Journal of Quality Measurement and Analysis JQMA 6(2) 2010, Jurnal Pengukuran Kualiti dan Analisis

Transcription

1 Journal o Quality Measurement and Analysis JQMA 6(), 5-7 Jurnal Pengukuran Kualiti dan Analisis UNSTEADY MAGNETOHYDRODYNAMIC MIXED CONVECTION STAGNATION POINT FLOW OF A VISCOELASTIC FLUID ON A VERTICAL SURFACE (Aliran Titik Genangan Olakan Campuran Magnetohidrodinamik Tak Mantap bagi Bendalir Likat-kenyal pada Permukaan Menegak) KARTINI AHMAD & ROSLINDA NAZAR ABSTRACT In this paper, the problem o unsteady magnetohydrodynamic (MHD) viscoelastic luid lowing towards a stagnation point on a vertical surace is studied. The temperature o the surace is assumed to vary linearly with the distance rom the stagnation point. The partial dierential equations which governed the low are transormed into a system o ordinary dierential equations, which are then solved numerically by an implicit inite-dierence scheme known as the Keller-box method. The eects o the viscoelastic parameter K, mixed convection parameter λ, magnetic parameter M and Prandtl number Pr on the low and heat transer characteristics are presented in this paper. The numerical solutions obtained are uniormly valid or all dimensionless time rom initial unsteady-state low to inal steady-state low in the whole spatial region. Keywords: magnetohydrodynamic (MHD); mixed convection; unsteady stagnation point low; vertical surace; viscoelastic luid ABSTRAK Dalam makalah ini, masalah aliran bendalir likat-kenyal magnetohidrodinamik (MHD) tak mantap yang mengalir ke arah suatu titik genangan pada permukaan tegak dikaji. Suhu permukaan dianggap berubah secara linear terhadap jarak daripada titik genangan. Persamaan pembezaan separa yang menakluk aliran kemudiannya dijelmakan kepada sistem persamaan pembezaan biasa, yang seterusnya diselesaikan secara berangka menggunakan skema beza terhingga tersirat yang dikenali sebagai kaedah kotak Keller. Kesan parameter likat-kenyal K, parameter olakan campuran λ, parameter magnetik M dan nombor Prandtl Pr terhadap ciriciri aliran dan pemindahan haba dipertimbangkan dalam makalah ini. Penyelesaian berangka yang diperoleh adalah sah secara seragam bagi julat masa tak bermatra bermula daripada aliran awal tak mantap kepada aliran akhir mantap. Kata kunci: magnetohidrodinamik (MHD); olakan campuran; aliran titik genangan tak mantap; permukaan tegak; bendalir likat-kenyal. Introduction Viscoelastic luid is one type o second grade luid (non-newtonian luid) that has received much attention recently, as this class o luid exhibits viscous and elastic-like characteristics when undergoing deormation. Honey, plastic ilms, and artiicial ibers are some examples o viscoelastic luid. Hence, industrially, this kind o luid has become one o the topics discussed. Pioneering work by Oldroyd (95), Beard and Walters (964a) and Rajagopal et al. (984), who have developed the boundary layer theory or the second grade luid have motivated many researchers to really explore this kind o luid with various situations. It seems that the boundary layer equations or this luid are an order higher than those or the Newtonian (viscous) luid and the adherence boundary conditions are insuicient to

2 Kartini Ahmad & Roslinda Nazar determine the solution o these equations completely. Thereore, Rajagopal (984, 995) and Rajagopal and Kaloni (989) have done urther investigation on this matter. The study o a non-newtonian luid low in the region o stagnation point has been done by several authors (Srivatsava 958; Rajeswari & Rathna 96; Beard & Walters 964b; Garg & Rajagopal 99; Ariel ; Mahapatra & Gupta 4). Ayub et al. (8) studied the stagnation point low o viscoelastic luid towards a stretching sheet, Li et al. (9) investigated the case o oblique stagnation point low o a viscoelastic luid with the eect o heat transer and very recently, Labropulu et al. () considered the two-dimensional stagnation point low o a viscoelastic second-grade luid over a stretching surace with heat transer and Robert et al. () established the existence and uniqueness results over the semi-ininite interval [,) or a class o nonlinear ourth order ordinary dierential equations arising in the hydromagnetic stagnation point low o a second grade luid over a stretching sheet. Further, the eect o mixed convection stagnation point low in a viscoelastic luid adjacent to a vertical surace has been studied by Hayat et al. (8), and Anwar et al. (8) considered the low o a viscoelastic luid over a horizontal circular cylinder, while the steady MHD mixed convection low o a viscoelastic luid in the vicinity o two-dimensional stagnation point with magnetic ield has been investigated by Kumari and Nath (9). Most o the recent researches on convective lows in non-newtonian luids have been directed to the problems o steady cases. However, unsteady convective boundary layer low problems have not, so ar, received as much attention. Thereore, there are ew literatures reported on problems that took into account the time actor. Several unsteady problems were done by, or example, Nazar et al. (4), Ishak et al. (6) and Hassanien and Al-Arabi (9) or the case o mixed convection low in viscous luid and porous medium. Lok et al. (6) studied the unsteady mixed convection low or the case o micropolar luid, whilst unsteady boundary layer or second-grade luid low can be ound in the paper by Labropulu et al. (3). Sujit and Pop (6) investigated the unsteady boundary layer ree convection low o an incompressible electrically conducting viscoelastic second-order luid over a vertically permeable lat plate, where temperature and concentration dierences are responsible or the convective buoyancy current while Qi and Xu (7) considered the unsteady low o viscoelastic luid with the ractional derivative Maxwell model in a channel. Problems o mixed convection low near the stagnation point with associated MHD has been pointed out by Abdelkhalek (6), who presented problem o steady two-dimensional laminar magnetohydrodynamic-mixed convection owing to the stagnation low against a heated vertical semi-ininite permeable surace or viscous luid. Ishak et al. (8) considered a steady MHD low towards a stagnation point on a vertical surace immersed in a micropolar luid and recently, Ishak et al. () investigated the problem o MHD mixed convection boundary layer low o a viscous and electrically conducting luid near the stagnation-point on a vertical permeable surace or viscous luid low. On the other hand, Hayat et al. () has presented the analytical analysis o steady MHD two-dimensional mixed convection boundary layer low o a viscous and incompressible luid near the stagnation-point on a vertical stretching surace embedded in a luid-saturated porous medium and thermal radiation using the homotopy analysis method (HAM). Motivated by the work o Hayat et al. (8) and Lok et al. (6), we investigate in this paper the unsteady MHD mixed convection boundary layer low o a viscoelastic luid near the stagnation point on a vertical surace. The transormed governing partial dierential equations in two variables are solved numerically using the Keller-box method or certain values o the viscoelastic parameter K, magnetic parameter M, mixed convection parameter λ and Prandtl number Pr. 6

3 Unsteady MHD mixed convection stagnation point low o a viscoelastic luid on a vertical surace. Basic equations Consider the mixed convection boundary layer stagnation point low over a semi-ininite vertical surace, which is placed in a viscoelastic luid o uniorm ambient temperature T. It is assumed that a uniorm magnetic ield o strength B o is applied in the positive y-direction and the surace o the plate is heated or cooled to a variable temperature T w (x), where T w (x) > T is or a heated plate and T w (x) < T is or a cooled plate. It is also assumed that at time t= the outside low starts in motion impulsively rom rest towards the plate with a steady velocity u e (x), as shown in Fig.. Figures (a) and (b) illustrate the assisting and opposing lows, respectively. T w (x) > T T w (x) < T x x u e (x) T y B o B o O u e (x) T y B o B o O T w (x) < T T w (x) > T (a) assisting low (b) opposing low Figure : Physical model and coordinate system The low in the neighbourhood o the stagnation line has the same characteristics irrespective o the shape o the body (Hiemenz 9). It is urther assumed that the temperature o the plate T w (x) varies linearly with the distance x along the plate. Thus the plate temperature and the condition ar rom the plate is assumed to be given by x x Tw ( x) T T, ue ( x) U e, L L () where U e is the reerence velocity, L is the characteristic length and T > is a reerence temperature. In the absence o heat generation and viscous dissipation, along with the Boussinesq approximation, the boundary layer equations are given by u v, x y 3 3 u u u due u u u u u v u u v ue k u v t x y dx x y 3 y x y y y y B g T T ue u, () (3) 7

4 Kartini Ahmad & Roslinda Nazar t x y y T u T v T T, (4) subject to the boundary conditions t : u x, y v x, y, T x, y T any x, y x :,,,,, x u x, u x, ue x Ue,, T x, T, x, L y t u x v x T x Tw x T T L where u and v are the velocity components in the x-and y-directions, respectively, t denotes the time, T,, g,,, σ, ρ, B o and k are the temperature, kinematic viscosity, gravity acceleration, thermal expansion coeicient, thermal diusivity, electrical conductivity, density, magnetic ield and viscoelastic parameter, respectively. The sign in Eq. (3) corresponds to the assisting and opposing lows, respectively. Introducing the new variables as ollows (Williams & Rhyne 98), where prime denotes dierentiation with respect to : (5) / / * * e y t t t u x y t Ue Ue U x, exp( ),, (,, ),, L L L / Ue / x (,, ),, (,, ),. v x y t T x y t T T L L Substituting (6) into Eqs. (3) and (4) yields M, iv K Pr, (6) (7) (8) 3 Gr gt L or. Here is the mixed convection parameter, where Gr and Re UeL Re are the Grasho and Reynolds numbers, respectively. It is worth mentioning that λ >, λ < and λ = correspond to the assisting low, opposing low and orced convection low, respectively. On the other hand, M and K are the dimensionless magnetic and viscoelastic parameters, respectively, given by The boundary conditions (5) become or. B L M, K k. (9) U L e U e (,) (,), (,),,, as, () 8

5 Unsteady MHD mixed convection stagnation point low o a viscoelastic luid on a vertical surace The physical quantities o interest in this problem are the skin riction coeicient C and the local Nusselt number Nu x, which are deined as where w and given by C u, Nu xq w w x, e Tw T () q w are the wall shear stress and the surace heat lux, respectively, which are w u u u u v T k u, qw k y x y y y y y y y y. () Substituting variables (6) into (), we obtain C / / Re x (), Nu (3) / / x Re x (). or. Equations (7) and (8) subject to boundary conditions () are coupled nonlinear parabolic partial dierential equations. Hence, we can obtain some particular cases o this problem... Initial unsteady low * For early unsteady low, we have ( t ), thus Eqs. (7) and (8) reduce to and the boundary conditions () become K iv, (4) (5) Pr () (), (),,, as (6).. Final steady-state low * For this case, ( t ), Eqs. (7) and (8) take the ollowing orm: iv (7) K M, subject to the same boundary conditions (6). 3. Results and Discussion Pr ( ), (8) The nonlinear partial dierential equations (7) and (8) subject to boundary conditions () have been solved numerically using an implicit inite-dierence scheme known as the Kellerbox method as described in the book by Cebeci and Bradshaw (988). 9

6 Kartini Ahmad & Roslinda Nazar Tables and present the numerical values o the skin riction coeicient Re / () / C x and the local Nusselt number Nu Re (), respectively, when ξ= (inal steady-state low) or various values o the viscoelastic parameter K when Pr=. and Pr= or the case o assisting low (λ > ) and opposing low (λ < ). In order to validate the accuracy o the present method, the results or the inal steady-state low obtained were compared with those o Hayat et al. (8) and it is ound that they are in good agreement. Thereore, we are conident that the developed code used in this study is suitable to solve the present problem discussed in this paper. x x Table : Values o / C or various values o K and Pr when λ=., -.. Results shown in ( ) are those o Re x Hayat et al. (8) M K Pr=. Pr= λ=. λ= -. λ=. λ= (.559).874 (.874).647 (.6474).956 (.9558).6844 (.6844).543 (.543) Table : Values o Nu x Re x -/ or various values o K and Pr number when λ=., -.. Results shown in ( ) are those o Hayat et al. (8) M K Pr=. Pr= λ=. λ= -. λ=. λ= (-.46) (-.39) (-.3698) (-.494) (-.3785) (-.3578) From Table, it can be seen that the values o / C Re x decrease as the viscoelastic parameter K increases regardless whether the low is assisting or opposing. The same phenomenon / happened or Nu x Re x as can be seen rom Table. Increasing Pr and heating the plate causing / C Re x to decrease and the opposite trends are observed or absolute values o

7 Unsteady MHD mixed convection stagnation point low o a viscoelastic luid on a vertical surace / Nu x Re x. On the other hand, cooling the plate will increase the values o () and (). The magnetic parameter M gives huge impact to the low as it increases the values o () and () or both assisting and opposing lows. Figures and 4 show the velocity proiles o the inal steady-state low (ξ=) or various values o the viscoelastic parameter K and magnetic parameter M when Pr= or assisting and opposing lows, respectively. For a particular value o K, the velocity boundary layer thickness increases monotonically with η, and becomes unity at the outside o the boundary layer, which actually satisies the boundary condition ( ). It is also illustrated that or a speciic value o K as the magnetic parameter M increases, the velocity thickness decreases. The temperature proiles o the inal steady-state low are shown in Figs. 3 and 5 or both assisting and opposing lows, respectively. It is observed that as K increases, the temperature proile increases. The thermal boundary layer thickness is slightly smaller when M= compared to M=. This justiy that the magnetic parameter M reduces the thickness o the thermal boundary layer. Comparatively, the velocity proiles or the assisting and opposing lows seem alike and thus, it can be concluded that the eects o the plate either being cooled or heated are not signiicant. The same trend also occurs or the temperature proiles..9.8 M= M=.7 ' () K=,,.3.. Pr=, = Figure : Velocity proiles o the inal steady-state low or various K and M when Pr= or λ=. (assisting low)

8 Kartini Ahmad & Roslinda Nazar.9.8 Pr=, =. M= M=.7.6 () K=,, Figure 3: Temperature proiles o the inal steady-state low or various K and M when Pr= or λ=. (assisting low).9.8 M= M=.7 ' () K=,,.3.. Pr=, = Figure 4: Velocity proiles o the inal steady-state low or various K and M when Pr= or λ=-. (opposing low)

9 Unsteady MHD mixed convection stagnation point low o a viscoelastic luid on a vertical surace.9.8 Pr=, = -. M= M=.7.6 () K=,, Figure 5: Temperature proiles o the inal steady-state low or various K and M when Pr= or λ=-. (opposing low) Figures 6 and 7 show the velocity and temperature proiles when Pr=.7 and K= by considering various values o the magnetic parameter M. It is observed that the thickness o the velocity and thermal boundary layer decreases as M increases, and valid or both assisting and opposing lows. Again, the eects o the plate either being cooled or heated are not signiicant, especially or large values o M..9.8 =. =-..7 ' () M=,,,.3.. Pr=.7, K= Figure 6: Velocity proiles o the inal steady-state low or various M when Pr=.7 and K= or λ=. (assisting low) and λ=-. (opposing low). 3

10 Kartini Ahmad & Roslinda Nazar Pr=.7, K= =. = -..6 () M=,,, Figure 7: Temperature proiles o the inal steady-state low or various M when Pr=.7 and K= or λ=. (assisting low) and λ=-. (opposing low) Pr=., M=,= C x.5 K=,, Figure 8: Variation o the skin riction coeicient with ξ or various K. 4

11 Unsteady MHD mixed convection stagnation point low o a viscoelastic luid on a vertical surace 8 7 Pr=.,M=,=. 6 5 K=,, 4 N ux Figure 9: Variation o the local Nusselt number with ξ or various K. Figures 8 and 9 present the variation o the skin riction coeicient / C Re x and the local / Nusselt number Nu x Re x, respectively, as a unction o ξ or various values o K. At the start o the motion (initial unsteady-state low), both the skin riction coeicient and the local Nusselt number have large magnitude (due to impulsive motion) and decrease monotonically and inally reach the steady-state value ξ.the low transition is very smooth rom small-time solution to the large-time solution. The values o / / C Re x and Nu x Re x seem to decrease when the viscoelastic parameter K increases, while the viscoelastic parameter K is not really aecting the value o changes in / Nu x Re x are comparatively small or variation in K. Nu x Re x / at ξ = since the 4. Conclusions In the present study, we have investigated theoretically the unsteady MHD mixed convection low o a viscoelastic luid near the stagnation point on a vertical surace. Both the assisting low and opposing low situations are considered. Numerical computation has been carried out to study the eects o the material parameter K, magnetic parameter M, mixed convection parameter λ and Prandtl number Pr on the skin riction coeicient, the local Nusselt number, as well as the velocity and temperature proiles. Results are presented in tables and igures or certain parameter conditions. From the solutions o the skin riction and heat transer coeicients it is ound that there is a smooth transition rom the small-time solution (initial unsteady-state) to the large-time solution (inal steady-state). 5

12 Kartini Ahmad & Roslinda Nazar Acknowledgment The inancial support received in the orm o a undamental research grant scheme (FRGS) rom the Ministry o Higher Education Malaysia is greatly acknowledged. The authors would also like to thank the reviewer or the valuable comments and suggestions. Reerences Abdelkhalek M.M. 6. The skin riction in the MHD mixed convection stagnation point with mass transer. Int. Comm. Heat Mass Transer 33(): Anwar I., Amin N. & Pop I. 8. Mixed convection boundary layer low o a viscoelastic luid over a horizontal circular cylinder. Int. J. Non-Linear Mech. 43(9): Ariel P. D.. On extra boundary condition in the stagnation point low o a second-grade luid. Int. J. Eng. Sci. 4: Ayub M., Zaman H., Sajid M. & Hayat T. 8. Analytical solution o stagnation-point low o a viscoelastic luid towards a stretching surace. Commun. Nonlinear Sci. Numer. Simulation 3(9): Beard D.W. & Walters K. 964a. Elasto-viscous boundary layer lows. Proc. Camb. Phil. Soc. 6: Beard D.W. & Walters K. 964b. Elastco-viscous boundary layer lows. Part I: two dimension low near a stagnation point. Proc. Camb. Phil. Soc. 6: Cebeci T. & Bradshaw P Physical and Computational Aspects o Convective Heat Transer. New York: Springer. Garg V.K. & Rajagopal K.R. 99. Stagnation point low o a non-newtonian luid. Mech. Res. Comm. 7: Hassanien I.A. & Al-Arabi T.H. 9. Non-Darcy unsteady mixed convection low near the stagnation point on a heated vertical surace embedded in a porous medium with thermal radiation and variable viscosity. Commun. Nonlinear Sci. Numer. Simulation 4(4): Hayat T., Abbas Z. & Pop I. 8. Mixed convection in the stagnation point low adjacent to a vertical surace in a viscoelastic luid. Int. J. Heat Mass Transer 5(-): Hayat T., Abbas Z., Pop I. & Asghar S.. Eects o radiation and magnetic ield on the mixed convection stagnation-point low over a vertical stretching sheet in a porous medium. Int. J. Heat Mass Transer 53(-3): Hiemenz K. 9. Der Grenzschit an einem in den gleichörmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dingl. Poly-tech. 3: 3-4. Ishak A., Nazar R. & Pop I. 6. Unsteady mixed convection boundary layer low due to a stretching vertical surace. Arabian J. Sci. & Engng. 3: Ishak A., Nazar R. & Pop I. 8. Magnetohydrodynamic (MHD) low o a micropolar luid towards a stagnation point on a vertical surace. Comp. & Math. with Appl. 56(): Ishak A., Nazar R., Bachok N. & Pop I.. MHD mixed convection low near the stagnation point on a vertical permeable surace. Physica A: Statistical Mech. & Appl. 389(): Kumari M. & Nath G. 9. Steady mixed convection stagnation-point low o upper convected Maxwell luids with magnetic ield. Int. J. Non-Linear Mech. 44(): Labropulu F., Xu X. & Chinichian M. 3. Unsteady stagnation point low o a non-newtonian second-grade luid. Int. J. Maths. & Mathematical Sci. 3(6): Labropulu F., Li D. & Pop I.. Non-orthogonal stagnation-point low towards a stretching surace in a non- Newtonian luid with heat transer. Int. J. Thermal Sciences 49(6): 4-5. Li D., Labropulu F. & Pop I. 9. Oblique stagnation-point low o a viscoelastic luid with heat transer. Int. J. Non-linear Mech. 44: 4-3. Lok Y.Y., Amin N. & Pop I. 6. Unsteady mixed convection low o a micropolar luid near the stagnation point on a vertical surace. Int. J. Thermal Sciences 45(): Mahapatra T.R. & Gupta A.S. 4. Stagnation-point low o a viscoelastic luid towards a stretching surace. Int. J. Non-Linear Mech. 39: 8-8. Nazar R., Amin N. & Pop I. 4. Unsteady mixed convection boundary layer low near the stagnation point on a vertical surace in a porous medium. Int. J. Heat Mass Transer 47(-3): Oldroyd J.G. 95. On the ormulation o rheological equations o state. Proc. Roy. Soc. London A : Qi H. & Xu M. 7. Unsteady low o viscoelastic luid with ractional Maxwell model in a channel. Mech. Res. Comm. 34(): -. Rajagopal K.R On the creeping low o the second-order luid. J. Non-Newtonian Fluid Mech. 5: Rajagopal K.R On boundary conditions or luids o the dierential type. In: Sequiera A. (Ed.). Navier Stokes Equations and Related Non-Linear Problems. New York: Plenum. 6

13 Unsteady MHD mixed convection stagnation point low o a viscoelastic luid on a vertical surace Rajagopal K.R., Na T.Y. & Gupta A.S Flow o a viscoelastic luid over a stretching sheet. Rheol. Acta 3: 3 5. Rajagopal K.R. & Kaloni P.N Some remarks on boundary conditions or lows o luids o the dierential type. In: Graham G.A.C. & Malik S.K. (Eds.). Continuum Mechanics and its Applications. New York: Hemisphere. Rajeswari G. & Rathna S.L. 96. Flow o a particular class o non-newtonian viscoelastic and visco-inelastic luids near a stagnation point. Z. Angew. Math. Phys. (ZAMP) 3: Robert A., Gorder V. & Vajravelu K.. Hydromagnetic stagnation point low o a second grade luid over a stretching sheet. Mech. Res. Comm. 37(): 3-8. Srivatsava A.C The low o a non-newtonian liquid near a stagnation point. Z. Angew. Math. Phys. 9: Sujit K.K. & Pop I. 6. Unsteady ree convection viscoelastic boundary layer low past a vertical porous plate with internal heat generation/absorption. Int. J. Fluid Mech. Res. 33(6): 5-5. Williams J.C. & Rhyne T.H. 98. Boundary layer development on a wedge impulsively set into motion. SIAM J. Appl. Math. 3: 5-4. Centre or Foundation Studies International Islamic University Malaysia 4635 Petaling Jaya Selangor DE, MALAYSIA katzdany@yahoo.com Pusat Pengajian Sains Matematik Fakulti Sains dan Teknologi Universiti Kebangsaan Malaysia 436 UKM Bangi Selangor DE, MALAYSIA rmn@ukm.my * * Corresponding author 7