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Research Article Mathematical Analysis of a Reactive Viscous Flow Through a Channel Filled with a Porous Medium

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Research Article Mathematical Analysis of a Reactive Viscous Flow Through a Channel Filled with a Porous Medium Hindawi Publishing Corporation Journal of Mathematics Volume 2016, Article ID 1350578, 8 pages http://dx.doi.org/10.1155/2016/1350578 Research Article Mathematical Analysis of a Reactive Viscous Flow through a Channel Filled with a Porous Medium Samuel O. Adesanya,1 J. A. Falade,2 J. C. Ukaegbu,1 and K. S. Adekeye1 1 Department of Mathematical Sciences, Redeemer’s University, Ede, Nigeria 2Department of Physical Sciences, Redeemer’s University, Ede, Nigeria Correspondence should be addressed to Samuel O. Adesanya; [email protected] Received 19 July 2016; Accepted 14 November 2016 Academic Editor: Ghulam Shabbir 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. 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 flux. Revellin et al. [8] addressed the thermal performance of adiabatic two-phase flow using two different methods. Studies on heat irreversibility in moving fluid find its rel- Hedayati et al. [9] utilized the thermodynamics analysis to evance in several geological, petrochemical, and industrial optimize flow on a nonstationary wedge. Butt and Ali [10] applications. In most flows at extremely high temperature, reported the irreversibility analysis of fluid slippage with heat irreversibility is unavoidable. This usually leads to convective boundaries. Other works that focused on the material waste due to reduced efficiency of the thermofluid minimization of energy losses in a fluid flow can be found machine. To conserve energy, Bejan [1] introduced an in references [11–19] and many more too numerous to be approach that is based on the thermodynamics second law listed. topredicttheperformanceofthermalsystemssoasto From applications’ point of view, studies on transport maximize scarce available energy for work and minimize reactivefluidsinporousmediaareveryimportantsince wastages. Following his analysis, Al-Zaharnah and Yilbas [2] they occur in many important areas like water treatment considered the irreversibility analysis in a viscous pipe flow. using fixed beds, agriculture, oil recovery, ground water flows, Haddad et al. [3] examined the heat irreversibility in forced geothermal engineering, exhaust systems in combustion, convective flow in concentric cylindrical annulus under material processing, and reservoir engineering. Recently, diverse flow conditions. Kahraman and Yur¨ usoy¨ [4] applied Rundora and his associates [20–22] documented several the same approach to study the heat irreversibility in non- investigations on unsteady reactive fluid flow in porous Newtonian fluid flow through pipes. Aksoy [5] considered mediumandhowtheflowevolvedtothesteadystate.Beg´ et the influence of couple stresses on the development of al.[23]examinedtheflowofviscoelasticfluidthroughanon- heat irreversibility in a channel with adiabatic surface and Darcian porous medium. Makinde [24] studied the inherent constant heat flux. Ting et al. [6] considered the irreversibility heat irreversibility in reactive fluid through a channel filled associated with nanofluids in a microchannel with porous with porous material. materials using water-alumina. Moreover, Khan and Gorla In all the studies above, the entropy productions in the [7] addressed the convective problem in non-Newtonian flow of viscous incompressible fluid flow through porous fluid flow through a channel with porous medium and heat medium have not been investigated. Therefore, the work done 2 Journal of Mathematics (− ) in [24] can be further extended to give more interesting = 0 , 2 results on the thermodynamics and heat transfer properties 0 ofthefluidflow.Thisisbecausehugeamountofmoney and effort could be wasted if the inherent irreversibility in = , Da 2 the fluid flow is not well addressed. Therefore, the specific ℎ objective of this article is to examine the rate at which ℎ22 entropy is produced in a viscous fluid flow system through = , 22 a porous medium. The problem under consideration is 0 nonlinear due to the exponential nature of the rate law in = 0 Arrhenius kinetics for combustible fluids. In view of this, exact solution for the temperature field may not be possible 2 −/ to get. To solve the problem, we seek Adomian series solution ℎ 0 = 0 , to avoid linearization of the exponential term. The Adomian 2 decomposition method is a straightforward way of solving 0 all kinds of differential equations arising from many physical ℎ2 =− , scenarios. It has been used extensively in the last few decades as reported in the bibliography by Rach [25], and, more 2 2 / recently,themethodhasbeenusedin[26–31].Theplanof 0 = , the article is as follows: the problem is formulated and the ℎ2 mathematical analysis is presented in Section 2. Section 3 of 0 1 the work gives the Adomian method of solution. Graphical 2 = results are presented and interpreted in Section 4 while, in Da Section 5, concluding remarks are given. (4) 2. Mathematical Analysis to get the dimensionless problems: 2 The steady flow of viscous incompressible reactive fluid 2 − =−1;( 0) =(1) =0, (5) through parallel-plate immersed in a porous medium is stud- 2 ied. The flow is assumed to be full-developed and driven by 2 2 an applied pressure gradient. The channel wall temperatures +{/(1+) +( ) +22}=0; 2 are kept constant. Then, the balanced governing equations are (6) [24] (0) =(1) =0 2 0=− + − 2 2 2 =( ) + (( ) +22). (7) (1) 2 2 2 0= + 0 −/ + ( ) + Setting 2 2 =( ) , with the following boundary conditions: 1 (8) (0) =(ℎ) =0 2 2 2 (2) 2 = (( ) + ), (0) =0= (ℎ) . Under these assumptions, entropy generation equation then, the irreversibility ratio becomes becomes 1 = 1 = , 2 2 Be 2 1 +2 1+Φ = ( ) + ( ) + . (3) (9) 2 0 0 0 Φ= 2 . 1 To nondimensionalize (1)–(3), we need the following param- eters and variables: From (9), it is evident that = , {0, 2 ≫1 ℎ { Be = {0.5 1 =2 (10) { = , { {12 ≪1. Journal of Mathematics 3 3. Adomian Method of Solution is expanded by Taylor’s series to get the following Adomian polynomials: A direct integration of (5)-(6) leads to the integral equations 0/(1+0) 0 = (0) 2 ()=∫ + ∫ ∫ ( ) (11) 0 0 0 /(1+ ) = 1 0 0 1 2 (1 + 1) with {(1 − 2 − 22 )2 +2(1+( )2 )} = 0 1 0 2 (17) () 2 4 2(1+1) (0) /(1+ ) = ∫ ⋅ 0 0 0 (12) . 2 . − ∫ ∫ {/(1+) +( ) +22}. 0 0 The zeroth-order components of the series solutions (14) and Due to the exponential nonlinearity in (12), we now define a (15) are series of functions defined by 2 () = − ∞ 0 0 2 (18) ()= ∑ () , =0 0 () = 0. (13) ∞ ()= ∑ () . Since the integral of a continuous function is continuous, =0 then each term of the series can be uniquely determined by Substituting (13) into the integral equations (11)-(12), we 2 obtain +1 () = ∫ ∫ ≥1 0 0 ∞ (0) 1 () ∑ () = ∫ =0 0 (14) = ∫ 1 ∞ 0 (19) + ∫ ∫ (2 ∑ ()) , 2 0 0 =0 2 2 −∫ ∫ (0 +( ) + ) 0 0 ∞ (0) ∑ () = ∫ =0 0 +1 () = − ∫ ∫ ≥1, 0 0 ∞ ∑=0 () − ∫ ∫ {Exp ( ∞ ) 0 0 1+∑=0 () where (0)/0 = and (0)/1 = are the parameters to be determined. 2 (15) ∞ Then, (17)–(19) are evaluated using MATHEMATICA +( (∑ ())) andthesolutionsareobtainedasfiniteseries: =0 ∞ 2 2 ()= ∑ () , + (∑ ()) }. =0 =0 (20) ()= ∑ () . The nonlinear term in (15) represented by =0 ∑∞ () = ( =0 ) The series solutions are shown to be convergent and twice Exp 1+∑∞ () (16) =0 differentiable (see Tables 1 and 2). Next, we establish the 4 Journal of Mathematics uniqueness solution of (20). It is well known that the Lipschitz With (22), (6) can now be written as condition is sufficient for the uniqueness of solution. There- 1 fore, we first seek for a Lipschitz constant such that 1 4 2 ( ) ( ) ( ) (3) = ( 5 ) 2 −1 4 2 () −() ≤ () −() (21) /(1+ ) 2 2 2 − ( 3 3 +( + )) ( 5) ( 4 2 ) 1 (1,2,...,5) is satisfied. To do this, the boundary-valued problems (6) 2 (1,2,...,5) are converted to system first-order differential equations by ( ) = (3 (1,2,...,5)) ; (23) introducing the following transformations: 4 (1,2,...,5) (5 (1,2,...,5)) 1 (0) 0 1 =, 2 (0) 0 2 =(), ( ) ( ) (3 (0)) = (0) , 0 3 =(), (22) 4 ( ) 1 (5 (0)) (2) 4 = () , where ( = 1, 2), the guess values that will ensure the = () . 5 boundary conditions, are satisfied. Then, 0, 0, 0, 0, 0 0, 0, 0, 1, 0 ( ) (0, 0, 0, 0, 1) = ( ) (24) ( 2 ) 0, 0, 1, 1 2 3 3/(1+3) 0, −2 2 − (1 − ) −23 0 ( 1+4 1+3 ) since /, , = 1,2,...,5, exist and are continuous in Let () and () be any two solutions of the integral equation the domain [0, 1]. Hence, the Lipschitz constant with the (12); then, property ()−() = − ∫ ∫ {() −()} . (27) 0 0 ≤ (25) This implies that exists. ()−() ≤ ∫ ∫ {() −()} . (28) 0 0 Uniqueness Analysis. The Adomian series solutions (20) of the nonlinear problem (6) converges if 0<≪1and |0()| < Inviewof(21),wethenhave ∞ = 2/2 ,where . ()−() ≤ ∫ ∫ ()−() 0 0 Proof. Let ([], ‖ ⋅ ‖) be a Banach space for all continuous ≤ ()−() ∫ ∫ (29) functions on with the norm 0 0 2 ‖ ()‖ = max | ()| . ≤ ()−() . ∀ (26) 2 Journal of Mathematics 5 Table 1: Uniqueness results =3, =1. { =0.1, =1, =1} -exact -Adomian Error 1.0 00 0 0 = 0.5 −8 0.8 0.1 0.0412846 0.0412847 8.58689 × 10 −7 0.2 0.072741 0.072742 1.72597 × 10 s 0.6 −7 N = 0.3 0.3 0.0953855 0.0953858 2.61030 × 10 −7 0.4 0.108743 0.108744 3.51816 × 10 0.4 −7 0.5 0.113181 0.113182 4.44491 × 10 −7 0.2 = 0.1 0.6 0.108743 0.108744 5.34572 × 10 −7 0.7 0.0953855 0.0953861 6.06498 × 10 −7 0.2 0.4 0.6 0.8 1.0 0.8 0.0729741 0.0729747 6.1915 × 10 −7 y 0.
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