Casson fluid flow toward a vertical plate embedded in porous medium in presence of heat source/sink

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1 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 Casson lid low toward a vertical plate embedded in poros medim in presence o heat sorce/sink Manish Raj, Abha Jha and Anil Sharma 3,3 Department o Mathematics, Universit o RajasthanJaipr-34, India Department o Mathematics, JECRC Universit, Jaipr Abstract Casson lid low throgh a poros medim towards a vertical plate in presence o heat generation or absorption is considered in this analsis. he governing bondar laer eqations are ormlated and transormed into set o ordinar dierential eqation sing similarit transormation. Ordinar dierential eqations solved nmericall b shooting techniqe with orth order Rnge-Ktta method. Soltions or the velocit and temperatre were obtained or some special cases. he eect o Casson parameter, thermal Grasho nmber, Prandtl nmber on the velocit bondar laer and on the thermal bondar laer are stdied and plotted. Ke words: Casson lid, Vertical plate, Poros medim, Stretching sheet, Heat Generation/absorption, Flid viscosit.. Introdction Casson lid lows past a srace embedded in a satrated poros medim have received considerable attention becase o nmeros applications in engineering and geophsics. Crane [] std the low cased b stretching o a sheet. Man researchers sch a [-4] extended the mark o Crane [] b inclding the eect o heat and mass transer analsis nder dierent phsical sitations. Several athor have considered varios aspects o this problem and obtained similarit soltions [5-3].All the above mentioned stdies contined their discssions b assming the niorm lid viscosit. However, it is known that the phsical properties o lid ma change signiicantl with temperatre [4-8] the variations o properties with temperatre has several practical applications in the ield o metallrg and chemical engineering in the extrsion process, the heat treated materials traveling between a eed roll and wind-p roll or on conveor belt possess the eatre o moving continos srace. he increase o temperatre leads to a local increase in the transport phenomena b redcing the viscosit across the momentm bondar laer and so rate o heat transer at the wall is also aected. hereore, to predict the low Abha Jha Corresponding Athor behavior accratel it is necessar to take into accont the viscosit variation or incompressible lids. Gar et al. [9] and Mehta and Sood [] showed that, when this eect is inclded the low characteristics ma change sbstantiall compared to constant viscosit assmption. For lbricating lids heat generated b internal riction and the corresponding rise in the temperatre aects the viscosit o the lid and so that the lid viscosit no longer be assmed constant. Mkhapadha et al. [] investigated the MHD bondar laer low with variable lid viscosit over a heated stretching sheet. he eects o temperatre dependent viscosit and thermal condctivit on low and heat transer over a stretching srace in dierent low sitations and or dierent lids were considered b El-Aziz [], Dandpat et al. [3], Salem [4], Mkhapadha and Laek [5], Prasad et al. [6] etc. he increasing o temperatre leads to the increase in the transport phenomena b redcing the viscosit across the momentm bondar laer and de to which the heat transer rate at the wall is also aected. A new dimension is added to the above mentioned std o Mkhapadha et al. [] b considering the eects o poros media. Flows throgh poros media are o principal interest becase these are qite prevalent in natre sch tpe o low inds its applications in a broad spectrm o disciplinar covering chemical engineering to geophsics. Flow throgh lidsatrated poros medim is important in man technological application and it has increasing importance with the growth o geo-thermal energ sage and in astrophsical problems. In certain poros media applications, working lid heat generation or absorption eects are important. Representative stdied dealing with these eects has been reported b Athors sch as Gpta and Sridhar [7]. Abel and Veena [8]. Abel et al. [8] stdied the low and heat transer o a viscoelastic lid immersed in a poros medim over a nonisothermal stretching sheet and the lid viscosit ma assmed to var as a nction o temperatre. he present work deals with Casson lid low towards a vertical plate embedded in poros ISSN: Page 4

2 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 medim in presence o heat sorce/sink. he governing bondar laer eqation have been transomeed to a two-point bondar vale problem sing a local similarit approach and these have been solved nmericall. he eects o varios embedded parameters on lid velocit, temperatre and concentration have been shown graphicall. It is hoped that the reslts obtained will not onl provide sel inormation or applications, bt also serve as a complement to the previos stdies.. Mathematical Model x w v Fig.Phsical model o a bondar laer low over a vertical stretching srace. We consider stead two-dimensional orced convection low o a viscos incompressible lid past heated stretching sheet immersed in a poros medim along a vertical plate in pressre o Cassen lid. he x-axis is taken along the plate and -axis is normal to the plate and low is conined in hal plane g > as shown in Fig.. he present paper is concerned primaril with mathematical ormlation based on Darc s law, where the eects o a solid bondar and the inertial eects are neglected. hese eects become more signiicant near bondar and in a media with high porosit [3]. In this paper, we shall limit or consideration to lows where the non-linear archheimer orm is neglected bt the linear dring term describing the distribted bod orce extracted b poros medim is retained. Here, we have assmed that Renolds nmber is ver small (tpicall < ) [3]. We assme that the rheological eqation o state or an isotropic and incompressible low o a csson lid can be written as ij ( B B c e ij e where B is plastic dnamics viscosit o the casson lid, is the ield stress o lid, is the prodct o component o deormation rate with itsel, namel, = e ij e ij, e ij is the (i,j) the component o the deormation rate o c is the critical vale. ij c c he governing eqations o continit and energ eqation are given b (with the application o Darc s law) x v x x v v he bondar conditions are cx ; v k, w at Q C p K g () () (3) (4) ISSN: Page 43

3 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 ; as where and v are components o velocit respectivel in x and directions, is the temperatre, K is the coeicient o thermal disivit, Q is the dimensional heat generation/absorption ( > / < ), C p is coeicient o the speciic heat, is the lid densit, is the coeicient o lid viscosit, k is the permeabilit o the poros medim, is the thermal expansion coeicient, is Casson parameter, c ( > ) is constant, w is the niorm wall temperatre, is the temperatre ar awa rom the sheet and is gravitational acceleration. Proceeding with the analsis, we introdce the ollowing dimensionless variable and as well as the similarit variable c x, ) c x ( w where is the stream nction. he temperatre dependent lid viscosit is given b [3] a b( )] w where * is the constant vale o coeicient o viscosit ar awa rom the sheet and a,b are constant and b( > ), stream nction deined b (5) (6) v x Eqation o continit is satisied and eqation () and (3) redce to (7) a and A( ' ' ' ' ' A ' ' ' ' k a A( ' G (8) '' ' Pr he bondar conditions are ' ; at (9) () and ' as () Here A b ( w A is viscosit parameter. * is the permeabilit parameter k ck Swati Mkhopadha et al. [37] Pr k Q C p is the Prandtl nmber and c is the heat-sorce or sink parameter ISSN: Page 44

4 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 3 g x w G hermal Grasho nmber It is noted that when G = and this paper redces to the Swati Mkhapadha [37]. In order to assess the accrac o the nmerical method, comparison with these obtained b Swati Mkhapadha [37] are shown in able I. he reslts show good agreement. 3. Reslts and Discssion he sstem o non-linear ordinar dierential eqation (8) and (9) with bondar condition () and () is solved nmericall sing the shooting techniqe with orth-order Rnge-Ktta scheme. We gessing o ' ' ' b showing techniqe ntil the bondar conditions at ininit are satisied. he step size and =. is sed while obtaining the nmerical soltion and accrac p to seventh decimal place which ver sicient or convergence. In this method, we choose sitable inite vale which dependent on the vales o the parameter sed. he comptations were done b a programme which ses a smbolic and comptational compter langage matlab. (a) (b) ISSN: Page 45

5 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 Fig. Velocit proiles or several vales o viscosit variable parameter A (a) in case o non-poros medim and in absence o heat sorce/sink (a=,k=,pr=.5, =,Gt=, = )(b)in case o poros medim and in presence o heat sorce /sink (a=,k=.,pr=.5, =.,Gt=, = ) Fig 3 Velocit proiles or several vales o viscosit variable parameter A in case o non-poros medim and in absence o heat sorce/sink (a=,k=,pr=.5, =,Gt=, = ) (a) ISSN: Page 46

6 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 (b) Fig 4 Velocit proiles or several vales o permeabilit parameter k (a) in case o niorm viscosit and in absence o heat sorce/sink (a=,a=,pr=.5, =,Gt=, = ) (b) In case o variable viscosit and in absence o heat sorce /sink (a=,a=,pr=.5, =.,Gt=, = ) (a) ISSN: Page 47

7 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 (b) Fig 5 emperatre distribtion or several vales o permeabilit parameter k (a) in case o niorm viscosit and in absence o heat sorce/sink(a=,a=,pr=.5, =,Gt=, = )(b) In case o variable viscosit and in absence o heat sorce /sink (a=,a=,pr=.5, =.,Gt=, = ) Fig 6 emperatre distribtion or several vales o heat sorce/sink parameter in case o poros medim and variable viscosit,(a=,a=,k=.,pr=.5,gt=, = ) ISSN: Page 48

8 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 Fig 7 Velocit proile or several vales o prandtl nmber Pr in presence o poros medim, variable viscosit, heat sorce/sink and(a=,k=.,a=, =.,Gt=, = ) Fig 8 emperatre distribtion or several vales o prandtl nmber Pr in presence o poros medim, variable viscosit, heat sorce/sink (a=,k=.,a=, =.,Gt=, = ) ISSN: Page 49

9 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 (a) (b) Fig 9 (a) Velocit proile (b) emperatre distribtion or several vales o Grasho nmber (Gt) in presence o poros medim, variable viscosit, heat sorce/sink with in both case (a=,k=.,pr=.5, =.,A=, = )) ISSN: Page 5

10 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 (a) (b) Fig (a)velocit proile (b) emperatre distribtion or several vales o casson parameter in presence o poros medim, variable viscosit, heat sorce/sink with in both cases (a=,k=.,pr=.5, =.,A=,Gt=) he eects o temperatre-dependent lid viscosit on velocit distribtion and heat transer in case o non-poros media in absence o an heat sorce/sink. In ig. (a) and (b) velocit proiles are shown or dierent vales o A. Fig. (a) shown that lid velocit is ond to decrease p to the crossing over point with the increase in A bt ater the crossing over point it increases with increasing A. In Fig. (b) velocit ield is ond deca with increasing vale o or all vales o A considered. he combined eects o heat sorce and permeabilit parameter, in absence o thermal ISSN: Page 5

11 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 Grasho nmber and Cassan parameter at ininit are considered in this case. Fig.3 shown temperatre distribtion or several vales o viscosit variation parameter A in case o Grasho nmber at zero and Cassan parameter at ininit with non-poros medim and absence o heat sorce/sink.in this case temperatre decrease with increase o A. Now we concentrate on velocit and temperatre distribtion or the variation o permeabilit parameter k o the poros medim withot heat generation or absorption with Pr =.5, G =, =. Fig. 4(a) and 4(b) demonstrate the eects o permeabilit parameter k on velocit ield in the absence (A = ) and presence (A = ) o temperatre dependent lid viscosit parameter A respectivel with increasing k, lid velocit is ond to decrease [ig. 4(a) and 4(b)] is the porosit o the medim increases, the vale o k decreases. For decreasing k lid gets more space to low as a conseqence its velocit increases. Fig. 5(a) and 5(b) exhibit that ( ) is bondar laer increases with increasing permeabilit parameter k in both the cases the thermal bondar laer thickness becomes thinner with the decreasing the permeabilit parameter k. he eect o increasing vales o k opposes the low in the bondar laer region, which reslts in more heat transer rom the sheet to the lid. his is becase o presence o the poros medim to increase the resistance to the lid motion, this cases the lid velocit to decrease (Fig. 4(a) and Fig. 4(b)) and de to which there is rise in the temperatre in the bondar laer (Fig. 5(a) and 5(b)). In Fig. 6, eects o heat sorce/sink on the temperatre ield is shown, taking ixed vales or the parameters A =, k =., Pr =.5, = and G = and varios vales or internal heat generation/absorption parameter. In this case, temperatre ield increasing with the increase o heat sorce parameter. It is observed that the thermal bondar laer generates energ, which ceases the temperatre proiles to increase with the increasing vales o. he internal heat generation/absorption enhances/damps the heat transport. he heat generation sorce leads to a larger thermal dision laer that ma increase thermal bondar laer thickness, on the contrar, the thermal bondar laer thickness decreases or heat absorption sink. Fig.7 and Fig.8 depict the velocit and temperatre proiles or the eects o Prandtl nmber Pr on momentm and heat transer. Flid velocit decreases with increasing Prandtl nmber. An increase in Prandtl nmber redces the thermal bondar laer thickness. Prandtl nmber signiies the ratio o momentm disivit to thermal disivit. It can be noticed that as Pr decrease the thickness o the thermal bondar laer becomes greater than the thickness o the velocit bondar laer according to the wellknown relation Pr where is the thickness o the velocit bondar laer and is the thickness o the velocit bondar laer, so the thickness o the thermal bondar laer increases as Prandtl nmber Pr decreases, and hence temperatre proile decreases with increase o Prandtl nmber Pr. Fig. 8 implies that the increase o Prandtl nmber Pr reslts in a decrease o temperatre distribtion at a particlar point. his is de to the act that there wold be a decrease o thermal bondar laer thickness with increasing vales o Prandtl nmber Pr. emperatre distribtion asmptoticall approaches to zero in the ree strem region. In heat transer problems, the Prandtl nmber Pr controls the relative thickening o momentm and thermal bondar laers when Prandtl nmber Pr is small, it means that heat dises qickl compared to the velocit (momentm), which means that or liqid metals, the thickness o the thermal bondar laer is mch bigger than the momentm bondar laer. Flids with lower Prandtl nmber will passes higher thermal condctivities (and thicker thermal bondar laer strctres) so that heat can dise rom the sheet aster than or higher Pr lids (thinner bondar laers). Hence Prandtl nmber can be sed to increase the rate o cooling in condcting lows. In Fig. 9(b) it is observed that an increase in thermal Grasho nmber case a decrease in the thermal bondar laer thickness and conseqentl the lid temperatre decreases de to boanc eect. However, opposite tre in velocit proiles.he inlence o the Cassan parameter on the velocit and temperatre proiles is shown in Fig. (a) and Fig. (b) when a = ; A = ; k =.; =.; G = and Pr =.5. he magnitde o the velocit is greater in the case o a Casson lid when compared with a viscos lid. Hence in general, with an increase in, the velocit o the lid decreases or a stretching sheet. However, or a stretching sheet the opposite is tre in temperatre proiles case. able : Vales o skin-riction [ ''()] and wall temperatre gradient [ '()]or vales o k with a = ; A = ; =, Pr =, = and G =. K [ ''()] Present [ '()] Present [37] Swati std [37] Swati std ISSN: Page 5

12 International Jornal o Mathematics rends and echnolog (IJM) Volme 4 Nmber - December6 4. Reerences. Crane L J (97) Flow past a stretching plate. Z Angew Math Phs : Gpta P S, Gpta A S (977) Heat and mass transer on a stretching sheet with sction or blowing. Can J ChemEng 55: Chen C K, Char M I (988) Heat transer o a continos stretching srace with sction or blowing. J Math Anal Atrp/35 : Datta B K, Ra P, Gpta A S (985) emperatre ield in the low over a stretching sheet with niorm heat lx. IntCommn Heat Mass rans : Ishak A, Nazar R, Pop I (6) Mixed convection bondar laer in the stagnation point low toward a stretching vertical sheet. Meccanica 4: Ishak A, Nazar R, Pop I (7) Mixed convection on the stagnation point low towards a vertical continosl stretching sheet. ASME J Heat transer 9: Ishak A, Nazar R, Pop I (8) Mixed convection stagnation point low o a micropolar lid towards a stretching sheet. 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Int J EngSci 4: Bird R B, Stewart W E, Lightoot E N (96) ransport Phenomena. Wile, New York 35. Cortell R (5) Flow and heat transer o a lid throgh a poros medim over a stretching srace with internal heat generation/absorption and sction/blowing. Flid Dn Res 37: Sbhas Abel M, Nandeppanavar M M, Malkhd M B() Hdromagnetic bondar laer low and heat transer in viscoelastic lid over a continosl moving permeable stretching srace with non-niorm heat sorce/sink embedded in lid satrated poros medim. ChemEngCommn 97(5): doi:.8/ Swati Mkhopadha, G C Laak () Eects o variable lid viscosit and low past a heated stretching sheet embedded in a poros medim in presence o heat sorce/sink. Meccanica () 47: DOI ISSN: Page 53