the velocity has been scaled up with U, the velocity of the moving plate while the horizontal distances; y^* and d^* were scaled up with h. All other dimensionless parameters are as described in eqn (6):

Using the dimensionless quantities (6), the momentum and energy Eqs.(1) - (4) are presented as:

subject to the boundary conditions:

where

The parameters of interest that appeared in the dimensionless eqn (10) are; heat generation/absorption, (S), viscous dissipation (Eckert number, Ec), Prandtl number (Pr), Grashof number and Reynold number (GrRe), viscosity and conductivity factor parameters (λ) and (ε) respectively. All other physical parameters are defined in nomenclature.

3 Method of solution

Several methods ranging from numerical, perturbations and many other approximate solution techniques are used to solve linear, nonlinear and coupled (partial or ordinary) differential equations. Homotopy Perturbation Method (HPM) is one of the easiest approximate solution since the first two iterations are of accuracy than the traditional perturbation techniques and it doesn’t require only a small parameter. The method was first initiated by He[54] where he presented the method by considering a nonlinear differential equation

subject to the boundary conditions

where A is the general differential operator, B is a boundary operator, f (r) is a known analytic function, Γ is the boundary of the domain Ω. The operator A can be divided into two parts, namely, L and N, where L is the linear and N nonlinear parts. Therefore, the nonlinear differential equation can be transformed as:

Now the convex homotopy can be constructed from thenonlinear differential equation and its boundary condition as:

v(r, p) : Ω × [0,1]→which satisfies

or

where p ∈ [0, 1] is an embedding parameter, u₀ is the initial approximation which satisfies the boundary conditions. Therefore, we have

Now the solution of the nonlinear differential equation is expressed as:

setting p = 1, the approximate solution can be written as

Therefore the convex homotopy of momentum and energy equations are expressed as:

Since the zeroth order equations are linear and can be solved exactly with the boundary conditions, then initial approximation v₀, is not required. Therefore we have

such that

Substituting eqn(15) into eqn(14) and comparing the coefficients of p⁰, p¹, p², p³,..., we have

similarly, eqns (8),(9) and (10) can be transformed as

and the boundary conditions are transformed as

Solving from the above eqns(16), (19), (22) and (25) and applying the boundary conditions (28), we have

Solving eqns(17), (20), (23) and (26) and applying the boundary conditions (28) we have

Solving eqns(18), (21), (24) and (27) and applying the conditions (28), we have

where