Magnetic Effect on Free Convection in a Non-Darcy Porous Medium Saturated with Doubly Stratified Power-Law Fluid

Transcription

1 D. Sriivasacharya Magetic Effect o Free ovectio i a No-Darcy Porous Medium Saturated with Doubly Stratified Power-Law Fluid J. Praitha Natural covectio heat ad mass trasfer alog a vertical plate embedded i a doubly stratified power-law fluid saturated o-darcy porous medium with uiform heat ad mass h. RamReddy Natioal Istitute of Techology Waragal- Departmet of Mathematics Waragal-5000 A.P. Idia flu is preseted. The goverig partial differetial equatios are trasformed ito ordiary differetial equatios usig similarity trasformatios ad the solved umerically. The effects of magetic parameter stratificatio parameter ad power-law ide o the velocity temperature ad cocetratio are illustrated graphically. Keywords: hydromagetic effects free covectio o-darcy porous medium power-law fluid double stratificatio Itroductio Free ad forced covectio flows i a fluid saturated porous media are of great iterest because of their various egieerig scietific ad idustrial applicatios i heat ad mass trasfer which occurs i the fields of desig of chemical processig equipmet formatio ad dispersio of fog distributios of temperature ad moisture over agricultural fields ad groves of fruit trees ad damage of crops due to freezig ad pollutio of the eviromet grai storage systems heat pipes packed microsphere isulatio distillatio towers io echage colums subterraea chemical waste migratio solar power absorbers etc. A umber of studies have bee reported i the literature focusig o the problem of combied heat ad mass trasfer i porous media. The aalysis of covective trasport i a porous medium with the iclusio of o- Darcia effects has also bee a matter of study i recet years. Due to its importat applicatios i may fields a full uderstadig for combied heat ad mass trasfer by o-darcy atural covectio from a heated flat surface embedded i fluid saturated porous medium is meaigful. The iertia effect is epected to be importat at a higher flow rate ad it ca be accouted for through the additio of a velocity squared term i the mometum equatio which is kow as the Forchheimer s etesio of the Darcy law. A detailed review of covective heat trasfer i Darcia ad o-darcia porous media icludig a ehaustive list of refereces ca be foud i the book by Nield ad Beja (00). The study of flow heat ad mass trasfer i o-newtoia fluids has gaied much attetio from the researchers because of its egieerig ad idustrial applicatios such as the thermal desig of idustrial equipmet dealig with molte plastics polymeric liquids foodstuffs or slurries. Also the o-liear behavior of o- Newtoia fluids i porous matri is quite differet from that of Newtoia fluids i porous media. The predictio of heat or mass trasfer characteristics about atural covectio of o-newtoia fluids i porous media is very importat due to its practical egieerig applicatios such as oil recovery ad food processig. Several ivestigators have eteded the covectio of heat ad mass trasfer problems to fluids ehibitig o-newtoia rheology. Differet models have bee proposed to eplai the behavior of o-newtoia fluids. Amog these the power-law model which is merely a empirical relatioship betwee the stress ad velocity gradiets has bee successfully applied to o-newtoia fluids eperimetally. he ad he (988) have studied the atural covectio of a o-newtoia fluid about a horizotal cylider ad sphere i a porous medium. Pascal ad Pascal (997) have cosidered the free covectio i a o-newtoia fluid saturated Paper accepted November 00. Techical Editor: Môica Feijó Naccache porous medium with lateral mass flu. Free covectio heat ad mass trasfer of o-newtoia power law fluids with yield stress from a vertical flat plate i a saturated porous media was studied by Rami ad Aa (000). The flow of atural covectio heat ad mass trasfer of o-newtoia power law fluids with yield stress i porous media from a vertical plate with variable wall heat ad mass flues was cosidered by heg (00). Buoyat covectio of power-law fluid i a eclosure filled with heat-geeratig porous media was cosidered by Kim ad Hyu (00). The study of free covectio i boudary layer flows of power law fluids past a vertical flat plate with suctio/ijectio was doe by Sahu ad Mathur (99). Free covectio from a horizotal lie heat source i a power-law fluid-saturated porous medium was studied by Nakayama (993). There has bee a reewed iterest i MHD flow ad heat trasfer i porous ad clear domais due to the importat effect of magetic field o the boudary layer flow cotrol ad o the performace of may systems usig electrically coductig fluid such as MHD power geerators the coolig of uclear reactors plasma studies purificatio of molte metals from o-metallic iclusio geothermal eergy etractios etc. May problems of MHD Darcia ad o-darcia flow of Newtoia as well as o- Newtoia fluid i porous media have bee aalyzed ad reported i the literature. No-Darcy mied covectio i power-law fluids alog a o-isothermal horizotal surface i a porous medium has bee aalyzed by Kumari ad Nath (00). The Effect of magetic field o o-darcy aisymmetric free covectio i a power-law fluid saturated porous medium with variable permeability has bee cosidered by Masour ad El-Shaer (00). Stratificatio of fluid arises due to temperature variatios cocetratio differeces or the presece of differet fluids. I practical situatios where the heat ad mass trasfer mechaisms ru parallel it is iterestig to aalyze the effect of double stratificatio (stratificatio of the medium with respect to the thermal ad cocetratio fields) o the covective trasport i a fluid. The aalysis of free covectio i a doubly stratified medium is a fudametally iterestig ad importat problem because of its broad rage of egieerig applicatios. The applicatios iclude heat rejectio ito the eviromet such as lakes rivers ad the seas; thermal eergy storage systems such as solar pods ad heat trasfer from thermal sources such as the codesers of power plats. heg (009) cosidered the combied heat ad mass trasfer i atural covectio flow from a vertical wavy surface i a power-law fluid saturated porous medium with thermal ad mass stratificatio. Lami Narayaa ad Murthy (007) have cosidered free covective heat ad mass trasfer i a doubly stratified porous medium saturated with a power law fluid. From the literature survey it seems that the problem of atural covectio heat ad mass trasfer from vertical plate i o-darcy 8 / Vol. XXXIII No. Jauary-March 0 ABM

2 Magetic Effect o Free ovectio i a No-Darcy Porous Medium Saturated with Doubly Stratified Power-Law Fluid porous media saturated with power-law fluids with double stratificatio ad magetic effects has ot bee ivestigated so far. Thus this work aims to study the effects of double stratificatio o atural covectio i a power-law fluid saturated o-darcy porous medium with uiform heat ad mass flu. Nomeclature A slope of ambiet temperature B slope of ambiet cocetratio B magetic field stregth 0 cocetratio ambiet cocetratio 0 c emperical costat D solutal diffusivity f reduced stream fuctio G modified Darcy parameter g gravitatioal acceleratio K permeability costat k thermal coductivity Le lewis umber M magetic parameter N buoyacy ratio power-law ide q q heat Mass trasfers from the plate w m Ra the Darcy-Rayleigh umber T temperature T ambiet temperature 0 u v Darcia Velocity compoets i ad y directios y coordiates alog ad ormal to the plate Greek Symbols α thermal diffusivity β T β coefficiets of thermal ad solutal epasio η similarity variable θ dimesioless temperature φ dimesioless cocetratio ν kiematic viscosity μ magetic permeability e ρ desity of the fluid ψ stream fuctio σ electrical coductivity of the fluid ε ε thermal ad solutal stratificatio parameters Subscripts ambiet coditio Superscript. differetiatio with respect to η Mathematical Formulatio osider the two dimesioal free covectio flow of a electrically coductig fluid from the vertical flat plate i a doubly stratified o-newtoia power-law fluid saturated o-darcy porous medium. The -ais is take alog the plate ad y- ais ormal to it. The physical model ad coordiate systems are show i Fig.. The porous medium is cosidered to be homogeeous ad isotropic (i.e. uiform with a costat porosity ad permeability) ad is saturated with a fluid which is i local thermodyamic equilibrium with the solid matri. The fluid has costat properties ecept the desity i the buoyacy term of the balace of mometum equatio. The fluid flow is moderate ad the permeability of the medium is low so that the Forchheimer flow model is applicable ad the boudary effect is eglected. The plate is maitaied at costat heat flu q ad costat mass flu. w qm The ambiet medium is assumed to be vertically o-liearly stratified with respect to both temperature ad cocetratio i the form m T ( ) T 0 + A ad l ( ) 0 + B respectively where A ad B are costats ad varied to alter the itesity of stratificatio i the medium. A uiform magetic field is applied ormal to the plate. The magetic Reyolds umber is assumed to be small so that the iduced magetic field ca be eglected. Figure. Physical model ad coordiate system. Usig the Boussiesq ad boudary layer approimatios the goverig equatios for the power-law fluid are give by u v + 0 σ B μ K c K Kg + e u u + u ρν ν ν T u + v u + v D ( β T ( T T ) + β ( ) 0 ) () () α (3) where u ad v are Darcia velocity compoets alog ad y directios respectively T is the temperature is the cocetratio is the power-law ide K is the permeability c is a empirical costat g is the acceleratio due to gravity ν is the kiematic viscosity ρ is the desity α is the thermal diffusivity ad D is the solutal diffusivity of the medium σ is the electrical coductivity of the fluid μ is the magetic permeability is the stregth of the e magetic field βt is the coefficiet of thermal epasio ad β is the coefficiet of solutal epasio. The boudary coditios are v 0 q w k q m D at y 0 (5a) B 0 u 0 T T ( ) ( ) as y () (5b) J. of the Braz. Soc. of Mech. Sci. & Eg. opyright 0 by ABM Jauary-March 0 Vol. XXXIII No. / 9

3 where k is the thermal coductivity of the fluid. I view of the cotiuity of Eq. () we itroduce the stream ( 0) 0 θ ( 0) φ ( 0) fuctio ψ by ( ) 0 θ( ) 0 ( ) 0 f (a) f φ (b) ψ u ψ v Substitutig Eq. () i Eqs. ()-() ad the usig the followig similarity trasformatios y η ( + ) Ra θ () η φ () η k ψ α Ra ( T T ( ) ) D q w ( ( ) ) q m we get the followig system of equatios ( ) + M f + G ( f ) θ + Nφ ( + ) f ( + ) Ra ( + ) Ra () η f (8) θ ( f θ ( + ) f θ + ε f ) ( + ) Le φ ( + ) () (7) (9) ( f φ ( + ) f φ + ε f ) (0) Numerical Procedure The flow Eq. (8) coupled with the eergy ad cocetratio Eqs. (9) ad (0) costitute a set of oliear o-homogeeous differetial equatio for which closed-form solutio caot be obtaied. Hece the problem has bee solved umerically usig shootig techique alog with fourth order Ruge-Kutta itegratio. The basic idea of shootig method for solvig boudary value problem is to try to fid appropriate iitial coditio for the computed solutio hit the target so that the boudary coditios at other poits are satisfied. Furthermore the higher order o-liear differetial equatios are coverted ito simultaeous liear differetial equatios of first order ad they are further trasformed ito iitial valued problem applyig the shootig method icorporatig fourth order Ruge-Kutta method. The iterative solutio procedure was carried out util the error i the solutio became less tha a predefied tolerace level. The o-liear differetial equatios (8)-(0) are coverted ito the followig system of liear differetial equatios of first order by the substitutio { f θ φ f θ φ } { z z z z z z } dz z dz dz dz 3 z5 z 5 z + N z ( z ) + M + G z 3 5 where the primes idicate partial differetiatio with respect to η aloe dz5 z3z ( + ) zz5 + εz ( + ) K g β q Ra T w is the Darcy-Reyleigh umber is the α kν dz Le[ z3z ( + ) zz + ε z ] ( + ) σ μ B K ( ) power-law ide parameter e 0 ( ) ( + ) M Ra ρ ν α The boudary coditios i terms of z z z3 z z5 z α is the magetic parameter Le is the Lewis umber D z 0) 0 z ( ) 0 z ( ) 0 G c K ν α parameter ( ) ( ) ( + ) ad Ra β N βt q k m q D w is the modified Darcy is the buoyacy ratio. ( + ) k ( + ) ( ) Ra ( + ) D ( + ) ( ) ε ad ε Ra q w q m are the thermal ad solutal stratificatio parameters. These parameters ε ad ε will be idepedet of oly whe m l i.e. whe ( + ) T ( ) T 0 + A ad ( + ) (+ ) ( ) 0 + B allowig similarity solutio. The boudary coditios (5) i terms of f θ ad φ become ( 3 are () z ) 0 z (0) z (0) (3) ( 5 Here η at is take as η ad chose large eough so that ma the solutio shows little further chage for η larger tha η ma. As the iitial values for z z 3 ad z are ot specified i the boudary coditios (3) assume some values for z (0) z 3 (0) ad th z (0). The the equatios () are itegrated usig the order Ruge-Kutta method from η 0 to η η over successive steps ma Δ η. The accuracy of the assumed iitial values z (0) z 3 (0) ad z is the checked by comparig the calculated values of (0) z (0) z (0) ad z (0) at with their give value at 3 η η ma η η i ma (3). If a differece eists aother set of iitial values for z (0) z (0) ad must be assumed ad the process is repeated. This 3 z (0) 0 / Vol. XXXIII No. Jauary-March 0 ABM

4 Magetic Effect o Free ovectio i a No-Darcy Porous Medium Saturated with Doubly Stratified Power-Law Fluid process is cotiued util the agreemet betwee the calculated ad the give coditio at η η ma is withi the specified degree of accuracy. I the preset study η ma has bee suitably chose at each time such that the velocity temperature ad cocetratio profiles approach zero at the outer edge of the boudary layer. Etesive calculatios have bee performed to obtai the wall velocity temperature ad cocetratio fields for a wide rage of parameters. The effect of thermal stratificatio parameter solutal stratificatio parameter magetic parameter o-darcy parameter ad power law ide parameter is studied o the velocity temperature ad cocetratio fields for uiform wall heat ad mass flu coditio is plotted for some selected combiatios of parameter values. Results ad Discussio The o-dimesioal velocity f '( temperature θ (η ) ad cocetratio φ ( are plotted for N 0.5 M Le G 0. ε 0. i Figs. - with varyig thermal stratificatio parameter. It ca be observed from Fig. that the velocity of the fluid decreases with the icrease of thermal stratificatio parameter. The thermal stratificatio reduces the effective covective potetial betwee the heated plate ad the ambiet fluid i the porous medium. Hece the thermal stratificatio effect reduces the velocity i the boudary layer. It ca be oted from Fig. 3 that the temperature of the fluid is decreased with the icrease i the value of the thermal stratificatio parameter. Whe the thermal stratificatio effect is cosidered the effective temperature differece betwee the plate ad the ambiet fluid will decrease. Therefore the thermal boudary layer is thickeed ad the temperature is reduced. It ca be foud from Fig. that the cocetratio of the fluid is icreased with icrease i the value of the thermal stratificatio parameter. The effect of solutal stratificatio parameter o the odimesioal velocity f '( temperature θ ( ad cocetratio φ( for N 0.5 M Le G 0. ε 0.8 is depicted i Figs It is observed from Fig. 5 that the fluid velocity is decreased with icrease i the value of solutal stratificatio parameter. It ca be see from Fig. that the temperature of the fluid i the medium is icreased with icrease i the value of the solutal stratificatio parameter. It ca be foud from Fig. 7 that the cocetratio of the fluid is decreased by icreasig the value of the solutal stratificatio parameter. The variatio of the o-dimesioal velocity f '( temperature θ ( ad cocetratio φ(η ) for N 0.5 Le G.0 ε 0. ε 0.8 with magetic parameter is show i Figs It ca be observed from Fig. 8 that the velocity of the fluid is decreased with icrease i the value of the magetic parameter. This is due to the fact that the itroductio of a trasverse magetic field ormal to the flow directio has a tedecy to create the drag kow as the Loretz force which teds to resist the flow. Hece the horizotal velocity profiles decrease as the magetic parameter M icreases. It ca be foud from Fig. 9 that icrease i the value of the magetic parameter icreases the temperature of the fluid i the medium. It ca be see from Fig. 0 that the cocetratio of the fluid is icreased by icreasig the value of the magetic parameter. As eplaied above the trasverse magetic field gives rise to a resistive force kow as the Loretz force of a electrically coductig fluid. This force makes the fluid eperiece a resistace by icreasig the frictio betwee its layers ad thus icreases its temperature ad cocetratio. The effect of o-darcy parameter o the o-dimesioal velocity f '( temperature θ ( ad cocetratio φ(η ) for N 0.5 M Le ε 0. ε 0.8 is plotted i Figs. -3. It is observed from Fig. that velocity of the fluid is decreased with icrease i the value of the o-darcy parameter. The icrease i o-darcy parameter implies that the porous medium is offerig more resistace to the fluid flow. This results i reductio i the velocity profiles. It ca be oted from Fig. that by icreasig the value of the o-darcy parameter there is a icrease i the temperature i the fluid medium. It ca be see from Fig. 3 that the cocetratio of the fluid is icreased if there is a icrease i the value of the o-darcy parameter. The icrease i o-darcy parameter reduces the itesity of the flow ad icreases the thermal ad cocetratio boudary layer thickesses. The o-dimesioal velocity f '( temperature θ (η ) ad cocetratio φ ( for N 0.5 M Le 0.5 G 0. ε 0. ε 0.8 with a variatio i power law ide parameter is plotted i Figs. -. It is observed from Fig. that the fluid velocity is icreased with icrease i the value of the power law ide parameter. The effect of the icreasig values of the power law ide is to icrease the horizotal boudary layer thickess. That is the thickess is much smaller for shear thiig (pseudo plastic; < ) fluids tha that of shear thickeig (dilatats; > ) fluids. I the case of a shear thiig fluid ( < ) the shear rates ear the walls are higher tha those for a Newtoia fluid. It ca be see from Fig. 5 that the temperature i the fluid is decreased with icrease i the value of the power law ide parameter. Icreasig the values of the power law ide leads to thiig of the thermal boudary layer thickess. It ca be foud from Fig. that the cocetratio of the fluid icreases with icrease i the value of the power law ide parameter. Icreasig the power-law ide () teds to retard the flow ad icrease the solutal boudary-layer thickess. oclusios I this paper a boudary layer aalysis for free covectio heat ad mass trasfer alog a vertical plate i a o-darcy porous media saturated with power-law fluid with uiform heat ad mass flu coditios i the presece of magetic field ad double stratificatio is preseted. Usig the similarity variables the goverig equatios are trasformed ito a set of ordiary differetial equatios where umerical solutio has bee preseted for a wide rage of parameters. The higher values of the thermal stratificatio parameter result i lower velocity ad temperature distributios but higher cocetratio distributio. The higher values of the solutal stratificatio parameter result i lower velocity ad cocetratio distributios but higher temperature distributio. A icrease i the values of the magetic parameter results i lower velocity distributio but higher temperature ad cocetratio distributios. The same ature ca be foud i the case of o-darcy parameter. Also the higher values of the power-law ide umber result i lower velocity ad temperature distributios but higher cocetratio distributio withi the boudary layer. Refereces rd Nield D.A. ad Beja A. 00 ovectio i porous media 3 editio New-York Spriger-Verlag. he H.T ad he.k. 988 Natural covectio of a o- Newtoia fluid about a horizotal cylider ad sphere i a porous medium It. ommu. Heat Mass Trasfer Vol. 5 pp Pascal J.P. ad Pascal H. 997 Free covectio i a o-newtoia fluid saturated porous medium with lateral mass flu It. J. No-Liear Mechaics Vol. 3 pp J. of the Braz. Soc. of Mech. Sci. & Eg. opyright 0 by ABM Jauary-March 0 Vol. XXXIII No. /

5 Rami Y.J. ad Aa S.M. 000 Free covectio heat ad mass trasfer of o-newtoia power law fluids with yield stress from a vertical flat plate i a saturated porous media It. ommu. Heat Mass Trasfer Vol. 7 pp hig-yag heg 00 Natural covectio heat ad mass trasfer of o-newtoia power law fluids with yield stress i porous media from a vertical plate with variable wall heat ad mass flues It. ommu. Heat Mass Trasfer Vol. 33 pp. 5-. Kim G.B ad Hyu J.M 00 Buoyat covectio of power-law fluid i a eclosure filled with heat-geeratig porous media Numerical Heat Trasfer Part A: Applicatios Vol. 5 pp Sahu A.K. ad Mathur M.N. 99 Free covectio i boudary layer flows of power law fluids past a vertical flat plate with suctio/ijectio Idia Joural of Pure ad Applied Mathematics Vol p. Nakayama A. 993 Free covectio from a horizotal lie heat source i a power-law fluid saturated porous medium It. J. of Heat ad Fluid Flow Vol. pp Figure. ocetratio profiles for various values of ε with N 0.5 M Kumari M. ad Nath G. 00 No-Darcy mied covectio i Le G 0. ε 0.. power-law fluids alog a oisothermal horizotal surface i a porous medium It. J. of Egieerig Sciece Vol. pp Masour M.A. ad El-Shaer N.A. 00 Effect of magetic field o o-darcy aisymmetric free covectio i a power-law fluid saturated porous medium with variable permeability Joural of Magetism ad Magetic Materials Vol. 50 pp hig-yag heg 009 ombied heat ad mass trasfer i atural covectio flow from a vertical wavy surface i a power-law fluid saturated porous medium with thermal ad mass stratificatio It. omm. Heat Mass Trasfer Vol. 3 pp Lakshmi Narayaa P.A ad Murthy P.V.S.N. 007 Free covective heat ad mass trasfer i a doubly stratified porous medium saturated with a power law fluid It. J. of Fluid Mechaics Research Vol. 3 pp Figure 5. Velocity profiles for various values of ε with N 0.5 M Le G 0. ε 0.8. Figure. Velocity profiles for various values of ε with N 0.5 M Le G 0. ε 0.. Figure. Temperature profiles for various values of ε with N 0.5 M Le G 0. ε 0.8. Figure 3. Temperature profiles for various values of ε with N 0.5 M Le G 0. ε 0.. Figure 7. ocetratio profiles for various values of ε with N 0.5 M Le G 0. ε 0.8. / Vol. XXXIII No. Jauary-March 0 ABM

6 Magetic Effect o Free ovectio i a No-Darcy Porous Medium Saturated with Doubly Stratified Power-Law Fluid Figure 8. Velocity profiles for various values of M with N 0.5 Le G ε 0. ε 0.8. Figure. Velocity profiles for various values of G with N 0.5 Le M ε 0. ε 0.8. Figure 9. Temperature profiles for various values of M with N 0.5 Le G ε 0. ε 0.8. Figure. Temperature profiles for various values of G with N 0.5 Le M ε 0. ε 0.8. Figure 0. ocetratio profiles for various values of M with N 0.5 Le G ε 0. ε 0.8. Figure 3. ocetratio profiles for various values of G with N 0.5 Le M ε 0. ε 0.8. J. of the Braz. Soc. of Mech. Sci. & Eg. opyright 0 by ABM Jauary-March 0 Vol. XXXIII No. / 3

7 Figure. Velocity profiles for various values of with N 0.5 Le 0.5 G 0. M ε 0. ε 0.8. Figure. ocetratio profiles for various values of with N 0.5 Le 0.5 G 0. M ε 0. ε 0.8. Figure 5. Temperature profiles for various values of with N 0.5 Le 0.5 G 0. M ε 0. ε 0.8. / Vol. XXXIII No. Jauary-March 0 ABM