Numerical Treatment of Heat and Mass Transfer of MHD Flow of Carreau Fluid with Diffusion and Chemical Reaction through a Non Darcy Porous Medium

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1 The Ope Mathematcs Joural 9-5 Ope ccess Numercal Treatmet of Heat ad Mass Trasfer of MHD Flow of Carreau Flud wth Dffuso ad Chemcal Reacto through a No Darcy Porous Medum Mohamed Y. bou-zed * Departmet of Mathematcs Faculty of Educato Shams Uversty Helopols Caro Egypt bstract: The umercal solutos of the system of o lear partal dfferetal equatos whch descrbe the usteady flow of MHD o-newtoa flud wth heat ad mass trasfer past a porous plate through a o-darcy porous medum are obtaed takg to accout the effects of vscous ad ohmc dsspatos ad the chemcal reacto. Fte dfferece method s used for solvg the system of equatos. The results of the velocty temperature cocetrato dstrbutos sk-frcto rate of heat ad mass trasfer are llustrated graphcally for dfferet values of physcal parameters. lso the stablty codto s studed. - INTRODUCTION Flow through porous meda s very prevalet ature ad therefore the study of flow through a porous medum has become of prcple terest may ad egeerg applcatos. Thermal ad solutal trasport by flud flowg through a porous matrx s pheomeo of great terest from the theory ad applcato pot of vew. Heat trasfer the case of homogeous flud-saturated porous meda has bee studed wth relato of dfferet applcatos lke dyamc of hot udergroud sprgs terrestral heat flow through aqufer hot flud ad gto frot dsplacemets reservor egeerg heat exchage betwee sol ad atmosphere ad heat exchages wth fludzed beds. Mass trasfer sothermal codto has bee studed wth applcatos to problems of mxg of fresh ad salt water a qufers spreadg of solules fludzed beds ad crystal washers salt leachg sols etc. []. There has bee a reewed terest Darcy ad o- Darcy flow a o-newtoa flud saturated porous medum. Ths cotug terest due to ts applcatos whch cludg heat trasfer ehacemet packed bed chemcal reactors buldg sulato ehaced ol recovery food techology ad fltrato processes []. Darcy's law was the frst model of trasport processes porous meda. It's vald oly for slow flows through porous meda of relatvely low permeablty. The Darcy's law breaks dow for flows at hgh velocty whch erta effects are o loger eglgble. Forchhemer [] proposed addg a velocty square term to the Darcy term that's to clude the erta effects the mometum equato. For flow through a hghly porous medum the boudary frctoal effect eeds to be cosdered so that Brkma [4] suggested addg a macroscopc vscous term. Thus The Brkma-Forchhemer- exteded Darcy model (The o-darcy effects) s the most geeral model goverg flow porous meda. The *ddress correspodece to ths author at the Departmet of Mathematcs Faculty of Educato Shams Uversty Helopols Caro Egypt; E-mal: master_math@yahoo.com o-darcy effects tally studed by Vafa [5]. The umercal solutos of the problem of heat ad mass trasfer of the pulsatle flow of magetohydrodyamc o- Newtoa flud (bvscosty flud) through a porous medum betwee two permeable parallel plates was obtaed by Mokhtar et al. [6]. Nakayama [7] vestgated The Newtoa ad o-newtoa Couette flows through elastc flud saturated porous meda due to a movg plate boudary. The steady MHD flow of a compressble electrcally coductg vscoelastc flud through a porous medum betwee two porous parallel plates uder the fluece of a trasverse magetc feld wth heat ad mass trasfer porous meda aalyzed by Eldabe ad Sallam [8]. I hs paper Elshehawy et al. [9] vestgated the problem of the effect of porous medum perstaltc moto of a geeralzed Newtoa flud. The o-darcy mxed covecto of a o-newtoa flud from a vertcal sothermal plate embedded a homogeeous porous medum the presece of surface jecto or sucto vestgated by Ibrahm et al. []. The o-darcy mxed covecto from a horzotal surface a porous medum saturated wth a power law flud studed by Kumar ad Nath []. I the preset work fte dfferece method s used to solve the system of o lear partal dfferetal equatos whch are obtaed from the usteady flow of a electrcally coductg o-newtoa (Carreau) flud past a porous plate through a o-darcy porous medum wth heat ad mass trasfer the presece of vscous ad Joulea dsspatos ad chemcal reacto effects. The plate s oscllatg ts ow plae wth supermposed jecto or sucto. Numercal solutos are obtaed for the velocty temperature ad cocetrato dstrbutos s llustrated graphcally. The Stablty codto of the umercal method s studed. - BSIC EQUTIONS The basc equatos of the MHD flow of o-newtoa flud through o-darcy porous meda wth heat ad mass trasfer usg Carreau model [9] are: /9 9 Betham Ope

2 No Darcy Porous Medum The Ope Mathematcs Joural 9 Volume Cotuty Equato.V () Mometum Equato dv dt. J B μ K V C E V V () K p Maxwell's Equatos.B (a) B μ m J E B t (b) (c) J ( E V B) (d) Heat Equato dt C p dt k V c T j x j (J.J ) D m K T C C s (4) Cocetrato Equato dc dt D m C D m K T T m T (5) j where The stress s defed by ( ) j (6) dfferetato d(.) dt j j ã j (.) (V.)(.) t ad the materal tme - PROBLEM FORMULTION The flow of a o-newtoa flud that follows Carreau model o porous plate through a o-darcy porous medum s cosdered. Choose the x-axs alog the plate ad the y- axs perpedcular to t. uform magetc feld B ( B ) s mposed ormal to the plate (see Fg. ()) ad the duced magetc feld s eglgble compared wth the mposed feld so that the magetc Reyolds umber s small. The quattes μ m ad are all costats throughout the flow feld. The electrc feld E s assumed to be zero. I vew of these assumptos Eq. (b) ca be gored retag the equato.j. For t < flud s at rest ad for t > the plate s movg perodcally. From cotuty equato () the velocty feld V(u(y; t) V ) V < s the sucto velocty ad V > s the blowg velocty. Fg. (). Sketch of the problem. The goverg equatos (-6) ca be smplfed to u t V u y u y u y y B v K u C E u. K p T t V T y K T y u C p u y y C C V D t y C ( C C ) y m B u C p Wth tal ad boudary codtos at t : u( yt) T ( yt) T C( yt) C () at t > : u(t) U e ( I)t T(t) T y m (T w T )e ( I)t C(t) C (C w C )e ( I)t y y (7) (8) (9) (a) lm u( y t) lm T ( y t) T lm C( y t) C f Itroduce the dmesoless quattes:- y v u U C T T T T w (b) V t d C C Cw C () usg the dmesoless quattes () to the system of equatos (7-b) gves

3 4 The Ope Mathematcs Joural 9 Volume Mohamed Y. bou-zed f d f () ˆ M D a f F s f W f f () d S c () ( ) m () d P r E c ˆ () W f (4) f ME c f m d (5) S c at at Wth tal ad boudary codtos : f () > : f e ( C I )( )) e ( C I )( ( )) e at (a) ( C I )( )) Wth tal ad boudary codtos at : f () (t) () (6) at > : f () e (C I ) () e (C I ) (7a) () e (C I ) lm f ( ) lm ( ) lm ( ) (7b) 4- METHOD OF SOLUTION We shall solve the system of o-lear partal dfferetal equatos umercally usg the fte dfferece techque whch s dscussed by Soudalgekar [] ad so the system of equatos (-7b) yeld f f d f f ˆ f f f () () W f f M D a f F s ( f ) (8) d P r () ME c ( f ) E c ˆ () W f f f f (9) f at (b) 4.- Stablty Codto of the Scheme The Vo Neuma method s used to study the stablty codto for the fte dfferece equatos. ssume a Fourer compoet for f ad as f F()e Ip() ()e Ip() ad ()e Ip() where p s the wave umber y-drecto. Let p() the f I I I F( ) e ( ) e ad ( ) e () Smlarly f ± ad f F()e I(±) ± ± ()e I(±) ad (4) ()e I(±) F ()e I è È()e I ad ø Ø()e I (5) Substtutg equatos (-5) to equatos (8-) we get F F d F(eI ) () ˆ W f f F (e I e I ) M () D a F F s f F (6)

4 No Darcy Porous Medum The Ope Mathematcs Joural 9 Volume 5 d (ei ) P r (e I e I ) () ME c F f E c ˆ () W f f f f F(e I ) (7) d (ei ) S c (e I e I ) () m (8) The equatos (6-8) ca be wrtte as F F d ( ei ) ˆ () () (cos) () W f f () M D a F s f (9) F E C ˆ (ei ) () W f f f f ()ME C f d ( ei ) () P r () (cos) () d ( ei ) () S C () (cos) m () ad equatos (9-) F F () F () 4 (4) where 4 Ec (ei ) f f ()M f P r 4 Sc m ) ( ) (cos ) ( ) ( ) ( ˆ ) ( 4 s a I f F D M f f W e d The equatos (-4) matrx form ca be expressed as follows F F 4 ; the amplfcato factor s a matrx 4 For stablty the modulus of each of the egevalue m of the amplfcato must ot exceed uty. The egevalues are gve by 4 r P m Sc. Hece the stablty codto s: 4 (5) The scheme s stable whe equalty (5) s satsfed. The local trucato error by employg the procedure used Smth [] s O() O() ad t ted to zero as ad. Hece the scheme s compatble ad the coverget because compatblty ad stablty are ecessary ad suffcet codtos for covergece.

5 6 The Ope Mathematcs Joural 9 Volume Mohamed Y. bou-zed 5- THE SKIN-FRICTION HET ND MSS TRNSFER The sk-frcto heat ad mass trasfer the o dmesoal form ca be defed as [] ˆ f () W f (6) Q (7) S t (8) We ca wrte Eqs. (6-8) by usg fte dfferece method as follows: ˆ f f f () () W f f (9) Q S t (4) (4) 6- NUMERICL RESULTS ND DISCUSSION I order to get a sght to the physcal stuato of the problem we have computed umercal values of the velocty temperature ad cocetrato. They are obtaed for varous values of the parameters goverg the problem. The umercal calculatos were performed by usg a mesh sze. ad tme step. Fgs. (-4) dsplay the results for velocty temperature ad cocetrato profles for dfferet values of the parameters assocated wth the goverg problem. We observe that the velocty creases as Darcy umber D a o dmesoal parameter ˆ ad parameter of flue-.5.5 D a. D a. D a.5 4 Fg. (). The velocty dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Fs Pr Ec Sc5 /5 ad m α. α.5 α.5 4 Fg. (). The velocty dstrbuto s plotted versus y for C d.5.4 W.7 M Da. Fs Pr Ec Sc5 /5 ad m.5.

6 No Darcy Porous Medum The Ope Mathematcs Joural 9 Volume d.5 d.5 d..5 4 Fg. (4). The velocty dstrbuto s plotted versus y for C ˆ.4 W.7 M Da. Fs Pr Ec Sc5 /5 ad m Fs Fs Fs.5 4 Fg. (5). The velocty dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Da. Pr Ec Sc5 /5 ad m M M M5.5 4 Fg. (6). The velocty dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 Da. Fs Pr Ec Sc5 /5 ad m.5. ce of sucto/blowg d crease as show Fgs. (-4) but the velocty decreases as Forchhemer umber F s Magetc parameter M ad Wesseberg umber W crease as show Fgs. (5-7). Fgs. (8-9) show the effect of dmesoless power-law dex ad the tme o velocty. It s observed that the velocty creases wth creasg values of ad. Both of temperature ad cocetrato crease as parameter of fluece of sucto/blowg d ad the tme crease as show Fgs. (-). The temperature decreases as Forchhemer umber Fs ad Wesseberg umber W crease as show Fgs. (4-5). The temperature creases as Darcy umber D a Eckert umber Ec dmesoless power-law dex o dmesoal parameter ˆ ad Magetc parameter M crease as show Fgs. (6-). lso the temperature creases for <.5 but t decreases for >.5 as Pradtl umber Pr creases as show Fg. ().

7 8 The Ope Mathematcs Joural 9 Volume Mohamed Y. bou-zed.5.5 W.4 W.7 W Fg. (7). The velocty dstrbuto s plotted versus y for C d.5 ˆ.4 M Da. Fs Pr Ec Sc5 /5 ad m Fg. (8). The velocty dstrbuto s plotted versus y for C d.5 ˆ W.7 M Da. Fs Pr Ec Sc5 /5 ad m τ5 5 π τ5 4 π τ π.5 4 Fg. (9). The velocty dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Da. Fs Pr Ec Sc5 ad m.5.

8 No Darcy Porous Medum The Ope Mathematcs Joural 9 Volume d.5 d.5 d Fg. (). The temperature dstrbuto s plotted versus y for C ˆ.4 W.7 M Da. Fs Pr Ec Sc5 /5 ad m d.5 d.5 d..5 4 Fg. (). The cocetrato dstrbuto s plotted versus y for C ˆ.4 W.7 M Da. Fs Pr Ec Sc5 /5 ad m τ5 5 π τ5 4 π τ π.5 4 Fg. (). The temperature dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Da. Fs Pr Ec Sc5 ad m.5.

9 The Ope Mathematcs Joural 9 Volume Mohamed Y. bou-zed.5.5 τ5 5 π τ5 4 π τ π.5 4 Fg. (). The cocetrato dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Da. Fs Pr Ec Sc5 ad m Fs Fs Fs.5 4 Fg. (4). The temperature dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Da. Pr Ec Sc5 /5 ad m W.4 W.7 W Fg. (5). The temperature dstrbuto s plotted versus y for C d.5 ˆ.4 M Da. Fs Pr Ec Sc5 /5 ad m.5.

10 No Darcy Porous Medum The Ope Mathematcs Joural 9 Volume.5.5 D a. D a. D a.5 4 Fg. (6). The temperature dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Fs Pr Ec Sc5 /5 ad m Ec Ec4 Ec.5 4 Fg. (7). The temperature dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Da. Fs Pr Sc5 /5 ad m Fg. (8). The temperature dstrbuto s plotted versus y for C d.5 ˆ W.7 M Da. Fs Pr Ec Sc5 /5 ad m.5.

11 The Ope Mathematcs Joural 9 Volume Mohamed Y. bou-zed.5.5 α. α.5 α.5 4 Fg. (9). The temperature dstrbuto s plotted versus y for C d.5.4 W.7 M Da. Fs Pr Ec Sc5 /5 ad m.5. 4 M M M5 4 Fg. (). The temperature dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 Da. Fs Pr Ec Sc5 /5 ad m.5..5 Pr Pr5 Pr Fg. (). The temperature dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Da. Fs Ec Sc5 /5 ad m.5. Fgs. (-4) are graphed to llustrate the effects of the effects of chemcal reacto Schmdt umber Sc ad reacto order m respectvely. It s foud that the cocetrato creases wth creasg m but decreases as ad Sc crease.

12 No Darcy Porous Medum The Ope Mathematcs Joural 9 Volume.5.5 δ δ δ6.5 4 Fg. (). The cocetrato dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Da. Fs Pr Ec Sc5 /5 ad m Sc Sc5 Sc.5 4 Fg. (). The cocetrato dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Da. Fs Pr Ec /5 ad m m.5 m.5 m Fg. (4). The cocetrato dstrbuto s plotted versus y for C d.5 ˆ.4 W.7 M Da. Fs Pr Ec Sc5 ad. CONCLUSION umercal study usg a explct fte dfferece scheme s made to obta the soluto of the usteady flow of MHD o-newtoa flud through a o-darcy porous medum wth heat ad mass trasfer. lso we take to our cosderato the effects of vscous ad Ohmc dsspato

13 4 The Ope Mathematcs Joural 9 Volume Mohamed Y. bou-zed ad chemcal reacto effects. I ths study a Carreau model s used as a o-newtoa flud ad the Darcy- Forchhemer-Brkma model s used to represet the flud trasport wth the porous medum. The covergece of the scheme s studed usg stablty method. The effect of the problem's parameters such as o-newtoa parameters ˆ ad W magetc parameter M Darcy umber D a Forchhemer umber F s Eckert umber E c Schmdt umber S c Pradtl umber P r reacto order m chemcal reacto ad the parameter of fluece of sucto \ blowg d are dscussed by a set of graphs. It s foud that the velocty ad temperature crease as Darcy umber D a creases but they decrease as Forchhemer umber F s creases. lso t s foud that the cocetrato decreases as Schmdt umber Sc ad chemcal reacto crease. For egeerg purpose the results of ths problem are of great terest petroleum applcatos as reservor of ol or gas flows atural gas producto ad ehaced ol producto. The flow of the Petroleum through the Porous groud represets a example of the moto of our flud especally the moto of the flud the earth's core. lso there are may applcatos of ths moto may felds such strophyscal Plasma MHD Metallurgcal processes ad Geophyscal. NOMENCLTURE reacto rate costat B Total magetc feld B o\ Costat C Cocetrato of the speces C E Ergu costat C p Specfc heat at costat pressure C s Cocetrato susceptblty D a Darcy umber E Electrc feld E c Eckert umber U C p (T w T ) F s Forchhemer umber C E U K p I J k c K p K T m Curret desty Thermal coductvty Costat permeablty Thermal dffuso rato Reacto order M Magetc parameter B P r S c Dmesoless power-lawdex Pradtl umber K T Schmdt umber D m t Tme T Temperature of the flud T m The mea temperature U o The referece velocty or free stream velocty V Velocty vector (u(y t) V o ) W Greek Symbols Wesseber g umber U ˆ No dmesoal parameter μ o The accelerato / decelerato parameter Chemcal reacto r e (C -C ) m- / U j Zero-shear-rate vscosty Stra-rate tesor K Thermal dffusvty μ μ m Tme costat Vscosty Magetc permeablty Kematc vscosty μ k c C p Frequecy of the oscllatg plate Secod varat of stra-rate tesor a j j Desty of the speces Electrcal coductvty of the flud Stress tesor the Carreau model. Superscrpts ad subscrpts w REFERENCES Free stream codto Wall or plate codto. [] Murthy SN. Effect of double dsperso o mxed covecto heat ad mass trasfer a o-darcy porous medum. SME J Heat Trasfer ; : 476.

14 No Darcy Porous Medum The Ope Mathematcs Joural 9 Volume 5 [] may S. Numercal treatmet for the solutos of problems of o- Newtoa fluds flow through dfferet geometrc surfaces. M. Sc. Thess. Faculty of Educato. Roxy Egypt: Shams Uversty 7. [] Forchhemer P. Wasserbewegug Durch Bode. Zet ver Deut Ig 9; 45: 78. [4] Brkma HC. calculato of the vscous force exerted by a flowg flud o a Dese swarm of partcles. ppl Sc Res 974; I: 7. [5] Vafa K. Boudary ad Ierta effects o flow ad heat trasfer porous meda. It J Heat Mass Trasfer 98; 4: 95. [6] Mokhtar Eldabe NT ad bou-zed MY. Numercal study of pulsatle MHD o-newtoa flud flow wth heat ad mass trasfer through a porous medum betwee two permeable parallel plates. Id J Mech Cot Math Sc 6; ():. [7] Nakayama. No-Darcy Couette flow a porous medum flled wth a Ielastc o-newtoa flud. Tras SME 99; 4: 64. [8] Eldabe NT Sallam SN. No-Darcy Couette flow through a porous medum of magetohydrodyamc vsco-elastc flud wth heat ad mass trasfer. Ca J Phys 5; 8: 4. [9] El-shehawy EF yma MF Elsayed ME. Perstaltc moto of a geeralzed Newtoa flud though a porous medum. J Phys Soc Jp ; 69(): 4. [] Ibrahm FS bdel Gad SM Subba R Gorla R. No-Darcy mxed covecto flow alog a vertcal plate embedded a o- Newtoa flud saturated porous medum wth surface mass trasfer. It J Num Method Heat Flud Flow ; : 97. [] Kumar M Nath G. No-Darcy mxed covecto power-law fluds alog a o-sothermal horzotal surface a porous medum. It J Eg Sc 4; 4: 5. [] Soudalgekar VM. Fte dfferece aalyss of traset free covecto wth mass trasfer o a sothermal vertcal flat plate. It J Eg Sc 98; 9: 757. [] Smth GD. Numercal soluto of partal dfferetal equatos. New York: Oxford Uversty press 985. Receved: November 8 Revsed: March 5 9 ccepted: Jue 9 Mohamed Y. bou-zed; Lcesee Betham Ope. Ths s a ope access artcle lcesed uder the terms of the Creatve Commos ttrbuto No-Commercal Lcese ( whch permts urestrcted o-commercal use dstrbuto ad reproducto ay medum provded the work s properly cted.