THE ANALYSIS OF A REACTIVE HYDROMAGNETIC FLUID FLOW IN A CHANNEL THROUGH A POROUS MEDIUM WITH CONVECTIVE COOLING

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1 U.P.B. Sci. Bull., Series D, Vol. 78, Iss. 4, 16 ISSN HE ANALYSIS OF A REACIVE HYDROMAGNEIC FLUID FLOW IN A CHANNEL HROUGH A POROUS MEDIUM WIH CONVECIVE COOLING Athoy Rotimi HASSAN 1 * ad Riette MARIZ his paper ivestigates the aalysis of a reactive hydromagetic fluid flowig betwee two parallel plates through a porous medium with covective boudary coditios. Neglectig the cosumptio of the material which is exothermic uder Arrheius kietics; it is assumed that the flow system exchages heat with the ambiet followig Newto s law of coolig. Approximate solutios of the oliear dimesioless equatios goverig the fluid flow are obtaied usig the traditioal perturbatio method ad Adomia decompositio method (ADM). Also, the diagoal Pade approximatio techique is used to determie the thermal criticality values as well as bifurcatio coditios. he etropy geeratio aalysis ad effects of all importat flow properties o the fluid flow are also preseted ad discussed. Keywords: Reactive fluids, porous medium, thermal criticality, etropy geeratio, covective coolig, Adomia decompositio method (ADM), Pade approximatio techique ad Arrheius kietics. 1. Itroductio Over the past few decades, studies relatig to aalysis of a reactive hydromagetic fluid flow are o the icrease due to its immese applicatios i may egieerig ad idustrial processes as described i [1] [6] such as, petroleum idustries, chemical egieerig, etc. I a reactig material udergoig a exothermic reactio i which reactat cosumptio is eglected, heat is beig produced i accordace with Arrheius rate law ad Newtoia coolig where covectio forms a itegral part of heat trasfer due to differeces i ambiet temperatures. he process of covectio ot oly affects heat trasfer, but also helps maitai comfort coditios. I additio to that, [7] metioed that thermal explosios occur whe the reactios produce heat too rapidly for a stable balace betwee heat productio ad heat loss to be preserved. Moreover, studies ivolvig the fluid properties i a chael through a porous medium have bee ivestigated i [8] [11], just to metio few. Also, studies i [1] [16] examied fluid flowig betwee walls with covective coolig effects because of its importace i techological applicatios, for 1 PhD, Dept. of Mathematics, ai Solari Uiversity of Educatio, Ijagu, Nigeria ad Postdoctoral Research Fellow, Dept. of Mathematical Scieces, Uiversity of South Africa, * Correspodig Author el: E mail: athoyhassa7@yahoo.co.uk Prof., Dept. of Mathematical Scieces, Uiversity of South Africa, Pretoria, South Africa

2 44 Athoy Rotimi Hassa ad Riette Maritz example, the coolig processes of uclear reactors ad refrigerators where ivestigatios were doe o covective boudary coditios of the flow system. However, as discussed i [11], it is eeded to fid out the property of porous medium which measures the capacity ad ability of the formatio to trasmit fluids. Hece, i the preset stu, the aalysis of [6] ad [17] are exteded to iclude ad ivestigate the effects of fluid flow through a porous medium ad symmetrical covective coolig o the overall flow structure i a reactive hydromagetic fluid betwee two parallel porous plates which was ot accouted for i the previously obtaied results. his preset stu has sigificat beefits i egieerig ad idustrial processes where there is a iheret simplicity for the applicatios just requirig some provisio for atural heat flow to the ambiet which is ofte achieve by adequate vetig o the system of flow rather tha forced covectio. I order to obtai approximate solutios for the oliear dimesioless equatios goverig the fluid flow, traditioal perturbatio method shall be used to determie the temperature profile. Also, etropy geeratio aalysis shall be ivestigated while Adomia decompositio method (ADM) together with the diagoal Pade approximatio techique shall be used to determie the thermal criticality values as well as bifurcatio coditios of the fluid flow system. I the rest of this paper, the problem is formulated i sectio. he goverig equatios are solved usig traditioal perturbatio method i sectio 3.he etropy geeratio aalysis were derived ad the thermal criticality coditios were determied usig ADM ad diagoal Pade approximatio techique i sectio 4. Presetatios of aalytical results of the problem are show i tables ad graphs i sectio 5; while sectio 6 gives the cocludig remarks.. Mathematical Formulatio Let us cosider the stea flow of a icompressible reactive fluid through a chael made up of two parallel porous plates distat a apart ad the fluid is subjected to covective coolig at the boudaries. he fluid is electrically coducted uder the ifluece of a trasversely applied magetic field, B. he x- ad y-axes are chose parallel ad perpedicular to the plates respectively as show i Fig. 1.

3 he aalysis of a reactive hydromagetic fluid [ ] porous medium with covective coolig 45 Neglectig the cosumptio of the reactat, the differetial equatios goverig the fluid flow i o dimesioless form as i [6] ad [17] may be writte as: dp d u B u u (1) dx K E d du R k QC Ae B u u () K he flow is symmetric about the vertical x axis. Hece the correspodig boudary coditios alog the chael cetrelie is give as du d o y ad d y d y d u, k h o y a. (3) I equatios (1) (3), u is the axial velocity, is the absolute emperature, P is the modified pressure, μ is the fluid viscosity, σ is the electrical coductivity, B is the magetic field, K is the porous permeability of the medium, k is the thermal coductivity, Q is the heat of reactio term, Co is the reactat species iitial cocetratio, A is the reactio rate costat, E is the activatio eergy, R is the uiversal gas costat, h is the heat trasfer coefficiet, a is the chael half width ad xy, is the coordiate system measured i the axial ad ormal directios respectively. It should be oted that the last term i equatios (1) ad () are due to the ifluece of porosity as i [8 11]. Also, the first term i equatio (4) is the rate of heat trasfer while other terms accout for viscous dissipatios ad magetic effect. Itroducig the followig dimesioless parameters ad variables:

4 46 Athoy Rotimi Hassa ad Riette Maritz y x u E a dp EU R ah y, x, u,, G, Br,, Bi, a a U R U dx kr E k QAa C E R U e, H Ba QEAa C R a, e, ad kr K E R (4) E he goverig boudary value problem equatios (1) (3) become the followig i dimesioless form: du G H u. (5) d du 1 e H u (6) together with the boudary coditios d du o y ad u, d Bi o 1 (7) I equatios (1) (7), other variables ad parameters like is the wall temperature, G is the pressure gradiet, U is the fluid characteristic velocity, δ is the activatio eergy parameter, γ is the viscous heatig parameter, α is the porous medium permeability parameter, Br is the Brikma umber, H is the Hartma umber, Bi is the Biot Number, λ is the Frak Kameettski parameter, is the wall temperature parameter ad Da is the Darcy umber. 3. Perturbatio Method he fluid velocity equatio (5) is a liear secod order o-homogeeous differetial equatio that has exact solutio with the appropriate boudary coditios as G GCosh[ y H ] 1 H ( G GCosh[ H ]) Cosh[ y H ] Sech[ H ] u( y) H. H ( G GCosh[ H ]) Cosh[ y H ] Sech[ H ] H (8) Substitutig (8) i (6), it will be coveiet to assume a series solutio i the Frak Kameettski parameter due to the o-liear ature of (6) i this form followig [18]:

5 he aalysis of a reactive hydromagetic fluid [ ] porous medium with covective coolig 47 ( y) ( y) (9) e Where < λ << 1, clearly, 1 ca be aylor s series expaded, usig the solutio series (9) i (6) ad equatig the orders of λ, we obtai ad solve the followig: d ( ), (1) such that '(), '(1) Bi( 1), such that 1 '(), 1 '( 1) Bi1 ( 1) 1 d1 1 du ( ) e H u ( ) 3 ( ) 1 d e 1 (1 ) such that '(), '( 1) Bi ( 1), (11) d 1 e (1 ) (1) ( 4 ) such that 3 '(), 3 '( 1) Bi3 (1) ad so o. Solvig equatios (1) (13) give us the fluid temperature profile ad the effects of physical aspects of the flow properties are discussed i sectio Etropy Geeratio Aalysis he total etropy chage observed i a closed system is the sum of the etropy chage which ca be attributed to reversible heat trasfer ad the etropy chage attributable to irreversibility. Although, it is difficult to directly measure the magitude of irreversibility i a closed system, but ca be calculated from the etropy geeratio equatio. he etropy productio is due to heat trasfer ad the combied effects of fluid frictio ad Joules dissipatio. Followig [3, 5, 6 ad ], the geeral equatio for the etropy geeratio per uit volume i the presece of a magetic field ad porous medium is give by: S m d du B u u d y d y K (13) k (14)

6 48 Athoy Rotimi Hassa ad Riette Maritz he first term i (14) is the irreversibility due to heat trasfer; the secod term is the etropy geeratio due to viscous dissipatio ad the last two are the local etropy geeratio due to the effects of magetic field ad porosity respectively. We express the etropy geeratio umber i dimesioless form usig the existig dimesioless variables ad parameter i (4) as: N s m S a E kr d Br du ( H ) u (15) d he first term, is assiged N1 which is the irreversibility due to heat Br du trasfer ad the secod term, ( H ) u referred to as N is the etropy geeratio due to the combied effects of viscous dissipatio, magetic R field ad porosity of the flow regime where is the wall temperature E parameter. We defied N (16) N1 as the irreversibility distributio ratio. Relatio (16) shows that heat trasfer domiates whe 1ad fluid frictio domiates whe ϕ > 1. his is used to determie the cotributio of heat trasfer i may egieerig desigs. As a alterative to irreversibility parameter, the Beja umber (Be) is defied as N1 1 Be where Be 1. (17) N 1 s 4.. hermal Criticality he aalysis of the thermal criticality for the fluid flow through a porous medium with covective coolig is doe by usig Adomia Decompositio Method (ADM) ad Pade approximatio to obtai the solutio of the o liear boudary value problem equatios goverig the fluid flow. Usig ADM, the solutio of the temperature profile is give as y y du 1 ( y) a e H u (18) where a () is to be determied by usig the boudary coditios. he ADM requires that the approximate solutio is the partial sum k ( y) ( y) (19a)

7 he aalysis of a reactive hydromagetic fluid [ ] porous medium with covective coolig 49 of the followig series ( y ) ( y ) (19b) where the compoets, 1,,..., k are to be determied. Writig the o liear term i (18) as a series of Adomia polyomials, we have ( y) 1 ( y) () A ( y) e such that (18) becomes y y du ( y) a A ( y) H u. (1) ad some of the Adomia polyomials obtaied from () are ( y) 1 ( ) A e y, (a) ( y) 1 ( y) e 1 ( y) 1 A A 1 ( y) ( y), (b) 1 ( y) e y y y y 1 ( ) ) 1( ) (1 ( ) ( )) 1 ( y) 4 (c) Followig [3, 4, 17 ad ] ad takig the zeroth compoets of (1), we have ( y) a (3) y y du 1( y) A( y) H u (4) y y 1( ) ( ), 1 y A y (5) o this ed, the diagoal form of the series solutios (19a) is evaluated usig the built i Pade approximat procedure i MAHEMAICA ad the boudary coditios i (7) give as: '(1) Bi (1) (6) akig the diagoal Pade approximat of (19a) at various values leads to a eigevalue problem. o show that the series coverge, the ukow costat a is evaluated usig values for the kow parameters. he critical values of the Frak Kameettski parameter ( c ) for the o existece of solutio or thermal ruaway for the fluid flow are preseted ad discussed i the ext sectio.

8 5 Athoy Rotimi Hassa ad Riette Maritz 5. Discussio of Results I this sectio, we discuss the solutios of velocity ad temperature profiles, solutio braches, etropy geeratio ad thermal criticality for hydromagetic fluid flow through a porous medium with covective coolig. he rapid covergece of the series solutios of the temperature profile which clearly shows the efficiecy ad reliability i the approximatio is show i able 1 while able displays the computatio of the etropy geeratio aalysis which idicates that the etropy geeratio rate is maximum at the plate surfaces ad miimum aroud the core regio of the chael. Also, the irreversibility distributio ratio ( ) shows that heat trasfer domiates at upper ad lower plate surfaces because 1ad fluid frictio domiates at the ceterlie of the regio because 1. y Rapid covergece of the series solutios of the emperature Profiles y H G 1, Bi 1,.5, k Computatio of the Etropy Geeratio Aalysis H G 1, N 1 N able 1 able Bi 1,.5, Br 1.1 N 1 s Be

9 he aalysis of a reactive hydromagetic fluid [ ] porous medium with covective coolig 51 Effect of differet parameters o the developmet of thermal ruaway Pade H G Bi / / / / / / / / / c able 3 Meawhile, able 3 shows the effects of differet parameters o the developmet of thermal ruaway. It shows that the magitude of thermal criticality icreases with icreasig values of porous medium term (α), covective coolig term (Bi) ad magetic field itesity (H) which stabilizes the fluid flow. he velocity profiles with variatios i porous medium term ad magetic field are respectively show i Figs. ad 3. It is show that the fluid velocity reduces with icreasig values of porous medium term (α) ad magetic field itesity (H) which is due to the retardig effect of the porosity ad magetic force preset i the chael. Fig. : Fluid velocity profile with variatios i porous medium term Fig. 3: Fluid velocity profile with variatios i magetic field itesity he temperature profiles are show i Figs I fig. 4, the fluid temperature icreases as the viscous heatig parameter icreases, this is caused by the coversio of kietic eergy i the movig fluid to iteral eergy. he maximum fluid temperature is obtaied at the miimum values of magetic field itesity parameter (H) as show i fig. 5. Also, i Fig. 6, the fluid temperature reduces as the porous medium term icreases; this is due to the reductio i fluid flow ad the time take for fluid to flow withi the porous medium thereby reduces the temperature.

10 5 Athoy Rotimi Hassa ad Riette Maritz he fluid temperature profile with variatios i covective coolig term (Bi) is show i figure 7; it is observed that the miimum value of temperature is obtaied at the maximum value of Biot umber due to the ifluece of thermal coductivity o the fluid temperature. Also, the fluid temperature icreases as Frak Kameettski parameter (λ) icreases as show i figure 8; this is due to a icrease i the heat geerated withi the flow chael. Fig. 4: Fluid temperature profile with variatios i viscous heatig parameter Fig. 5: Fluid temperature profile with variatios i magetic field itesity Fig. 6: Fluid temperature profile with variatios i porous medium parameter Fig. 7: Fluid temperature profile with variatios i covective coolig term Fig. 8: Fluid temperature profile with variatios i Frak Kameettski parameter

11 he aalysis of a reactive hydromagetic fluid [ ] porous medium with covective coolig 53 Figs. 9 to 1 display the variatio of parameters o etropy geeratio rate. Geerally, it is oticed that the etropy geeratio rate is at maximum at the surfaces ad at miimum aroud the core regio of the chael of fluid flow. I figure 9, the ifluece of porous medium parameter ( ) is clearly oticed as it yields a iterestig result with respect to the etropy geeratio rate with icreasig value of over movig surfaces. O the other hads, Figs. 1 ad 11 showed that the etropy geeratio rate icreases respectively with icreasig values of Frak Kameettski parameter (λ) ad wall temperature parameter 1 ( Br ) i the thermoamic performace of the flow system. I Fig. 1, the rate of disorder is reduced with a icrease i magetic field itesity (H) Fig. 9: Etropy geeratio rate for various values of porous medium parameter Fig. 1: Etropy geeratio rate for various values of wall temperature parameter Fig. 11: Etropy geeratio rate for various values of Frak Kameettski parameter Fig. 1: Etropy geeratio rate for various values of magetic field itesity However, Figs show the Beja umber (Be) for various parametric values i the chael width. he geeral observatio is that the fluid frictio over irreversibility domiates at the chael core regio while heat trasfer rate over irreversibility domiates at both upper ad lower wall surfaces. It is clearly oticed that, the domiat ifluece of heat irreversibility of the plate icreases with icreasig values of porous medium parameter ( ) ad Frak

12 54 Athoy Rotimi Hassa ad Riette Maritz Kameettski parameter (λ) i Figs. 13 ad 14, while the reverse is the case i Fig. 15 where heat irreversibility of the plate decreases with a icreasig value of the 1 wall temperature parameter ( Br ) Fig. 13: Beja umber for various values of porous medium parameter Fig. 14: Beja umber for various values of Frak Kameettski parameter Fig. 15: Beja umber for various values of wall temperature parameter Fig. 16: A slice of approximate bifurcatio diagram Fially, aother iterestig aspect of the problem is the critical poit show i figure 16, a slice of approximate bifurcatio diagram, it is oticed that the problem has upper ad lower solutios at, a sigle solutio at c ad o solutio at c. 6. Coclusio he aalysis of a reactive hydromagetic fluid flow betwee two parallel plates through a porous medium with covective boudary coditios is ivestigated usig the traditioal perturbatio method together with Adomia Decompositio Method (ADM) ad diagoal Pade Approximat to determie the thermal criticality values as well as bifurcatio coditios. It is observed that the fluid velocity reduces with icreasig values of porous medium ad magetic itesity c

13 he aalysis of a reactive hydromagetic fluid [ ] porous medium with covective coolig 55 parameters. he fluid temperature decreases with icreasig values of activatio eergy, porous medium, magetic itesity ad covective coolig terms. Also, a icrease i the covective coolig, porous medium ad magetic itesity fields o the fluid flow will improve stability ad this will help to brig about a delay i the appearace of thermal ruaway. R E F E R E N C E S [1]. J. A. Gbadeya ad A.R. Hassa, Multiplicity of solutios for a reactive variable viscous Couette flow uder Arrheius kietics, i Mathematical heory ad Modellig Vol., o.9, 1, pp []. A. R. Hassa ad J. A. Gbadeya, he effect of heat absorptio o a variable viscosity reactive Couette flow uder Arrheius kietics, i heoretical Mathematics & Applicatios. Vol. 3, o.1, 13, pp [3]. A. R. Hassa ad J. A. Gbadeya, Etropy geeratio aalysis of a reactive hydromagetic fluid flow through a chael, i U. P. B., Series A, Vol. 77, Iss., 15, pp [4]. A. R. Hassa ad J. A. Gbadeya, hermal stability of a reactive hydromagetic Poiseuille fluid flow through a chael, i America Joural of Applied Mathematics, Vol., o.1, 14, pp 14. [5]. A. R. Hassa ad J. A. Gbadeya, A reactive hydromagetic iteral heat geeratig fluid flow through a chael, i Iteratioal Joural of Heat ad echology, Vol. 33, o 3, 15, pp [6]. O. D. Makide ad O. Awar Beg, O iheret irreversibility i a reactive hydromagetic Chael Flow, i Joural of hermal Sciece, Vol. 19, o 1, 1, pp [7]. K aira, A mathematical aalysis of thermal explosios, i IJMMS, Vol. 8, o 1, 1, pp [8]. P. K. Sigh, Effects of variable fluid properties ad viscous dissipatio o mixed covectio fluid flow past a vertical plate i porous medium, i Iteratioal Joural of Scietific ad Egieerig Research, Vol. 3, o 1, 1, pp [9]. O. D. Makide, hermal stability of a reactive viscous flow through a porous saturated chael with covective boudary coditios, i Iteratioal Joural of Numerical Methods for Heat & Fluid Flow, Vol. 17, o. 8, 7, pp [1]. O. D. Makide, hermal stability of a reactive viscous flow through a porous saturated chael with covective boudary coditios, i Applied hermal Egieerig, Vol. 9, 9, pp [11]. A. R. Hassa ad R. Maritz, he aalysis of a variable viscosity fluid flow betwee parallel porous plates with o uiform wall temperature, I Italia Joural of Pure ad Applied Mathematics, Vol. 36, o 1, 16, pp 1 1. [1]. O. D. Makide ad M. Maserumule, Iheret irreversibility ad thermal stability for stea flow of variable viscosity liquid film i a cylidrical pipe with covective coolig at the surface, I Iteratioal Joural of Numerical Methods for Heat & Fluid Flow, Vol., o. 1, 1, pp [13]. O. D. Makide, O hermal Stability of a reactive third - grade fluid i a chael with covective coolig the walls, I Applied Mathematics ad Computatio, Vol. 13, 9, pp [14]. P. O. Olarewaju, J. A. Gbadeya,. Hayat. ad A. A. Hedi, Effects of iteral heat geeratio thermal radiatio ad buoyacy force o a boudary layer over a vertical plate with a covective surface boudary coditio, i South Africa Joural of Sciece, Vol. 17, o. (9/1), 11, pp

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