The Effect of Temperature Dependent Viscosity on MHD Natural Convection Flow from an Isothermal Sphere

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1 Journl o Applied Fluid echnics, Vol. 5, No., pp. 5-3, 0. Avilble online t.jmonline.net, ISSN , EISSN The Eect o Temperture Dependent Viscosit on HD Nturl Convection Flo rom n Isotherml Sphere.. oll, S.C. Sh nd.a. Hossin 3 Deprtment o Electricl Engineering & Computer Science, North South Universit, Dhk9, Bngldesh School o Chemistr, Phsics & echnicl Engineering, Queenslnd Universit o Technolog, George St., GPO Bo 434, Brisbne QLD 400, Austrli 3 Deprtment o themtic, COSATS Institute o Inormtion Technolog, Islmbd, Pkistn Corresponding Author Emil: mmmun@northsouth.edu, mmmun@gmil.com (Received Februr 8, 00; Accepted Jul 9, 00) ABSTRACT Lminr mgnetohdrodnmic (HD) nturl convection lo rom n isotherml sphere immersed in luid ith viscosit proportionl to liner unction o temperture hs been studied. The governing boundr ler equtions re trnsormed into non-dimensionl orm nd the resulting nonliner sstem o prtil dierentil equtions re reduced to convenient orm hich re solved numericll b to ver eicient methods, nmel, (i) Implicit inite dierence method together ith Keller bo scheme nd (ii) Direct numericl scheme. Numericl results re presented b velocit nd temperture distribution, stremlines nd isotherms o the luid s ell s het trnser chrcteristics, nmel the locl skin-riction coeicients nd the locl het trnser rte or ide rnge o mgnetohdrodnmic prmgnet nd viscosit-vrition prmeter. Keords: Nturl convection, HD, Temperture dependent viscosit, Finite dierence, Sphere. NOENCLATURE C p C g Gr k Nu Pr q T T T rdius o the sphere speciic het t constnt pressure skin-riction coeicient dimensionless strem unction ccelertion due to grvit Grsho number therml conductivit o the luid HD prmeter Nusselt Number Prndtl Number het lu t the surce temperture o the luid temperture o the mbient luid temperture t the surce u, v dimensionless velocit components û,vˆ dimensionl velocit components, is direction volumetric coeicient o therml epnsion strem unction shering stress viscosit-vrition prmeter * constnt densit o the luid reerence kinemtic viscosit (T) viscosit o the luid dimensionless temperture unction β 0 strength o mgnetic ield σ 0 electric conduction * constnt. INTRODUCTION A stud o the lo o electricll conducting luid in presence o mgnetic ield is importnt rom the technicl point o vie nd such tpes o problems hve received much ttention b mn reserchers. Vjrvelu nd Hdjinolou (997) studied the convective het trnser in n electricll conducting luid t stretching surce. As mentioned b Vjrvelu nd Hdjinolou (997), the rte o cooling nd, thereore, the desired properties o the end product cn be controlled b the use o electricll conducting luids nd the ppliction o mgnetic ield. The use o mgnetic ield hs been lso used in the process o puriiction o molten metls rom non-metllic inclusions. HD ree convection lo o visco-elstic luid pst n ininite porous plte s investigted b Chodhur nd Islm (000). Rptis nd Kousin

2 .. oll et l. / JAF, Vol. 5, No., pp. 5-3, 0. (98) hve investigted the problem o mgnetohdrodnmic ree convection lo nd mss trnser through porous medium bounded b n ininite verticl porous plte ith constnt het lu.. FORULATION OF PROBLE A sted to-dimensionl HD lminr ree convective lo rom uniorml heted sphere o rdius, hich is immersed in viscous nd incompressible luid hving temperture dependent viscosit. Here viscosit is the liner unction o the luid temperture. It is ssumed tht the surce temperture o the sphere T >T here T is the mbient temperture o the luid. The conigurtion considered is s shon in Fig.. here is the viscosit o the mbient luid nd * is deined s ollos * T (5b) here denotes the ilm temperture o the luid. We no introduce the olloing non-dimensionl vribles:, ˆ / 4 / v Gr Gr g T Gr / vˆ, T ˆ, u Gr T T, T T 3 (6) here (= /) is the reerence kinemtic viscosit nd Gr is the Grsho number nd is the nondimensionl temperture. Substituting Eq. (6) into Eqs. ()-(3) leds to the olloing non-dimensionl equtions Fig.. Phsicl model nd coordinte sstem. The equtions governing the lo re ˆ ˆ r rvˆ 0 () vˆ ˆ ˆ ˆ ˆ ˆ 00 g T T sin () T T k v ˆ ˆ ˆ C ˆ (3) p The boundr conditions o Eqs. () to (3) re v ˆ 0, T T, ˆ t 0 (4) 0, T T s ˆ (4b) here r ˆ sinˆ /,, vˆ long the ˆ, ˆ û re velocit components es, g is the ccelertion due to grvit, is the densit, (T) is the viscosit o the luid depending on the luid temperture T, is the coeicient o therml epnsion, k is the therml conductivit o the luid, 0 is the electricl conduction, β 0 is the strength o mgnetic ield. Out o the mn orms o viscosit vrition, hich re vilble in the literture, e ill consider onl olloing orm proposed b Chrrudeu (975) * T T (5) u u u v ru rv 0 u u sin u (7) (8) u v (9) Pr With the boundr conditions (4) become u v 0,, t 0 (0) u 0, 0, s (0b) here is the viscosit-vrition prmeter, is the mgnetic prmeter nd Pr is the Prndtl number hich re deined s in Eq. (), respectivel T c nd Pr k p T T, 00 / Gr 3. SOLUTION ETHODOLOGY () To solve the bove non-liner dierentil equtions e hve emploed to numericl methodologies: nmel, implicit inite dierence (IFD) method ith the Keller bo scheme nd the direct numericl scheme (DNS), hich re individull presented belo. 3. Implicit Finite Dierence ethod (IFD) Implicit inite dierence (IFD) method s irst introduced b Keller (978) nd elbortel described b Cebeci nd Brdsh (984). To solve Eqs. (7)-(9) using IFD subject to the boundr conditions (0), e ssume the olloing vribles,, r, () 6

3 .. oll et l. / JAF, Vol. 5, No., pp. 5-3, 0. here is the non-dimensionl strem unction deined in the usul s u, v (3) r r Substituting Eqs. ()-(3) into Eqs. (8)-(9) e get, ter some lgebr, the olloing trnsormed equtions 3 cos 3 sin sin - Pr sin (4) cos Along ith boundr conditions (5) 0,, t 0 (6) 0, 0, s (6b) The phsicl qulities o principle interest re shering stress in terms o the skin-riction coeicient nd the rte o het trnser in terms o the Nusselt number, hich cn be ritten, in non-dimensionl orm s C, U q Nu k T T (7) T here, ˆ ˆ q k (8) ˆ 0 ˆ 0 Using Eqs. (6), () nd (6), e get / 4 C Gr,0 NuGr, / Direct Numericl Scheme (9) (0) To incorporte the DNS method solving Eqs. (7)-(9) subject to the boundr conditions (0), e introduce the olloing ne set o trnsormtions. u X, Y, U, V v () Using Eq. () into Eqs. (7)-(0), e get U cos X V X X U 0 X sin X Y U U XU V U X Y U U sin X Y Y Y X U () (3) XU V X Y Pr Y The corresponding boundr conditions re (4) U V 0, t X 0 n Y (5) U V 0, t Y 0, X 0 (5b) U 0, 0 s Y, X 0 (5c) No Eqs. ()-(4) subject to the boundr conditions (5) re discretised or direct numericl scheme (DNS) using centrl-dierence or diusion terms nd the orrd-dierence or the convection terms, inll e get sstem o tri-digonl lgebric equtions olloing s: A D (6) i i Bii Cii i here, is the dependent vrible, A i, B i, C i re the tridigonl mtri elements nd D i re the nonhomogeneous terms o the ssocited mtri. The lgebric Eq. (6) hve been solved b Gussin elimintion technique. In computtion, the continuit eqution hs been solved directl or the norml velocit V b the olloing discretistion: Y U V j V j j j sin X (7) U cos X X Y X i U i, j U i, j X The computtion is strted rom X =, nd then mrches up implicitl. Here = /80 nd = re used or the X-nd Y- grids respectivel. No e re t the position to mesure o the phsicl quntities, nmel the shering stress nd the rte o het trnser rom the olloing dimensionless reltions: C Gr / 4 U X Y Y 0 NuGr / 4 Y Y 0 4. RESULT AND DISCUSSION (8) (9) In this stud e hve investigted the problem o HD lminr nturl convection lo nd het trnser rom n isotherml sphere ith temperture dependent viscosit. Here e hve considered the viscosit o the luid is proportionl to the liner unction o temperture tht mens i the temperture o the luid increses, the viscosit o the luid lso increses. This phenomenon generll occurs or smll Prndtl number Pr. For emple, the viscosit o ir is kg.m -.s -,.389 kg.m -.s -,.86 kg.m -.s - nd 3.65 kg.m -.s - t 00 0 K, 00 0 K, K nd K temperture respectivel. The viscosit o mmon NH 3 is kg.m -.s -,.886 kg.m -.s -, nd 6.49 kg.m -.s - t 0 0 K, K nd K respectivel (see Cebeci nd Brdsh, 984). Equtions (4)-(5) subject to the boundr conditions (6) re solved numericll using implicit inite- 7

4 .. oll et l. / JAF, Vol. 5, No., pp. 5-3, 0. Tble The results o C Gr /4 or dierent vlues o the viscosit vrition prmeter hile = nd Pr = C Gr /4 = =.0 = = 5.0 IFD DNS IFD DNS IFD DNS IFD DNS Tble The results o NuGr -/4 or dierent vlues o the viscosit vrition prmeter hile = nd Pr= NuGr -/4 = =.0 = = 5.0 IFD DNS IFD DNS IFD DNS IFD DNS dierence (IFD) together ith Keller bo scheme nd the Eqs. ()-(5) re solved b DNS. The numericl solutions strt t the loer stgntion point o the sphere, 0 nd proceed round the sphere up to the upper stgntion point,. Solutions re obtined or HD prmeter (=, 0.,,,.0) nd or ide rnge o vlues o the vrible viscosit prmeter (=,.0,, 5.0.). Since the vlues o (,0) or (U/Y) Y=0 nd (,0) or (/Y) Y=0 re knon rom the solutions o the coupled Eqs. (4)-(5) or ()-(4), numericl vlues o the shering stress in terms o skinriction coeicient C Gr /4 /(+) rom Eq. (9) or Eq. (8) nd the het trnser rte in terms o the Nusselt number Nu rom Eq. (0) or Eq. (9) re clculted rom loer stgntion point to upper stgntion point o the sphere. Numericl vlues o C Gr /4 /(+) nd NuGr /4 re depicted in Tbles - nd Fig.. It should be noted tht or constnt viscosit e recover the problem tht discussed b Hung nd Chen (987) considering Pr =.0 hich is not shon here. For the riting simplicit, e hve used the smbol C Gr /4 insted o C Gr /4 /(+). The eect o dierent vlues o mgnetic prmeter (=, 0.,,,.0) on locl skin riction coeicient C Gr /4 nd the locl Nusselt number NuGr /4 ith the viscosit-vrition prmeter =.0 nd or Pr = 0.73 re illustrted in Fig. Here e notice tht the greement beteen the results obtined b using the Keller bo method (IFD) nd the direct numericl scheme (DNS) is ecellent. It cn esil be seen tht ith the eect o mgnetic prmeter leds to decrese the locl skin riction coeicient C Gr /4 nd the locl Nusselt number NuGr /4. This phenomenon cn esil be understood rom the ct tht the incresing vlues o mgnetic prmeter, the Lorentz orce, hich oppose the lo, tht mens, decrese the velocit nd temperture grdient nd hence the locl skin-riction coeicient C Gr /4 nd the locl Nusselt number NuGr /. () C Gr / IFD DNS.0.0 NuGr -/ IFD DNS.0.0 Fig.. () Skin-riction coeicient Rte o het trnser or dierent vlues o hile Pr = 0.73 nd =.0 Oing to increse the vlues o increse the luid temperture ithin the boundr ler nd the ssocite 8

5 .. oll et l. / JAF, Vol. 5, No., pp. 5-3, 0. therml boundr ler becomes thicker. For incresing luid temperture, the temperture dierence beteen luid nd surce decreses nd the corresponding rte o het trnser NuGr /4 decreses. Here it should be noted tht the point o seprtion o the lo occurs t the upper stgntion point. The numericl vlues o the skin-riction coeicient C Gr /4 nd the locl Nusselt number NuGr /4, ginst the curvture prmeter or dierent vlues o viscosit-vrition prmeter (=,,.0,.0, 5.0) hile Pr = 0.73 (ir t 0 0 C nd tm pressure) re entered in Tble nd respectivel. We urther notice tht, the greement beteen the results obtined b the IFD method nd the DNS is ecellent. With the incresing vlues o the viscosit-vrition prmeter, it is seen tht the vlues o skin-riction coeicient C Gr /4 nd the Nusselt number NuGr /4 decrese. For incresing vlues o, the viscosit o the luid ithin the boundr ler increses hich retrds the luid motion, s results the corresponding skin-riction coeicient C Gr /4 decreses. For incresing vlues o the viscosit-vrition prmeter, the temperture o the luid increse hich is shon in Fig. 4b. Since the temperture o the luid increses nd hence the corresponding temperture dierence beteen the surce nd the luid enhnces. Due to higher temperture o the luid the rte o het trnser tht mens the Nusselt number NuGr /4 decreses. It is importnt to clculte or the eperimentlist tht t = /, the skin riction coeicient C Gr /4 nd the locl Nusselt number NuGr /4 decrese b % nd.38 % respectivel s increses rom to 5.0. Attention is no given to the eects o pertinent prmeters on the dimensionless velocit nd temperture in the lo ield, computed onl b the implicit inite dierence (IFD) method, nd these re presented grphicll in Figs 3 nd 4. Figure 3-3b displ results or the velocit nd temperture proiles, bsed on Eqs. (4) nd Eq. (5) ith the boundr conditions (6), or dierent vlues o mgnetic prmeter (=, 0.,,,.0) plotted ginst t = /3 hving Prndtl number Pr = 0.73 ith =.0. From the Fig. 3 it is seen tht, s the mgnetic prmeter increses, the velocit proile decrese nd the temperture proile increses slightl. The reson o this prcticl scenrio is tht the interction o the mgnetic ield nd the moving electric chrge crried b the luid induces orce hich tends to oppose the luid motion. But ner the surce o the clinder, velocit increses nd then decreses slol nd inll pproches to zero ccording to outer boundr condition. This implies tht there eists locl mimum o the velocit ithin the boundr ler. Figure 4-4b illustrte the velocit nd temperture distribution ginst the vrible or dierent vlues o the viscosit-vrition prmeter (=,.0,, 5.0) t = /3 hile Pr =0.73 nd =. It cn be observed tht the velocit decreses nd temperture distribution increses ith the incresing vlues o the viscosit-vrition prmeter,. It should be noted tht t ech vlue o the viscosit-vrition-prmeter, the velocit proile hs locl mimum 0.3 () / Fig. 3. () Velocit nd Temperture distribution or dierent vlues o hile Pr = 0.73 t = / () / Fig. 4. () Velocit nd Temperture distribution or dierent vlues o hile Pr = 0.73 t = /3. 9

6 .. oll et l. / JAF, Vol. 5, No., pp. 5-3, 0. vlue ithin the boundr ler. The mimum vlues o the velocit re , , 00, t =.3054,.65930,.7434, or =,.0,, 5.0 respectivel. The mimum velocit decreses b 3.66% s increses rom to 5.0. It lso be concluded tht the velocit boundr ler nd the therml boundr ler thickness enhnce or lrge vlues o. () strightorrd s cn be seen in this rme or = 5.0 in Fig. 5b nd in this cse the luid temperture increses slightl hich s lso noticed in Fig. 3b nd the surce het trnser rte reduced (see in Tble ). () Fig. 5. Stremlines or () = nd = 5.0 hile Pr = 0.73, = nd = 6. Figures 5 nd 6 illustrte the eect o the viscositvrition prmeter, on the development o stremlines nd isotherms, hich re plotted or Pr = 0.73 nd =. From Fig. 5, it is seen tht ithout eect o viscosit-vrition (i.e. = ) the nondimensionl vlue o m ithin the computtionl domin is bout.50 ner t the upper stgntion point ( ) o the sphere nd hen the boundr ler thickness is the highest, but m decreses ith the increment o nd it ttins bout.0 or = 5.0 (see Fig. 5b). This phenomenon ull coincides ith the erl discussion mde on Fig. 4, the luid slo don s increses nd the thickness o the velocit boundr ler lso decreses. The isotherm ptterns or corresponding vlues o re shon in Fig. 6. From these to rmes, e cn see tht the groth o therml boundr ler over the surce o the sphere in signiicnt. As increses rom the loer stgntion point ( ), the hot luid rises due to the grvit hence the thickness o the therml boundr ler,, increses. This phenomenon is ver Fig. 6. Isotherms or () = nd = 5.0 hile Pr = 0.73, = nd = CONCLUSION The eect o temperture-dependent viscosit on the HD nturl convection boundr ler lo rom n isotherml sphere hs been investigted theoreticll. Numericl solutions o the equtions governing the lo re obtined b using the ver eicient implicit inite dierence (IFD) method together ith Keller bo scheme nd b direct numericl scheme (DNS). From the present investigtion the olloing conclusions m be drn: Incresing the vlues o the mgnetic prmeter nd viscosit-vrition prmeter led to decrese the locl skin-riction coeicient C Gr /4 nd the locl Nusselt number, NuGr /4. It is seen the velocit distribution decrese s ell s the temperture distribution increses ith the incresing vlues o the mgnetic prmeter nd viscosit-vrition prmeter. The results hve demonstrted tht the ssumption o constnt luid properties m introduce severe errors in the prediction o the surce shering stress nd the rte o het trnser. 30

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