Research Article  Open Access
Samuel O. Adesanya, J. A. Falade, J. C. Ukaegbu, K. S. Adekeye, "Mathematical Analysis of a Reactive Viscous Flow through a Channel Filled with a Porous Medium", Journal of Mathematics, vol. 2016, Article ID 1350578, 8 pages, 2016. https://doi.org/10.1155/2016/1350578
Mathematical Analysis of a Reactive Viscous Flow through a Channel Filled with a Porous Medium
Abstract
An investigation has been carried out to study entropy generation in a viscous, incompressible, and reactive fluid flowing steadily through a channel with porous materials. Approximate solutions for both velocity and temperature fields are obtained by using a rapidly convergent Adomian decomposition method (ADM). These solutions are then used to determine the heat irreversibility and Bejan number of the problem. Variations of other important fluid parameters are conducted, presented graphically, and discussed.
1. Introduction
Studies on heat irreversibility in moving fluid find its relevance in several geological, petrochemical, and industrial applications. In most flows at extremely high temperature, heat irreversibility is unavoidable. This usually leads to material waste due to reduced efficiency of the thermofluid machine. To conserve energy, Bejan [1] introduced an approach that is based on the thermodynamics second law to predict the performance of thermal systems so as to maximize scarce available energy for work and minimize wastages. Following his analysis, AlZaharnah and Yilbas [2] considered the irreversibility analysis in a viscous pipe flow. Haddad et al. [3] examined the heat irreversibility in forced convective flow in concentric cylindrical annulus under diverse flow conditions. Kahraman and Yürüsoy [4] applied the same approach to study the heat irreversibility in nonNewtonian fluid flow through pipes. Aksoy [5] considered the influence of couple stresses on the development of heat irreversibility in a channel with adiabatic surface and constant heat flux. Ting et al. [6] considered the irreversibility associated with nanofluids in a microchannel with porous materials using wateralumina. Moreover, Khan and Gorla [7] addressed the convective problem in nonNewtonian fluid flow through a channel with porous medium and heat flux. Revellin et al. [8] addressed the thermal performance of adiabatic twophase flow using two different methods. Hedayati et al. [9] utilized the thermodynamics analysis to optimize flow on a nonstationary wedge. Butt and Ali [10] reported the irreversibility analysis of fluid slippage with convective boundaries. Other works that focused on the minimization of energy losses in a fluid flow can be found in references [11–19] and many more too numerous to be listed.
From applications’ point of view, studies on transport reactive fluids in porous media are very important since they occur in many important areas like water treatment using fixed beds, agriculture, oil recovery, ground water flows, geothermal engineering, exhaust systems in combustion, material processing, and reservoir engineering. Recently, Rundora and his associates [20–22] documented several investigations on unsteady reactive fluid flow in porous medium and how the flow evolved to the steady state. Bég et al. [23] examined the flow of viscoelastic fluid through a nonDarcian porous medium. Makinde [24] studied the inherent heat irreversibility in reactive fluid through a channel filled with porous material.
In all the studies above, the entropy productions in the flow of viscous incompressible fluid flow through porous medium have not been investigated. Therefore, the work done in [24] can be further extended to give more interesting results on the thermodynamics and heat transfer properties of the fluid flow. This is because huge amount of money and effort could be wasted if the inherent irreversibility in the fluid flow is not well addressed. Therefore, the specific objective of this article is to examine the rate at which entropy is produced in a viscous fluid flow system through a porous medium. The problem under consideration is nonlinear due to the exponential nature of the rate law in Arrhenius kinetics for combustible fluids. In view of this, exact solution for the temperature field may not be possible to get. To solve the problem, we seek Adomian series solution to avoid linearization of the exponential term. The Adomian decomposition method is a straightforward way of solving all kinds of differential equations arising from many physical scenarios. It has been used extensively in the last few decades as reported in the bibliography by Rach [25], and, more recently, the method has been used in [26–31]. The plan of the article is as follows: the problem is formulated and the mathematical analysis is presented in Section 2. Section 3 of the work gives the Adomian method of solution. Graphical results are presented and interpreted in Section 4 while, in Section 5, concluding remarks are given.
2. Mathematical Analysis
The steady flow of viscous incompressible reactive fluid through parallelplate immersed in a porous medium is studied. The flow is assumed to be fulldeveloped and driven by an applied pressure gradient. The channel wall temperatures are kept constant. Then, the balanced governing equations are [24]with the following boundary conditions:Under these assumptions, entropy generation equation becomesTo nondimensionalize (1)–(3), we need the following parameters and variables: to get the dimensionless problems:Settingthen, the irreversibility ratio becomes From (9), it is evident that
3. Adomian Method of Solution
A direct integration of (5)(6) leads to the integral equationswithDue to the exponential nonlinearity in (12), we now define a series of functions defined bySubstituting (13) into the integral equations (11)(12), we obtainThe nonlinear term in (15) represented byis expanded by Taylor’s series to get the following Adomian polynomials:The zerothorder components of the series solutions (14) and (15) areSince the integral of a continuous function is continuous, then each term of the series can be uniquely determined bywhere and are the parameters to be determined.
Then, (17)–(19) are evaluated using MATHEMATICA and the solutions are obtained as finite series:The series solutions are shown to be convergent and twice differentiable (see Tables 1 and 2). Next, we establish the uniqueness solution of (20). It is well known that the Lipschitz condition is sufficient for the uniqueness of solution. Therefore, we first seek for a Lipschitz constant such thatis satisfied. To do this, the boundaryvalued problems (6) are converted to system firstorder differential equations by introducing the following transformations:With (22), (6) can now be written aswhere , the guess values that will ensure the boundary conditions, are satisfied. Then,since , , exist and are continuous in the domain . Hence, the Lipschitz constant with the propertyexists.


Uniqueness Analysis. The Adomian series solutions (20) of the nonlinear problem (6) converges if and , where .
Proof. Let be a Banach space for all continuous functions on with the normLet and be any two solutions of the integral equation (12); then,This implies thatIn view of (21), we then haveLet ; then,orHence, the problem will have a unique solution whenever and as suchKindly see Tables 1 and 2 for the numerical results.
4. Results and Discussion
In this section, the effects of pertinent fluid parameters on the velocity and temperature profiles are shown graphically. Figure 1 shows the effect of medium porosity on the entropy generation rate. Since an increase in the porous permeability parameter implies a decrease in the medium porosity, this caused a decreased flow and heat trapping strategy. The net effect is seen here; that is, entropy production increases only in the centerline of the channel while it decreases at the walls. Figure 2 represents the effect of activation energy parameter on the entropy generation rate. Since activation energy decreases the fluid temperature, it is therefore expected to decrease the entropy generated in the flow region. This is true since an increase in the FrankKamenetskii parameter is known to enhance the fluid temperature. Therefore, by increasing values of this parameter, entropy generated is expected to be on the increase as shown in Figure 3. In Figure 4, the effect of the viscous heating parameter is shown. The result shows that entropy generation rate increases with increasing values of the viscous heating parameter due to frictional interaction in the fluid layers. Figure 5 represents the Bejan number’s variations with the activation energy. From the plot, as the activation energy parameter increases, heat irreversibility due to heat transfer decreases. As a result, fluid friction irreversibility dominates over heat transfer irreversibility within the porous medium. Moreover, as the porous permeability parameter increases in Figure 6, there are reduced flow and fluid temperature rises. The net effect shows that heat transfer irreversibility dominates over irreversibility due to fluid friction. Similar behaviour is observed in Figure 7 as the exothermic FrankKamenetskii parameter increases. Finally, as viscous heating parameter increases, the viscous heat dissipation increases in the flow channel, and fluid friction irreversibility dominates over heat transfer irreversibility as observed in Figure 8.
5. Conclusion
The entropy generation rate in the flow of reactive fluid through a porous medium has been investigated. Adomian decomposition method is used to obtain approximate solution to the strongly nonlinear boundaryvalued problem for the dimensionless energy equation. The main contributions to knowledge from the present analysis are as follows: a reactive fluid flowing through a porous medium, entropy generation is minimum at the centerline of the channel; as a result, fluid friction irreversibility dominates over heat transfer irreversibility in the centerline. Secondly, porous medium with low permeability is one major factor that depletes the useful available energy in a thermofluid.
Nomenclature
:  Fluid temperature 
:  Pressure 
:  Wall temperature 
:  Thermal conductivity of the material 
:  Porous permeability respectively 
:  Dynamic viscosity 
:  Heat of reaction 
:  Rate constant 
:  Activation energy 
:  Universal gas constant 
:  Initial concentration of the reactant species 
:  Channel half width 
:  Cartesian coordinates system 
:  Fluid characteristic velocity 
:  Dimensionless axial pressure gradient 
:  FrankKamenetskii parameter 
:  Activation energy parameter 
:  Viscous heating parameter 
:  Porous medium permeability parameter 
:  Darcy number, respectively 
:  Dimensional and dimensionless entropy generation. 
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
References
 A. Bejan, “Second law analysis in heat transfer,” Energy, vol. 5, no. 89, pp. 720–732, 1980. View at: Publisher Site  Google Scholar
 I. T. AlZaharnah and B. S. Yilbas, “Thermal analysis in pipe flow: influence of variable viscosity on entropy generation,” Entropy, vol. 6, no. 3, pp. 344–363, 2004. View at: Publisher Site  Google Scholar
 O. M. Haddad, M. K. Alkam, and M. T. Khasawneh, “Entropy generation due to laminar forced convection in the entrance region of a concentric annulus,” Energy, vol. 29, no. 1, pp. 35–55, 2004. View at: Publisher Site  Google Scholar
 A. Kahraman and M. Yürüsoy, “Entropy generation due to nonnewtonian fluid flow in annular pipe with relative rotation: constant viscosity case,” Journal of Theoretical and Applied Mechanics, vol. 46, no. 1, pp. 69–83, 2008. View at: Google Scholar
 Y. Aksoy, “Effects of couple stresses on the heat transfer and entropy generation rates for a flow between parallel plates with constant heat flux,” International Journal of Thermal Sciences, vol. 107, pp. 1–12, 2016. View at: Publisher Site  Google Scholar
 T. W. Ting, Y. M. Hung, and N. Guo, “Entropy generation of viscous dissipative nanofluid flow in thermal nonequilibrium porous media embedded in microchannels,” International Journal of Heat and Mass Transfer, vol. 81, pp. 862–877, 2015. View at: Publisher Site  Google Scholar
 W. A. Khan and R. S. R. Gorla, “Second law analysis for free convection in nonNewtonian fluids over a horizontal plate embedded in a porous medium: (prescribed heat flux),” Brazilian Journal of Chemical Engineering, vol. 29, no. 3, pp. 511–518, 2012. View at: Publisher Site  Google Scholar
 R. Revellin, S. Lips, S. Khandekar, and J. Bonjour, “Local entropy generation for saturated twophase flow,” Energy, vol. 34, no. 9, pp. 1113–1121, 2009. View at: Publisher Site  Google Scholar
 F. Hedayati, A. Malvandi, and D. D. Ganji, “Secondlaw analysis of fluid flow over an isothermal moving wedge,” Alexandria Engineering Journal, vol. 53, no. 1, pp. 1–9, 2014. View at: Publisher Site  Google Scholar
 A. S. Butt and A. Ali, “Effects of magnetic field on entropy generation in flow and heat transfer due to a radially stretching surface,” Chinese Physics Letters, vol. 30, no. 2, Article ID 024701, 2013. View at: Publisher Site  Google Scholar
 S. O. Adesanya and O. D. Makinde, “Thermodynamic analysis for a third grade fluid through a vertical channel with internal heat generation,” Journal of Hydrodynamics, vol. 27, no. 2, pp. 264–272, 2015. View at: Publisher Site  Google Scholar
 S. O. Adesanya and O. D. Makinde, “Entropy generation in couple stress fluid flow through porous channel with fluid slippage,” International Journal of Exergy, vol. 15, no. 3, pp. 344–362, 2014. View at: Publisher Site  Google Scholar
 S. O. Adesanya and O. D. Makinde, “Effects of couple stresses on entropy generation rate in a porous channel with convective heating,” Computational & Applied Mathematics, vol. 34, no. 1, pp. 293–307, 2015. View at: Publisher Site  Google Scholar  MathSciNet
 S. Mahmud and R. A. Fraser, “Thermodynamic analysis of flow and heat transfer inside channel with two parallel plates,” Exergy, vol. 2, no. 3, pp. 140–146, 2002. View at: Publisher Site  Google Scholar
 S. H. Tasnim, M. Shohel, and M. A. H. Mamun, “Entropy generation in a porous channel with hydromagnetic effect,” Exergy, vol. 2, no. 4, pp. 300–308, 2002. View at: Publisher Site  Google Scholar
 I. Ozkol, G. Komurgoz, and A. Arikoglu, “Entropy generation in laminar natural convection from a constant temperature vertical plate in an infinite fluid,” Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, vol. 221, no. 5, pp. 609–616, 2007. View at: Publisher Site  Google Scholar
 S. O. Adesanya and O. D. Makinde, “Irreversibility analysis in a couple stress film flow along an inclined heated plate with adiabatic free surface,” Physica A, vol. 432, pp. 222–229, 2015. View at: Publisher Site  Google Scholar  MathSciNet
 J. Srinivas, J. R. Murthy, and A. J. Chamkha, “Analysis of entropy generation in an inclined channel flow containing two immiscible micropolar fluids using HAM,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26, no. 34, pp. 1027–1049, 2016. View at: Publisher Site  Google Scholar
 J. Srinivas and J. V. Ramana Murthy, “Second law analysis of the flow of two immiscible micropolar fluids between two porous beds,” Journal of Engineering Thermophysics, vol. 25, no. 1, pp. 126–142, 2016. View at: Publisher Site  Google Scholar
 O. D. Makinde, T. Chinyoka, and L. Rundora, “Unsteady flow of a reactive variable viscosity nonNewtonian fluid through a porous saturated medium with asymmetric convective boundary conditions,” Computers & Mathematics with Applications, vol. 62, no. 9, pp. 3343–3352, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 L. Rundora and O. D. Makinde, “Effects of suction/injection on unsteady reactive variable viscosity nonNewtonian fluid flow in a channel filled with porous medium and convective boundary conditions,” Journal of Petroleum Science and Engineering, vol. 108, pp. 328–335, 2013. View at: Publisher Site  Google Scholar
 L. Rundora and O. D. Makinde, “Effects of Navier slip on unsteady flow of a reactive variable viscosity nonNewtonian fluid through a porous saturated medium with asymmetric convective boundary conditions,” Journal of Hydrodynamics, vol. 27, no. 6, pp. 934–944, 2015. View at: Publisher Site  Google Scholar
 O. A. Bég, H. S. Takhar, R. Bhargava, S. Rawat, and V. R. Prasad, “Numerical study of heat transfer of a third grade viscoelastic fluid in nonDarcy porous media with thermophysical effects,” Physica Scripta, vol. 77, no. 6, Article ID 065402, 2008. View at: Publisher Site  Google Scholar
 O. D. Makinde, “Thermal ignition in a reactive viscous flow through a channel filled with a porous medium,” Journal of Heat Transfer, vol. 128, no. 6, pp. 601–604, 2006. View at: Publisher Site  Google Scholar
 R. Rach, “A bibliography of the theory and applications of the Adomian decomposition method, 1961–2011,” Kybernetes, vol. 41, no. 78, pp. 1087–1153, 2012. View at: Google Scholar  MathSciNet
 G. Adomian and R. Rach, “Modified decomposition solution of nonlinear partial differential equations,” Applied Mathematics Letters, vol. 5, no. 6, pp. 29–30, 1992. View at: Publisher Site  Google Scholar  MathSciNet
 G. Adomian and R. Rach, “Analytic solution of nonlinear boundary value problems in several dimensions by decomposition,” Journal of Mathematical Analysis and Applications, vol. 174, no. 1, pp. 118–137, 1993. View at: Publisher Site  Google Scholar  MathSciNet
 M. Sheikholeslami, D. D. Ganji, H. R. Ashorynejad, and H. B. Rokni, “Analytical investigation of JefferyHamel flow with high magnetic field and nanoparticle by Adomian decomposition method,” Applied Mathematics and Mechanics, vol. 33, no. 1, pp. 25–36, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 M. R. Mohyuddin, “Resonance and viscoelastic poiseuille flow in a porous medium,” Journal of Porous Media, vol. 9, no. 8, pp. 799–806, 2006. View at: Publisher Site  Google Scholar
 M. R. Mohyuddin, “MHD flow and heat transfer for second grade fluid in a porous medium with modified darcy's,” International Journal of Fluid Mechanics Research, vol. 34, no. 5, pp. 462–474, 2007. View at: Publisher Site  Google Scholar
 S. Asghar, M. R. Mohyuddin, and T. Hayat, “Effects of Hall current and heat transfer on flow due to a pull of eccentric rotating disks,” International Journal of Heat and Mass Transfer, vol. 48, no. 34, pp. 599–607, 2005. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2016 Samuel O. Adesanya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.