Transient Laminar MHD Free Convective Flow past a Vertical Cone with Non-Uniform Surface Heat Flux

Transcription

1 Nonlinear Analysis: Modelling and Conrol, 29, Vol. 4, No. 4, Transien Laminar MHD Free Convecive Flow pas a Verical Cone wih Non-Uniform Surface Hea Flux Bapuji Pullepu, A. J. Chamkha 2 Deparmen of Mahemaics, Sahyabama Universiy Chennai, India-69 bapujip@yahoo.com 2 Manufacuring Engineering Deparmen The Public Auhoriy for Applied Educaion and Training Shuweikh, 7654 Kuwai achamkha@yahoo.com Received: Revised: Published online: 29-- Absrac. Numerical soluion of unseady laminar free convecion from an incompressible viscous fluid flow pas a verical cone wih non-uniform surface hea flux q w(x) = ax m varying as a power funcion of he disance from he apex of he cone (x = ) in he presence of a ransverse magneic field applied normal o he surface is considered. The dimensionless governing coupled parial differenial boundary layer equaions are formulaed and solved numerically using an efficien and uncondiionally sable finie-difference scheme of he Crank-Nicolson ype. The numerical resuls are validaed by comparisons wih previously published work and are found o be in excellen agreemen. The velociy and emperaure fields have been sudied for various combinaions of physical parameers (Prandl number P r, exponen and magneic parameer M). The local as well as he average skin-fricion parameer and he Nussel number are also presened and analyzed graphically. Keywords: finie-difference mehod, free convecion, MHD, non-uniform surface hea flux, verical cone, unseady flow. a consan M magneic parameer B magneic filed srengh Nu X non-dimensional local Nusel number f () local skin-fricion in [6] NU non-dimensional average Nusel f (η) dimensionless velociy number in X-direcion in [6] Pr Prandl number F () local skin fricion in [2] q w rae of hea ransfer per uni area Gr L Grashof number R dimensionless local radius of he cone g acceleraion due o graviy r local radius of he cone k hermal conduciviy T emperaure L reference lengh T dimensionless emperaure m exponen in power law variaion ime in surface hea flux dimensionless ime 489

2 Bapuji Pullepu, A. J. Chamkha U dimensionless velociy X dimensionless spaial co-ordinae in X-direcion along cone operaor u velociy componen x spaial co-ordinae along cone in x-direcion generaor V dimensionless velociy Y dimensionless spaial co-ordinae in Y -direcion along he normal o he cone generaor v velociy componen y spaial co-ordinae along he normal in y-direcion o cone generaor Greek symbols α hermal diffusiviy /Φ () local Nussel number in [2] β volumeric hermal expansion µ dynamic viscosiy η dimensionless independen ν kinemaic viscosiy variable in [6] τ X dimensionless local skin-fricion ρ densiy parameer σ elecrical conduciviy of τ dimensionless average he fluid skin-fricion parameer φ semi verical angle of he cone θ() emperaure in [6] Subscrips w condiion on he wall free sream condiion Inroducion Naural convecion flows under he influence of a graviaional force have been invesigaed mos exensively because hey occur frequenly in naure as well as in science and engineering applicaions. When a heaed surface is in conac wih he fluid, he resul of emperaure difference causes buoyancy force, which induces naural convecion hea ransfer. From a echnological poin of view, he sudy of convecion hea ransfer from a cone is of special ineres and has wide range of pracical applicaions. Mainly, hese ypes of hea ransfer problems deal wih he design of spacecrafs, nuclear reacor, solar power collecors, power ransformers, seam generaors and ohers. Since 953, many invesigaions [ 2] have developed similariy and non-similariy soluions for axi-symmerical problems for naural convecion flows over a verical cone in seady sae. Recenly, Bapuji and Ekambavanan [3] have numerically sudied he soluions of seady flows pas plane and axi-symmerical shape bodies. Also, Bapuji e al. [4, 5] have numerically sudied he problem of ransien naural convecion from a verical cone wih isohermal and non-isohermal surface emperaure using an implici finie-difference mehod. Recenly hea flux applicaions are widely used in indusries, engineering and science fields. Hea flux sensors can be used in indusrial measuremen and conrol sysems. Examples of few applicaions are deecion fouling (Boiler Fouling Sensor), monioring of furnaces (Blas Furnace Monioring/General Furnace Monioring) and flare monioring. Use of hea flux sensors can lead o improvemens in efficiency, sysem safey and mod- 49

3 Transien Laminar MHD Free Convecive Flow eling. The sudies [6 3] have considered problems of flow pas a verical cone/frusum of a cone in he case of uniform/non-uniform surface hea flux wih porous/non-porous medium. Recenly, Bapuji e al. [3] have numerically sudied he problem of ransien naural convecion from a verical cone wih non-uniform surface hea flux using an implici finie-difference mehod. All he above invesigaions [ 3] do no include deal wih MHD effec. MHD flow and hea ransfer is of considerable ineres because i can occur in many geohermal, geophysical, echnological, and engineering applicaions such as nuclear reacors and ohers. The geohermal gases are elecrically conducing and are affeced by he presence of a magneic field. Vajravelu and Nayfeh [32] sudied hydromagneic convecion from a cone and a wedge wih variable surface emperaure and inernal hea generaion or absorpion. Chamkha [33] considered he problem of seady-sae laminar hea and mass ransfer by naural convecion boundary layer flow around a permeable runcaed cone in he presence of magneic field and hermal radiaion effecs, non-similar soluions were obained and solved numerically by an implici finie-difference mehodology. Takhar e al. [34] developed he problem of unseady mixed convecion flow over a verical cone roaing in an ambien fluid wih a ime-dependen angular velociy in he presence of a magneic field. The coupled nonlinear parial differenial equaions governing he flow have been solved numerically using an implici finie-difference scheme. Afify [35] sudied he effecs of radiaion and chemical reacion on seady free convecive flow and mass ransfer of an opically dense viscous, incompressible and elecrically conducing fluid pas a verical isohermal cone in he presence of a magneic field. Afify s similariy equaions were solved numerically using a fourh-order Runge-Kua scheme wih he shooing mehod. Laer, Chamkha and Al-Mudhaf [36] focused on he sudy of unseady hea and mass ransfer by mixed convecion flow over a verical permeable cone roaing in an ambien fluid wih a ime-dependen angular velociy in he presence of a magneic field and hea generaion or absorpion effecs wih he cone surface is mainained a variable emperaure and concenraion. Numerical soluions obained, solving by he parial differenial equaions using an implici, ieraive finie-difference scheme. Recenly, Elkabeir and Modaher [37] sudied chemical reacion, hea and mass ransfer on MHD flow over a verical isohermal cone surface in micropolar fluids wih hea generaion and absorpion. Their numerical soluions were obained by using he fourh-order Runge-Kua mehod wih shooing echnique. The presen work is devoed o he sudy of ransien laminar free convecion flow pas a verical cone wih non-uniform surface hea flux in he presence of a magneic field. In order o check he accuracy of he numerical resuls, he presen resuls are compared wih he available resuls of Lin [6], Pop and Waanabe [7], Na and Chiou [23], Hossain and Paul [2] and are found o be in excellen agreemen. 2 Mahemaical analysis The problem of axi-symmerical, unseady, laminar free convecion flow of a viscous incompressible elecrically-conducing fluid pas a verical cone wih non-uniform sur- 49

4 Bapuji Pullepu, A. J. Chamkha face hea flux under he influence of ransversely applied magneic field is formulaed mahemaically in his secion. The following assumpions concerning he magneic field and he geomery are made:. The magneic field is consan and is applied in a direcion perpendicular o he cone surface. 2. The magneic Reynolds number is small so ha he induced magneic field is negleced and herefore, does no disor he magneic field. 3. The coefficien of elecrical conduciviy is a consan hroughou he fluid. 4. The Joule heaing of he fluid (magneic dissipaion) and viscous dissipaion are negleced. 5. The Hall effec of magneohydrodynamics is negleced. 6. The sysem is considered as axi-symmerical. 7. The effec of pressure gradien is assumed negligible. The coordinae sysem is chosen (as shown in Fig. ) such ha x measures he disance along surface of he cone from he apex (x = ) and y measures he disance normally ouward. x u r v g B y Fig.. Physical model and co-ordinae sysem. Here, φ is he semi verical angle of he cone and r is he local radius of he cone. Iniially ( ), i is also assumed ha he cone surface and he surrounding fluid, which is a res, have he same emperaure T. Then a ime >, i is assumed ha hea is supplied from cone surface o he fluid a he rae q w (x) = ax m and i is mainained a his value wih m being a consan. The fluid properies are assumed consan excep 492

5 Transien Laminar MHD Free Convecive Flow for densiy variaions, which induce buoyancy force erm in he momenum equaion. The governing boundary layer equaions of coninuiy, momenum and energy under Boussinesq approximaion are as follows: Equaion of coninuiy x (ru) + (ru) =, () y equaion of momenum u + u u x + v u y = gβ( T T ) 2 u cosφ + ν y 2 σb2 ρ equaion of energy T + u T x + v T y = T α 2 The iniial and boundary condiions are u, (2) y 2. (3) : u =, v =, T = T for all x and y, > : u =, v =, T / y = q w (x)/k a y =, u =, T = T a x =, u, T T as y. Furher, we inroduce he following non-dimensional variables: X = x L, Y = y ( ) ν L, = L 2 Gr2/5 L, R = r L, U = ( L ν Gr 2/5 L ) u, V = ( L ν Gr /5 L ) v, T = T T L[q w (L)/k] Gr/5 L, M = σb2 L2 Gr 2/5, µ (4) (5) where Gr L = gβ[q w (L)]L 4 cosφ/ν 2 k is he Grashof number based on L, Pr = ν/α is he Prandl number and r = xsin φ. Equaions () (3) can hen be wrien in he following non-dimensional form: X (RU) + (RV ) =, Y (6) U + U U X + V U Y = T + 2 U MU, Y 2 (7) T + U T X + V T Y = 2 T Pr Y 2, (8) 493

6 Bapuji Pullepu, A. J. Chamkha where M is he magneic parameer. The corresponding non-dimensional iniial and boundary condiions are : U =, V =, T = for all X and Y, > : U =, V =, T/ Y = X m a Y =, U =, T = a X =, U, T T as Y. (9) Once he velociy and emperaure profiles are known, i is ineresing o sudy he local as well as he average skin-fricion parameer and he rae of hea ransfer a seady sae and ransien levels. The local non-dimensional skin-fricion parameer τ X and he local Nussel number Nu X are given by τ X = Gr 3/5 L ( ) U Y Y =, Nu X = X Gr/5 L T Y = ( T ). () Y Y = Also, he non-dimensional average skin-fricion parameer τ and he average Nussel number Nu can be wrien as τ = 2Gr 3/5 L X ( ) U Y Y = dx, Nu = 2Gr /5 L X T Y = ( T ) dx. () Y Y = The derivaives involved in equaions () and () are obained using five-poin approximaion formula and hen he inegrals are evaluaed using Newon-Coes closed inegraion formula. 3 Soluion procedure The governing parial differenial equaions (6) (8) are unseady, coupled and non-linear wih iniial and derivaive boundary condiions (9). They are solved numerically by an implici finie-difference mehod of Crank-Nicolson ype as described in deail by Bapuji e al. [4, 5]. The region of inegraion is considered as a recangle wih sides X max =. and Y max = 26, where Y max corresponds o Y = which lies very well ouside he momenum and hermal boundary layers. The finie-difference scheme is uncondiionally sable as explained by Bapuji e al. [5]. Sabiliy and compaibiliy ensure he convergence. 4 Resuls and discussion In order o prove he accuracy of our numerical resuls, he presen resuls for he seadysae flow condiions a X =. when M = (i.e. he absence of magneic field effec) are compared wih available soluions from he open lieraure. The velociy and 494

7 Transien Laminar MHD Free Convecive Flow emperaure profiles of he cone for Pr =.72 are displayed in Fig. 2 and he numerical values of local skin-fricion τ X and emperaure T for differen values of Prandl number are shown in Table and are compared wih similariy soluions of Lin [6] in seady sae using suiable ransformaion (Y = (2/9) /5 η, T = (2/9) /5 ( θ()), U = (2/9) 3/5 f (η), τ X = (2/9) 2/5 f ()). In addiion, he local skin-fricion τ X and he local Nussel number Nu X for differen values of Prandl number when hea flux gradien power m =.5 a X =. in seady sae are compared wih he non-similariy resuls of Hossain and Paul [2] in Table 2. I is observed ha he resuls are in good agreemen wih each oher. I is also noiced ha he presen resuls agree well wih hose of Pop and Waanabe [7], Na and Chiou [23] (as poined ou in Table. ). Table. Comparison of seady sae local skin-fricion parameer and emperaure values a X=. wih hose of Lin [6] Temperaure Local skin fricion M = Lin resuls [6] Presen resuls Lin resuls [6] Pr θ() ( 2 9 )/5 θ() T f () ( 2 9 )/5 f () τ X Presen resuls a.8893 a b a Values aken from Pop and Waanabe [7] when sucion/injecion is zero. b Values aken from Na and Chiou [23] when and soluions for flow over a full cone. Table 2. Comparison of seady-sae local skin-fricion parameer and local Nussel number values a X =. wih hose of Hossain and Paul [2] for differen values of Pr when m =.5 and sucion is zero Local skin-fricion Local Nussel number M = Hossain resuls [2] Presen resuls Hossain resuls [2] Presen resuls Pr F () τ X/(Gr L) 3/5 /Φ () Nu X/(Gr L) /

8 Bapuji Pullepu, A. J. Chamkha Figures 3 hrough 6 presen ransien velociy and emperaure profiles a X =. for various parameers Pr, m and magneic parameer M. The value of wih sar ( ) symbol denoes he ime aken o reach seady sae. In Figs. 3 and 4, ransien velociy and emperaure profiles are ploed for various values of Pr and M. Applicaion of a magneic field normal o he flow of an elecrically conducing fluid gives rise o a resisive force ha acs in he direcion opposie o ha of he flow. This force is called he Lorenz force. This resisive force ends o slow down he moion of he fluid along he cone and causes an increase in is emperaure and a decrease in velociy as M increases. This is clear from Figs. 3 and 4. Also, i is observed from hese figures ha he momenum and hermal boundary layers become hick when he values of Pr decrease or he values of M increase. The viscous force increases and hermal diffusiviy reduces wih increasing values of Pr, causing a reducion in he velociy and emperaure. I is also noiced ha he ime aken o reach seady-sae condiions increases wih increasing values of P r or M. I is noiced from he Figs. 3 and 4 ha he emporal maximum value of velociy reaches seady sae only when he value of increases and ha here is insignifican effec on he emperaure profiles. In Figs. 5 and 6, ransien velociy and emperaure profiles are ploed for various values of m wih M =. and Pr =.7. Impulsive forces are reduced along he surface of he cone near he apex for increasing values of m (i.e. he gradien of hea flux along he cone near he apex reduces wih increasing values of m). Due o his, he difference beween emporal maximum velociy values and seady-sae values reduces wih increasing values of m and ha here is no significan effec on emperaure profiles as noiced from Fig. 6. I is also observed ha increasing m reduces he velociy as well as he emperaure and akes more ime o reach seady-sae condiions Lin [6] Presen resuls.4 Velociy profile Pr = T U Temperaure profile Y Fig. 2. Comparison of seady sae emperaure and velociy profiles a X=.. 496

9 Transien Laminar MHD Free Convecive Flow.4.35 c a b m =.25 M = m =.25 M = d.5.25 U.2.5. e f g h Pr a) b) * c) * d) * e).7. f).7. g) * h).7. i) 6.7. T.5 a Pr a) * b) * c) * d).7. e) * f) 6.7. b.5 c f i * Seady Sae Value e d * Seady Sae Value Fig. 3. Transien velociy profiles a X=. for various values of P r and M. Y Fig. 4. Transien emperaure profiles a X=. for various values of Pr and M. Y * * 4.48* Pr =.7 M =. m = * 4.46* Pr =.7 M =. m = * *.2.25 U T * Seady Sae Value * Seady Sae Value Fig. 5. Transien velociy profiles a X=. for various values of m. Y Fig. 6. Transien emperaure profiles a X=. for various values of m. Y 497

10 Bapuji Pullepu, A. J. Chamkha Figures 7 hrough depic he variaions of he ransien local skin-fricion parameer τ X and he local Nussel number Nu X a various posiions on he surface of he cone (X =.25 and.) for conrolling parameers m, Pr and M. The local skin-fricion parameer τ X and he local Nussel number Nu X for differen values of Pr and M a various posiions on he surface of he cone (X =.25 and.) in he ransien period are shown in Figs. 7 and 8, respecively. I is observed ha he local skin-fricion parameer and he local Nussel number decreases wih increasing values of M and he effec of M on he local skin-fricion parameer τ X and he local Nussel number Nu X is less near he apex of he cone and increases gradually wih increasing he disance along he surface of he cone from he apex. Also, i is noiced from Fig. 7 ha he local wall shear sress decreases as Pr increases because he velociy decreases wih an increasing value of Pr as shown in Fig. 3. The local Nussel number Nu X increases wih increasing values of Pr and his is clear from Fig. 8. The variaion of he local skin-fricion parameer τ X and he local Nussel number Nu X in he ransien period a various posiions on he surface of he cone (X =.25 and.) and for differen values of m are shown in Figs. 9 and. I is observed from Fig. 9 ha he local skin-fricion parameer decreases wih increasing values of m and ha he effec of m on he local skin-fricion τ X is more near he apex of he cone and reduces gradually wih increasing he disance along he surface of he cone from he apex. From Fig., i is noiced ha near he apex, he local Nussel number Nu X reduces wih increasing values of bu his rend is slowly changed and reversed as he disance increases along he surface from apex..2 Pr =.7 M =., 2., 3. m =.25 X = m =.25 X = X/Gr L 3/5.6 Pr =.7 M =., 2., 3. Nu X/GrL / a) Pr = 6.7, M =. b) Pr =.7, M =., 2., 3. Pr = 6.7, M =..2 a b b a Fig. 7. Local skin fricion a X =.25 and. for various values of Pr and M in ransien period Fig. 8. Local Nussel number a X =.25 and. for various values of Pr and M in ransien period. 498

11 Transien Laminar MHD Free Convecive Flow.2.8 m = Pr =.7 M =. m = X/Gr L 3/ Nu X /Gr L / Pr =.7 M =. X = Fig. 9. Local skin fricion a X =.25 and. for various values of m in ransien period Fig.. Local Nussel number a X =.25 and. for various values of m in ransien period M =.6 Pr =.7 m = /Gr L 3/5.5.4 Nu/Gr L /5.8 M =..3.6 M = Pr =.7 m = Fig.. Average skin fricion for various values of m and M Fig. 2. Average Nussel number for various values of m and M. 499

12 Bapuji Pullepu, A. J. Chamkha Finally, Figs. and 2 illusrae he effecs of m and M on he average skin-fricion parameer τ and he average Nussel number Nu in he ransien period. The average skinfricion parameer τ is more for lower values of m. I is observed from Figs. and 2 ha he values of he average skin-fricion parameer τ and he average Nussel number Nu decrease wih increasing values of M. In addiion, i is clear from he Fig. 2, he effec of m is almos negligible on he average Nussel number Nu. 5 Conclusions This paper deals wih unseady laminar free convecion flow of an elecrically-conducing fluid pas a verical cone wih non-uniform surface hea flux in he presence of a ransverse magneic field. The dimensionless governing boundary-layer equaions are solved numerically using an implici finie-difference mehod of he Crank-Nicolson ype. Presen resuls are compared wih available resuls from he open lieraure and found o be in very good agreemen. The following conclusions are drawn: The ime aken o reach seady sae increases wih increasing values of Pr, m and M. The velociy U increases when he conrolling parameers P r, m and M are reduced. The surface emperaure T reduces as he values of M decrease and he values of Pr, m increase. The momenum and hermal boundary layers become hick for lower values of Pr or higher values of M. The values of he local skin-fricion parameer τ X and he local Nussel number Nu X reduce as M increases. The local skin-fricion parameer τ X decreases while he local Nussel number Nu X increases wih increasing values of P r. The local and average skin-fricion parameers increase when he value of m is reduced. The average skin-fricion parameer and he average Nussel number reduce when he values of M increases. The effec of m on he average Nussel number Nu is almos negligible. References. H. J. Merk, J. A. Prins, Thermal convecion laminar boundary layer, I, II, Appl. Sci. Res. A4, (24), pp , 953,

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