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

Abstract This study theoretically investigates the effects of variable viscosity, thermal conductivity and wall conduction on a steady mixed convection flow of heat generating/absorbing fluid passes through a vertical channel. One of the channel plates moves with a constant velocity while the other is stationary. The governing flow equations are solved analytically using homotopy perturbation method (HPM). The effects of the thermophysical and hydrodynamics parameters are captured in graphs and tables. It has been observed that, both the velocity and temperature distributions decrease with increase in viscosity and boundary plate thickness near the heated plate while a reverse cases were observed near the cold plate. Increase in thermal conductivity ε decreases the fluid flow near the heated plate. When the boundary plate thickness is increased, the critical value of Gre to onset the reverse flow increases while increase in thermal conductivity reduces the critical value of Gre. It’s also noticed that the skin friction and rate of heat transfer at the heated plate decrease with increase in boundary plate thickness d.


Introduction
Heat transfer and uid ow have been matters of research for many years due to their applications in modern industries. Fluid ows are captured in many mathematical geometries. Such ows 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 uid ows are driven either by natural, forced or mixed convection.
Ajay and Fe rey [1] studied combined forced and natural convection heat transfer where they showed that the velocity pro le reduces near the channel plate (y = ) with increasing mixed convection while the reverse case was observed at the other plate (y = ). Jha et al. [2] examined mixed convection ow in a vertical tube lled 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 ow is more prominent than the transient ow. He further discovered that the ow and thermal distributions are enhanced as the strength of the mixed convection increases. Jha et al. [4] studied mixed convection ow in an inclined channel lled with porous material and concluded that the ow is reversal type, since the velocity pro le 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 ow of a non-Newtonian power-law nano uid was investigated by Machal and Naikoti [5]. They concluded that, the velocity pro le increases with increase in mixed convection while the reverse case was observed on the uid temperature. Recently, Jha and Aina [6] studied e ect of heat generation/absorption on MHD mixed convection ow 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 ow in a vertical porous channel was investigated by Ajibade and Onoja [7]. They revealed that the velocity pro le increases with increase in mixed convection while a reverse case was observed for temperature distribution. Jha and Aina [8] analysed the e ect of induced magnetic eld on MHD mixed convection ow in vertical microchannel. In lubrication industries, viscous dissipation e ects cannot be overlooked since internal energy is generated as a result of uid particles' interaction which a ects the uid ow and thermal behaviour in a medium. Pantokratoras [9] studied e ects of viscous dissipation in natural convection ow along a heated vertical plate. They concluded that the viscous dissipation has a strong e ect on velocity pro le as it helps the upward and opposes the downward ow. Jha and Ajibade [10] studied e ect of viscous dissipation on natural convection ow between vertical parallel plates. They showed that the viscous dissipation heating within the channel increases the uid temperature above the plate temperature when the Prandtl number of the working uid is relatively small. Kumar and Sanchayan [11] investigated e ects of viscous dissipation on the limiting value of Nusselt numbers for a shear driven ow between two asymmetrically heated plates. They concluded that the temperature pro le and the driven temperature di erence increase with an increase in viscous dissipation. In a related work, Fransisca et al. [12] examined heat transfer with viscous dissipation in Couette-Poiseuille ow and found that viscous dissipation has signi cant in uence on the uid temperature. Recently, Kabir et al. [13] investigated e ects of viscous dissipation on MHD natural convection ow with heat generation and found that the velocity and temperature pro les increase with increasing viscous dissipation. Mohamed [14] examined mixed convection ow on a micropolar uid over an unsteady stretching surface in the presence of viscous dissipation. He concluded that the velocity and the temperature pro les increase with increase in Eckert number. Prabhakar [15] examined e ects of thermal di usion and viscous dissipation on an unsteady Magnetohydrodynamics free convection ow past a vertical porous plate in the presence of heat sink. Their work concluded that the velocity and temperature pro les increase with increase in viscous dissipation. RaihanulHaque et al. [16] examined e ects of viscous dissipation on natural convection ow over a sphere with temperature dependent thermal conductivity in the presence of heat generation. They found that the velocity and temperature pro les 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 insigni cant role in the hydrodynamics and thermal behaviour within the channel. The boundary plate thickness of a material has signi cant e ects on the heat ux and uid ow 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 uid decreases with growing thickness in the boundary plates. In the study of e ects 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 crosssectional 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 trans-fer. 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 e ects of wall conduction on a mixed convection heat transfer in an externally nned pipes were studied by Moukalled et al. [22]. They observed that the axial wall conduction in the pipe wall is found to have strong e ect on the ow and thermal elds. Other works that investigated the e ect of boundary plates' thickness include the work of Kevin and Barbaros [23]. Convective ow in nature and engineering phenomena requires that viscosity and thermal conductivity of uids vary with temperature. For instance, the viscosity of dry air at C is . × − kg/ms while at C, it is .
× − kg/ms . The thermal conductivity of a dry air at C is . × − W /mK while at C, it is .
× − W /mK ( Adrian et al. [24]). Tarakaramu et al. [25] investigated the e ects of variable thermal conductivity on 3D ow of nano uid over a stretching sheet using numerical simulation while Dulal and Hiranmony [26] analysed the e ects of temperature-dependent viscosity and variable thermal conductivity on mixed convective di usion ow. They found that velocity pro le increases while temperature decreases with increase in mixed convection. Subhas and Mehesha [27] studied heat transfer on a magnetohydrodynamics viscoelastic uid ow over a stretching sheet with variable thermal conductivity and radiation and found that the variable thermal conductivity has an in uence in enhancing the temperature and uid ow. The works of Rahman et al. [28] and Naseem [29] have shown that the velocity and temperature pro les 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 pro les increase with increase in variable viscosity. MHD e ects 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 ow past an in nite vertical plate with variable thermal conductivity, heat source and chemical reaction. They concluded that, velocity and temperature pro les increase with increase in thermal conductivity. Sumayya et al. [35] investigated e ciency analysis of a straight n with variable heat transfer and thermal conductivity. They con-cluded that, temperature pro le increases with increase in thermal conductivity. Devi and Prakash [36] examined temperature-dependent viscosity and thermal conductivity e ects on hydromagnetic ow over a slendering stretching sheet. They concluded that, increase in the viscosity decreases the velocity pro le. The work of Animasaun [37] concluded that uid 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 pro les decrease with increasing viscosity while the temperature pro les increase with increase in viscosity. Recently, Sreenivasulu et al. [43] studied variable thermal conductivity in uence on hydromagnetic ow past a stretching cylinder in a thermally stratied medium with heat source/sink. They concluded that the velocity pro le increases with increasing mixed convection and the temperature pro le increases with increase in thermal conductivity. Muthtamilselvan et al. [44] investigated an internal heat generation of a dusty uid 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 e ects of viscous dissipation, wall conduction, boundary plate thickness, variable viscosity and thermal conductivity on a steady mixed convection ow. 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 uidparticle interactions are either under-determined or overdetermined 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 ow of viscous incompressible heat generating/absorbing uid. Obtaining a closed form solution of this model is a very difcult 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 adopted the Homotopy Perturbation Method to carry out its investigations.

Mathematical analysis
We consider a two-dimensional steady mixed ow of an incompressible viscous uid passes through vertical parallel plates of some thickness d * . One of the plates moves in the direction of the uid ow while the other is stationary (see gure 1). The plate at y * = −d * which moves uniformly with velocity U is heated to temperature Tw while the other at y * = h + d * is kept at ambient temperature T . In addition, an external pressure is applied to the ow which gave rise to mixed convection and also the presence of viscous dissipation was considered. The ow is assumed to be along x * direction and y * is normal to the plates. Internal energy generation is considered as a result of uid 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: The rst, second and third terms of the momentum equation (1) are the viscosity, thermal buoyancy, and pressure gradient e ects while the rst, second and third term of the energy equation (3) of the uid are the thermal conductivity, heat generation/absorption and viscous dissipation e ects respectively. Equations (2) and (4) are the energy equations of the bounding slabs p and p respectively. Since the ow is steady, we assume that the appropriate boundary conditions of the model are: The viscosity of the uid and its thermal conductivity are both assumed to vary linearly with temperature so that where µ and k represent the viscosity and thermal conductivity of uid temperature T , 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: 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 ( Gr Re ), viscosity and conductivity factor parameters (λ) and (ε) respectively. All other physical parameters are de ned in nomenclature.

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) di erential equations. Homotopy Perturbation Method (HPM) is one of the easiest approximate solution since the rst two iterations are of accuracy than the traditional perturbation techniques and it doesn't require only a small parameter. The method was rst initiated by He [54] where he presented the method by considering a nonlinear di erential equation where A is the general di erential 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 di erential equation can be transformed as: Now the convex homotopy can be constructed from the nonlinear di erential equation and its boundary condition as: v(r, p) : Ω × [ , ] → which satis es where p ∈ [ , ] is an embedding parameter, u is the initial approximation which satis es the boundary conditions. Therefore, we have Now the solution of the nonlinear di erential equation is expressed as: setting p = , 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 v , is not required. Therefore we have (14) such that u = u + pu + p u + p u + ..., Substituting eqn(15) into eqn (14) and comparing the coefcients of p , p , p , p ,..., we have . . .
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 A = , Now, the approximate solution of the momentum and energy eqns (7) and (9) of the uid as p → are: The interesting phenomena to be investigated in this study are onset of reverse ow, pressure gradient, skin friction, rate of heat transfer between the uid and boundary plates and the pressure gradient. In order to obtain the pressure gradient required to drive the ow formation for a constant mass ux q, the integral is evaluated for di erent ow parameters (see table 1). The pressure gradient can be expressed as: where and The critical values of mixed convection parameter Gre which signalled the onset of reverse ow at y = , can be obtained from the turning point of u, i.e du dy | y= = . Therefore, the in uence of di erent parameters as they control the critical Gre are presented in table 2. The skin friction at the plate surfaces can be de ned as: where µ * is the dimensional variable viscosity. Thus, the local skin friction factor can be expressed as: while the local surface heat ux is expressed as: where k * is the dimensional variable thermal conductivity, together with the expression of Nusselt number, Therefore, the Nusselt number can be expressed as:

Results and discussion
Viscous dissipation and boundary thickness e ects on a steady mixed convection ow in the presence of variable viscosity and thermal conductivity have been theoretically analysed. The ow is governed by six basic parameters, which are: Gre, thermal buoyancy, S heat generating/absorbing, λ, viscosity variation parameter and ε, thermal conductivity variation parameter, Br Brinkman number, q mass ux and d the boundary thickness. The value of Pr is chosen to be . throughout the work for air as the working uid. The solutions for momentum and energy equations in the uid section have been simulated for carefully selected values of the controlling parameters and the results are presented in gures 2-15. Also, the pressure gradient required to drive the ow to a prescribed mass ux is presented in table 1 while the critical values of the mixed convection parameter after which ow reversal is achieved near the stationary plate is given in table 2. The skin friction and the rate of heat transfer on the uid-plate interface were tabulated in tables 3 and 4. Figures 2 and 3 show the e ects of Brinkman number Br -. -.
-. Table 2: Critical values of Gre at the stationary plate y = for di erent values of S, λ, d and ε where q = and Br = .
. . . Table 3: skin friction τ at the heated plate y = and cold plate y = , for di erent values of S, d and λ where Br = . , ε = − . , Gre = and q = . . Table 4: Rate of heat transfer at the heated plate y = and cold plate y = for di erent values of S, d and ε where Br = . , Gre = , q = and λ = − .  on velocity and temperature distributions. It is seen that the velocity increases at the heated plate with increase in Br. This is due to the fact that, Brinkman number is the ratio of heat produced by viscous dissipation to heat transported by molecular conduction. However, a decrease is observed near the cold plate with growing Br. This clearly reveals the e ect of the pressure gradient and the impulsive motion by the heated plate to increase the velocity near the plate y = . On the other hand, gure 3 reveals that the temperature distribution decreases with increase in Brinkman number near the heated plate while an increase is observed near the cold plate as a result of energy di usion towards the cold plate which increases the temperature of the uid. It is also observed that the inter-facial temperature decreases at the heated plate with an increase in Br while an increase is observed near the cold plate. The e ect of Br is more pronounced at the cold, thus increases the temperature at the region. Figures 4 and 5 show the e ects of heat generating/absorption parameter S on velocity and temperature distributions. The velocity increases with increase in heat generation S < near the heated plate while it decreases near the cold plate. However, near the heated plate, the velocity distribution decreases with increase in heat absorption S > due to the fact that the heat generated is absorbed by the uid particles which makes the uid more dense, thus reduces the temperature and slow down the uid ow. It is also observed that the temperature distribution increases across the channel with increase in heat generation S < . This is physically true since heat generated strengthens the convection current of the uid thereby reduces the density of the uid, thus increases the uid temperature. This is attributed to the fact that, viscous dissipation, wall conduction and thermal conductivity add more heat to the uid and the heat di uses by pressure gradient. It is further noticed that the temperature distribution decreases with increase in heat absorption S > . This physically true, when heat is absorbed, then the uid becomes dense and the temperature reduces. The e ect of S on the inter-facial temperature is more pronounced at the heated plate than the cold plate. Figures 6 and 7 show the e ects of mixed convection Gre on the velocity and temperature distributions respectively. The velocity pro le increases with increase in Gre near the heated plate. This is attributed to the role of heat generated from the wall and the internal heat generated which makes the uid less dense and moves easily by the buoyancy forces. Near the cold plate, it has been observed that the velocity distribution decreases with increase in Gre as the uid is more dense near the cold plate. On gure 7, the temperature pro le increases with increase in Gre. This is physically true, since the heat generated by wall conduction and viscous dissipation di uses by the buoyancy forces which strengthens convection current, which in return increases uid velocity and consequently the sheared heating at every uid sections. The e ect of Gre on the inter-facial temperature is insigni cant at the plates. The e ects of variable viscosity λ on velocity and temperature distributions are captured on gures 8 and 9. It is observed from gure 8 that the velocity distribution decreases near the heated plate with increase in viscosity while an increase has been observed near the cold plate. Decrease in velocity near the heated plate is due to the fact that there is an increase in convection current which in return increases the viscosity of the uid, thereby increasing the viscous drag that decreases the velocity. However, the increase of velocity near the cold plate is attributed to the decrease in heat which reduces the uid viscosity, and eventually increases the uid velocity. The temperature distribution decreases near the heated plate with increase in viscosity λ while it increases near the cold plate. The change in viscosity a ects the inter-facial temperature on both the plates. It can be seen that the temperature at the heated plate decreases with increase in viscosity while a reverse case was observed at the cold plate. The e ects of thermal conductivity ε on velocity and temperature are depicted on gures 10 and 11. It is observed that the velocity distribution decreases with increase in thermal conductivity near the heated plate while it increases near the cold plate. On the other hand, the temperature distribution decreases with increase in thermal conductivity. This is attributed to increase in thermal conductivity which enhances heat transport within the system so that heat from the internal heating mechanism di uses towards the cold plate at a rate proportional to the thermal conductivity. Consequently, convection current is weakened near the heated plate and strengthened near the cold plate. The e ect of the thermal conductivity on the inter-facial temperature is little bit pronounced at the cold plate than the heated plate as a result of the energy generated which di uses towards the plate thus increases the temperature. Figures 12 and 13 show the di erent boundary layer thickness d e ects on velocity and temperature distributions. It can be seen from gure 12 that the velocity distribution decreases with increase in boundary plate thickness d near the heated plate. This is physically true, since when the thickness of a material is increased, the heat ux reduces and cools down the uid. Towards the cold plate, the velocity distribution increases. This is attributed to the fact that the energy dissipated di uses towards the cold plate which strengthens the convection current, hence increases the uid ow. The temperature distribution decreases near the heated plate with increase in boundary plate thickness. This is due to the fact that, increase in the thickness of a material reduces the heat ux through the system, which causes decrease in the temperature of the uid. It is also observed that the boundary plate thickness has more e ect on the inter-facial temperature at the heated plate than the cold plate. Figures 14 and 15 display the velocity and temperature distributions for di erent values of mass ux q. The velocity distribution increases with increasing q. This is due to the fact that, increase in mass ux is attributed to increase in pressure gradient to boost the uid particles' ow. The temperature distribution decreases near the heated plate with increase in q while a reverse case is observed near the cold plate. Since the mass ux increases with growing pressure gradient, the e ect of natural convection decreases so as to maintain the required ux within the channel. However, the e ect of channel boundary motion is not there so that an increase is required in uid temperature to drive the ow to the prescribed ow. Table 1 shows the e ect of dimensionless ow parameters on pressure gradient in the channel. It is observed that the pressure gradient increases with increase in mass ux q. That is, an increase in mass ux requires a corresponding increase in pressure gradient to drive the ow. This is obtained when Gre is relatively small. However, when Gre is large, the natural convection dominated the heat transfer, pressure gradient acts against the ow so as to obtain the required mass ux, thus the positive values of pressure gradient at large Gre. The table also shows that, a decrease in pressure gradient is obtained by increasing heat absorption S < . This is due to the fact that, increase in heat generation leads to increase in convection current by buoyancy forces. It is further observed that the pressure gradient acts against the ow when the boundary thickness is increased for relatively small mass ux. However, when the mass is high, the required pressure gradient decreases for a corresponding increase in boundary thickness. Table 2 shows the critical values of Gre which signal the onset of reverse ow at the cold plate y = . It can be seen that smaller critical values can be obtained by en- hancing the thermal conductivity ε or the heat generation parameter S < . This is physically true, since increase in thermal conductivity or heat generation enhances the convection current which in return strengthens the boundary layer thickness of the uid, hence reduces the density of the uid thereby needs smaller value of Gre to nullify the boundary friction on the plate. However, when boundary plate thickness or heat absorption is increased, the convection current within the channel weakens which raises the required critical values of Gre to nullify the boundary friction. The critical value of Gre increases with increase in viscosity. A general view of this table indicates that, each physical quantity that boosts convection current requires a decrease in Gre while the reverse trend is observed for each physical quantity that weakens the convection current. Table 3 displays the skin friction τ between the uid and the plates. It is observed that the skin friction increases at the heated plate with increase in viscosity λ while it decreases with an increase in boundary plate thickness. However, the skin friction decreases on the cold plate with increase in viscosity λ while a reverse trend is observed when the boundary plate thickness d is increased. On the other hand, when heat generation S < is increased, then an increase in skin friction is observed on both the plates. It is further observe that the skin friction decreases with increase in heat absorption S > on both boundary surfaces.
The rate of heat transfer Nu is depicted on table 4. It shows that the rate of heat transfer at both the plates increases with increase in thermal conductivity ε. This physically true, since when the thermal conductivity of a uid is increased, thermal di usivity becomes high thereby increasing the thermal boundary layer thickness of the uid, thus increasing the rate of heat transfer. However, the rate of heat transfer decreases with increase in boundary plate thickness d on both the plates. This is due to the fact that the increase in plate thickness reduces the heat ux through the plate to the uid which in return weakens the convection current of the uid. It is also observed that the rate of heat transfer decreases on the heated plate with increase in heat generation S < while a reverse trend is observed on the cold plate. It is further observed that the rate of heat transfer increases on the heated plate with increase in heat absorption S > but a decrease is observed on the cold plate. This is physically true since uid temperature decreases with heat absorption thereby reducing the temperature di erence between the uid and the cold plate while it increases same on the uid interface with the heated plate.

Conclusion
The present work theoretically investigated the e ects of viscous dissipation and boundary plate thickness on a steady mixed convection ow in the presence of variable thermal conductivity and viscosity. The work concluded that the velocity pro le increases near the heated plate with increase in Br, Gre, and S < while the reverse trends were observed near the cold plate. A decrease in velocity distribution is observed near the heated plate with increase in λ, d, ε and S > while reverse cases were observed near the cold plate. It is also concluded that the temperature distribution decreases near the heated plate with increase in Br, λ, q and d while a reverse cases were observed near the cold plate. It's further concluded that the skin friction and rate of heat transfer decrease with increase in boundary plate thickness d. The required critical value of Gre can be obtained by increasing the thermal conductivity while the pressure gradient increases with increase in mass ux. When Brinkman number (Br), viscosity (λ), thermal conductivity (ε), pressure gradient ( dP dx ) and boundary plate thickness (d) are set to zero, the work of Jha and Ajibade [55] is recovered.