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The present work examines the eﬀect of temperaturedependent fluid density on the reactive couple stress fluid flowing steadily in a vertical channel saturated with porous materials. The thermal effect gave rise to a linked differential system of equations. In the dimensionless form, the boundaryvalued problems (BVP) are uncoupled and solved by using Adomian decomposition method (ADM). The solutions are used to obtain expressions for entropy generation and Bejan heat irreversibility number in the porous medium. The relevant fluid parameters are varied and discussed physically. The present result shows good agreement with the previously obtained result when buoyancy force is neglected. The present study is relevantal in the thermal treatment of combustible fluids in chemical and petrochemical engineering.
reactive fluid, couple stress fuid, porous medium, buoyancy eﬀect and adomian decomposition method (ADM)
In recent times, studies involving fluid flow with convective heat transfer through parallel plates have attracted huge consideration because of its importance in many engineering and industrial processes. Several researchers [15] (and references therein) have investigated several aspects of the convective flow problems under different flow conditions. However, all the studies above are limited to the Newtonian case in which the classical NavierStokes equation is valid. In reality, most fluid of mechanical, medicinal and industrial significance contain several additives in the form of tiny microstructure like polymer additives in synthetic fluids, red blood cells in blood, drug particles in doping by athletes, targeted drug delivery in medicine and lots more. As shown in [6], Kaladhar et al., described the forced convective flow of reactive couple stress fluids undergoing destructive chemical reaction. Adesanya et al. [7] reported the inherent irreversibility in the couple stress fluid flow through horizontal channel. Asad et al. [8], highlighted the significance of the flow and heat transfer of couple stress fluid having temperaturedependent thermal conductivity. Nayak et al., [9] reported the hydromagnetic couple stress fluid flow and heat transfer in a barrierfilled rotating channel. Zelalem et al., [10] discussed the fluid flow and heat transfer of couple stress fluid through parallel walls endowed with extended porous materials. Interested readers can see [1116] for more significant results on couple stress fluid flow.
Motivated by the study in [6], the primary concern here is to investigate the impact of buoyancy force on the entropy generation due to convective flow and heat transfer. To the best our understanding, the flow situations reported here has not been conducted in the literature. The study is necessary due to the strong exothermic nature of some reactive fluid. Hence, a more accurate approximation would be achieved if the variation of fluid density with temperature is taken into consideration. The next section describes the flow assumptions in detail with the mathematical analysis. In section three, the Adomian method of solution will be presented. Section four of the work describes the tabular and graphical results while section five gives the summary of the work.
Consider the steady flow of an hydrodynamically and thermally developed combustible couple stress fluid through a vertical channel of distance h apart. The channel walls are maintained isothermally at a constant temperature T_{0}. As shown in igure 1 below, the noslip and the stress free conditions are imposed on the channel walls. The constant pressure gradient is in the direction of the flow and variation in density due to the presence of buoyancy force eﬀect is taken into account in the momentum equation.
Therefore, the linearized governing momentum and energy equations following the Boussinesq approximation can be written as [6]:
$0=\frac{d P}{d x}+\mu \frac{d^{2} u^{\prime}}{d y^{\prime 2}}\frac{\mu u^{\prime}}{K}\eta \frac{d^{4} u^{\prime}}{d y^{4}}+\rho g \alpha\left(TT_{0}\right)$ (1)
$0=\frac{d^{2} T}{d y^{\prime 2}}+\frac{Q C_{0} A}{k} e^{\frac{E}{R T}}+\frac{\mu}{k}\left(\frac{d u^{\prime}}{d y^{\prime}}\right)^{2}$
$+\frac{\mu u^{2}}{k K}+\frac{\eta}{k}\left(\frac{d^{2} u^{\prime}}{d y^{2}}\right)^{2}$ (2)
The last term in (1) is due to the effect of buoyancy force as in [15] together with the suitable boundary conditions at the plate ends are given to be:
$T(0)=T(h)=T_{0}$
$u^{\prime}(0)=\frac{d^{2} u^{\prime}}{d y^{\prime 2}}(0)=0=\frac{d^{2} u^{\prime}}{d y^{\prime 2}}(h)=u^{\prime}(h)$ (3)
Also, the entropy generation rate in the couple stress fluid flow can be represented with this expression as given by [6] as:
$E_{G}=\frac{k}{T_{0}^{2}}\left(\frac{d T}{d y^{\prime}}\right)^{2}+\frac{\mu}{T_{0}}\left(\frac{d u^{\prime}}{d y^{\prime}}\right)^{2}+\frac{\mu u^{\prime 2}}{T_{0} K}+\frac{\eta}{T_{0}}\left(\frac{d^{2} u^{\prime}}{d y^{2}}\right)^{2}$ (4)
With the underlisted nondimensional quantities:
$y=\frac{y^{\prime}}{h}, y=\frac{u^{\prime}}{U M}, \theta=\frac{E\left(TT_{0}\right)}{R T_{0}^{2}}, M=\frac{h^{2}}{\mu U} \frac{d P}{d x}, \gamma=\frac{h}{l}$
$\varepsilon=\frac{R T_{0}}{E}, \lambda=\frac{Q E A C_{0} h^{2}}{R T_{0}^{2} k} e^{\frac{E}{R T_{0}}}, \delta=\frac{U^{2} \mu M^{2}}{Q A C_{0} h^{2}} e^{\frac{E}{R T_{0}}}, l=\sqrt{\frac{\eta}{\mu}}$
$\beta=\sqrt{\frac{1}{D a}}, D a=\frac{K}{h^{2}}, G_{r}=\frac{\rho g \alpha R T_{0}^{2} h^{2}}{M U \mu E}, N_{s}=\frac{h^{2} E^{2} E_{G}}{k R^{2} T_{0}^{2}}$ (5)
we procure the underlisted boundary valued problems regulating the fluid flow from equations (1 – 4) as:
$1+\frac{d^{2} u}{d y^{2}}\beta^{2} u\frac{1}{\gamma} \frac{d^{4} u}{d y^{4}}+G_{r} \theta=0$ (6)
$\frac{d^{2} \theta}{d y^{2}}+\lambda\left(e^{\frac{\theta}{1+\varepsilon \theta}}+\delta\left[\left(\frac{d u}{d y}\right)^{2}+\frac{1}{\gamma}\left(\frac{d^{2} u}{d y^{2}}\right)^{2}+\beta^{2} u^{2}\right]\right)=0$ (7)
together with the boundary conditions:
$\begin{aligned} \theta(0) &=\theta(1)=0 \\ u(0) &=\frac{d^{2} u}{d y^{2}}(0)=0=\frac{d^{2} u}{d y^{2}}(1)=u(1) \end{aligned}$ (8)
and the entropy generation rate in dimensionless form is:
$N_{s}=\left(\frac{d \theta}{d y}\right)^{2}+\frac{\delta \lambda}{\varepsilon}\left(\left(\frac{d u}{d y}\right)^{2}+\frac{1}{\gamma}\left(\frac{d^{2} u}{d y^{2}}\right)^{2}+\beta^{2} u^{2}\right)$ (9)
Furthermore, let N_{1} be $\left(\frac{d \theta}{d y}\right)^{2}$ which represents the irreversibility due to heat transfer.
Also, let N_{2b }he $\frac{\delta \lambda}{\varepsilon}\left(\left(\frac{d u}{d y}\right)^{2}+\frac{1}{\gamma}\left(\frac{d^{2} u}{d y^{2}}\right)^{2}+\beta^{2} u^{2}\right)$which stands for the entropy generation due to the combined effects of the viscous dissipation and porous permeability. Then, Bejan number (Be) is defined as the heat irreversibility ratio as follows:
$B e=\frac{N_{1}}{N_{s}}=\frac{1}{1+\Phi}$ where $\Phi=\frac{N_{2}}{N_{1}}$ (10)
The coupled nonlinear equations for momentum in (6) and energy in (7) with the boundary conditions in (8) are solved by making use of ADM [1719]. First, the BVP are converted to the integral forms as follows:
$u(y)=a_{0} y^{3}+b_{0} y+\frac{\gamma y^{4}}{24}$$+\gamma \int_{0}^{y} \int_{0}^{y} \int_{0}^{y} \int_{0}^{y}$$\left[\frac{d^{2} u}{d Y^{2}}\beta^{2} u+G_{r} \theta\right] d Y d Y d Y d Y$ (11)
$\theta(y)=c_{0} y\lambda \int_{0}^{y} \int_{0}^{y}\left(e^{\frac{\theta}{1+s \theta}}+\delta\left[\left(\frac{d u}{d Y}\right)^{2}+\frac{1}{\gamma}\left(\frac{d^{2} u}{d Y^{2}}\right)^{2}+\beta^{2} u^{2}\right]\right) d Y d Y$ (12)
where a_{0},, b_{0} and c_{0} are constants that will be determined by employing the appropriate requirements stated in (8). By ADM, we take an infine series solutions of the form:
$u(y)=\sum_{n=0}^{\infty} u_{n}(Y)$ and $\theta(y)=\sum_{n=0}^{\infty} \theta_{n}(Y)$ (13)
substituting (13) into the coupled equations (11) and (12) to have
$\sum_{n=0}^{\infty} u_{n}(Y)=a_{0} y^{3}+b_{0} y+\frac{\gamma y^{4}}{24}+$$\gamma \int_{0}^{y} \int_{0}^{y} \int_{0}^{y} \int_{0}^{y}\left[\frac{d^{2}}{d Y^{2}}\left(\sum_{n=0}^{\infty} u_{n}(Y)\right)\beta^{2}\left(\sum_{n=0}^{\infty} u_{n} 0\right)\right] d y Y d Y d Y d Y$$+\gamma \int_{0}^{y} \int_{0}^{y} \int_{0}^{y} \int_{0}^{y}$$\left[G_{r}\left(\sum_{n=0}^{\infty} \theta_{n}(Y)\right)\right] d y Y d Y d Y d Y$ (14)
and
$\sum_{n=0}^{\infty} \theta_{n}(y)=c_{0} y\lambda \int_{0}^{y} \int_{0}^{y}$$\left(\begin{array}{c}{\sum_{n=0}^{\infty} \theta_{n}(y)} \\ {e^{1+\varepsilon \sum_{n=0}^{\infty} \theta_{n}(y)}}\end{array}\right.$$+\delta\left[\left(\frac{d}{d Y}\left(\sum_{n=0}^{\infty} u_{n}(Y)\right)\right)^{2}\right] d Y d Y$$\lambda \int_{0}^{y} \int_{0}^{y}$$\left(\delta\left[\frac{1}{\gamma}\left(\frac{d^{2}}{d Y^{2}}\left(\sum_{n=0}^{\infty} u_{n}(Y)\right)\right)^{2}\right.\right.$$+\beta^{2}\left(\sum_{n=0}^{\infty} u_{n}(Y)\right)^{2} ] d Y d Y$ (15)
let the nonlinear terms in (15) be represented by:
$\sum_{n=0}^{\infty} A_{n}(y)=e^{\frac{\sum_{n=0}^{\infty} \theta_{n}(y)}{1+\varepsilon \sum_{n=0}^{\infty} \theta_{n}(y)}}$
$\sum_{n=0}^{\infty} B_{n}(y)=\left[\left(\frac{d}{d Y}\left(\sum_{n=0}^{\infty} u_{n}(y)\right)\right)^{2} \frac{1}{\gamma}\left(\frac{d^{2}}{d Y^{2}}\left(\sum_{n=0}^{\infty} u_{n}(y)\right)\right)^{2}\right]$$+\left[\beta^{2}\left(\left(\sum_{n=0}^{\infty} u_{n}(y)\right)\right)^{2}\right]$ (16)
The Adomial polynomials can be obtained by the Taylor’s expansion of the series in (16). Then the zeroth order approximation becomes:
$u_{0}(y)=a_{0} y^{3}+b_{0} y+\frac{\gamma y^{4}}{24}$
$\theta_{n}(y)=0$ (17)
While other terms of the series can be obtained by using the relations:
$\begin{aligned} u_{n+1}(y) &=\gamma \int_{0}^{y} \int_{0}^{y} \int_{0}^{y}\left[\frac{d^{2} u_{n}}{d Y^{2}}\beta^{2} u_{n}+G_{r} \theta_{n}\right] d Y d Y d Y d Y \\ \theta_{n+1}(y) &=c_{0} y\lambda \int_{0}^{y} \int_{0}^{y}\left(A_{n}+\delta B_{n}\right) d Y d Y \end{aligned}$ (18)
such that the approximate solution of the coupled BVP can be obtained in the form of the partial sums:
$u(y)=\sum_{n=0}^{k} u_{n}(Y)$ and $\theta(y)=\sum_{n=0}^{k} \theta_{n}(Y)$ (19)
The partial sums in (19) are then utilized in (9)(10) to compute the entropy generation and the Bejan profiles provided in the following section. For easy iteration of the tedious calculations, equations (9)(10), (16)(19) are programmed in the Mathematica software package.
In this section, the velocity, temperature, entropy generation and heat irreversibility ratio profiles are presented. In Table 02, the rapid convergence of the Adomian series solution to the couple problem is presented. It is interesting to note that, convergence of the series is obtained with just few iterations. Also, the validation of the present result in the absence of Grashof number is presented in Table 02. From the table,a perfect agreement is observed bewteen the exact result presented in [6] and the the Adomian series approach.
Table 1. Rapid convergence of the series solution
ε = δ = 0.1, λ = β = 0.2, γ = G_{r} = 0.5 

n 
a_{0} 
b_{0} 
c_{0} 
0 
 0.250000 
0.0208333 
0 
1 
0.240422 
0.0198889 
0.100037 
2 
0.242108 
0.0200432 
0.101764 
3 
0.242143 
0.0200482 
0.101755 
4 
0.242143 
0.0200482 
0.101755 
Table 2. Validation of numerical results of the velocity profile
β = 0.1, γ = 1, G_{r} = 0 

u(y) 
Exact solutions from [6] 
ADM 
Absolute Error 
0.0 
0. 
0 
0 
0.1 
0.00371505 
0.00371574 
6.86111× 10^{6} 
0.2 
0.00702528 
0.00702658 
1.30462× 10^{6} 
0.3 
0.00961357 
0.00961536 
1.78656× 10^{6} 
0.4 
0.01125560 
0.01125770 
2.12648× 10^{6} 
0.5 
0.01181780 
0.01182000 
2.18983× 10^{6} 
0.6 
0.01125560 
0.01125770 
2.12648× 10^{6} 
0.7 
0.00961357 
0.00961536 
1.78656× 10^{6} 
0.8 
0.00702528 
0.00702658 
1.30462× 10^{6} 
0.9 
0.00371505 
0.00371574 
6.86111× 10^{6} 
1.0 
0 
2.45939 × 10^{18} 
2.45939 × 10^{18} 
Figure 2 represents the velocity profile with variations in the Grashof number. From the result, an increase in the Grashof number is seen to enhance the fluid temperature since an increase in fluid temperature increases the thermal expansion of the fluid, and the upthrust force will increase significantly in the vertical channel. Figure 3 shows the influence of couple stress inverse parameter on the fluid flow. As observed from the graphical result, an increase in the couple stress inverse parameter shows a corresponding increase in the flow velocity as a consequence of the shear thinning of the fluid as the couple stress inverse parameter increases. However, as the couple stress inverse parameter decreases, the shear thickening property of the fluid becomes prominent in the flow channel. As revealed in Figure 4, an increase in Grashof number is seen to enhance the temperature profile. The result is correct since Grashof number increases the flow velocity. Hence a rise in the fluid temperature is expected due to volumetric expansion. The result in Figure 5 revealed the effect of couple stress inverse parameter on the fluid temperature distribution within the flow channel. As observed, as the couple stress inverse parameter increases, the fluid becomes thinner, as a result, interparticle collisions becomes higher, and the fluid temperature rises. Furthermore, since both Grashof number and couple stress inverse parameter enhances both flow and temperature profiles, it is expected to enhance entropy generation within the channel. In the real sense, the couple stress inverse parameter will ultimately minimize entropy generation as it decreases as reported in Figures 67. Finally, in Figures 89, it is observed that as both Grashof number and couple stress inverse parameter increases, heat irreversibility due to fluid friction dominates over heat transfer irreversibility in the flow channel.
Figure 2. Effect of G_{r} on u(y)
Figure 3. Effect of γ on u(y)
Figure 4. Effect of G_{r} on ϴ(y)
Figure 5. Effect of γ on ϴ(y)
Figure 6. Effect of G_{r} on N_{s}
Figure 7. Effect of γ on N_{s}
Figure 8. Effect of G_{r} on Be
Figure 9. Effect of γ on Be
The effect of temperaturedepend density on the steady flow of a highly combustible couple stress fluid through a channel, due to convective flow and heat transfer, was presented. The nonlinear equations governing the fluid flow were formulated, nondimensionalized and solved by using Adomian decomposition method. The result of the computations highlights the significance of couple stresses in reducing the fluid temperature when undergoing exothermic chemical reactions and minimizing entropy generation. This is important in improving yields and enhancing the safety of the environment.
P 
Modified fluid pressure 
K 
Darcy's permeability constant. 
g 
Acceleration due to gravity 
T_{0} 
The wall temperature 
k 
Thermal conductivity of the fluid 
Q 
Heat of the reaction term 
C_{0} 
Reactant species initial concentration, 
h 
The channel width 
(x,y) 
Distance measured in the axial and normal direction 
A 
Reaction rate constant 
E 
Activation energy 
U 
Fluid velocity 
R 
Universal gas constant 
U 
Fluid characteristic velocity 
l 
Fluid molecular dimension 
M 
Pressure gradient 
Da 
Darcy number 
G_{r} 
Grashof number 
E_{G} 
Dimensional entropy generation rate 
N_{s} 
Dimensionless entropy generation rate 
Greek symbols 

μ 
Dynamic viscosity coefficient 
$\eta$ 
Couple stress coefficient 
$\rho$ 
Fluid density 
$\varepsilon$ 
Activation energy parameter 
$\lambda$ 
Frank—Kamenettski parameter 
$\gamma$ 
Couple stress inverse parameter 
$\delta$ 
Viscous heating parameter 
$\beta$ 
Porous medium permeability parameter 
$\alpha$ 
Thermal expansion coefficient 
$\theta$ 
Fluid temperature 
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