MHD Flow of Nanofluids over an Exponentially Stretching Sheet Embedded in a Stratified Medium with Suction and Radiation Effects

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1 Journal o Applied Fluid Mechanics, Vol. 8, No. 1, pp , 215. Available online at.jamonline.net, ISSN , EISSN MHD Flo o Nanoluids over an Exponentially Stretching Sheet Embedded in a Stratiied Medium ith Suction and Radiation Eects P. Loganthan 1 and C.Vimala 2 1, 2 Department o Mathematics, Anna University, Chennai- 6 25, India Corresponding Author vimala_1@redimail.com (Received December 11, 213; accepted February 23, 214) ABSTRACT An analysis has been carried out to investigate the inluence o the combined eects o MHD, suction and radiation on the orced convection boundary layer lo o a nanoluid over an exponentially stretching sheet, embedded in a thermally stratiied medium. The governing boundary layer equations o the problem are ormulated and transormed into ordinary dierential equations, using a similarity transormation. The resulting ordinary dierential equations are solved numerically, by the shooting method. The eects o the governing parameters on the lo and heat transer characteristics are studied and discussed in detail. Dierent types o nanoparticles, namely, Cu, Ag, Al 2 O 3 and TiO 2, ith ater as the base luid, are studied. It is ound that the eects o the radiation parameter, volume raction and suction are the same on the temperature proiles, in contrast to the eects o thermal stratiication. Comparisons ith previously published orks are perormed in some special cases, and ound to be in good agreement. Keyords: Exponentially stretching sheet, Suction, Nanoluid, MHD, Thermal stratiication, Thermal radiation. NOMENCLATURE b, c constants B magnetic ield strength C p speciic heat C local skin riction coeicient ƒ dimensionless stream unction suction parameter k thermal conductivity L reerence length M magnetic parameter N R radiation parameter Nu x local Nusselt number Pr Prandtl number qr radiation heat lux Re x local Reynolds number St stratiication parameter T temperature o the luid U o reerence velocity U velocity o stretching surace u, v velocity components in the x, y directions x, y dimensionless Cartesian co-ordinates Greek symbols α ƒ thermal diusivity o the nanoluid density ψ stream unction dimensionless thermal unction η similarity variable dynamic viscosity kinematic viscosity τ skin riction or shear stress Subscripts luid s solid n nanoluid stretching surace conditions luid conditions adjacent to the stretching surace luid conditions ar aay rom the stretching surace Superscripts ' dierentiation ith respect to 1. INTRODUCTION The study o the boundary layer lo and heat transer over a stretching surace, has gron considerably during the last e decades, due to its enormous applications in several industries and environments, such as the extrusion o plastic sheets, thinning and annealing o copper ires, making electronic chips, cooling o metallic sheets, hot

2 rolling, metal extrusion, glass iber production, ire draing, paper production, etc. The problem o the lo along a moving plate in a quiescent ambient luid as irst studied by Sakiadis (1961). Later, Crane (197) examined the problem o an incompressible boundary layer lo over a stretching sheet, hich moves ith a velocity varying linearly ith the distance rom a ixed point. Ater this pioneering ork, rapidly increasing number o papers have reported various aspects o this problem. Most o the available studies are restricted to the linear stretching o the sheet. Hoever, it is knon that realistically, the stretching o the sheet need not necessarily be linear (Gupta and Gupta, 1977). For example, in polymer processing and metal spinning processing, the quality o the inal product depends on the rate o heat transer at the stretching continuous surace ith exponential variations o the stretching velocity and temperature distribution. Magyari and Keller (1999) irst considered the boundary layer lo due to an exponentially stretching sheet, and analyzed the heat and mass transer in the boundary layers ith an exponentially varying all temperature. Partha et al. (25) obtained a similarity solution or a mixed convection lo past an exponentially stretching surace, by taking into account the inluence o the viscous dissipation on the convective transport. Sajid and Hayat (28) shoed the inluence o thermal radiation on the lo over an exponentially stretching sheet, and solved the problem analytically, using the homotopy analysis method. Later, Bidin and Nazar (29) presented a numerical solution to the same problem. El-Aziz (29) studied the viscous dissipation eect on the mixed convection lo o a micropolar luid over an exponentially stretching sheet. Loganathan and Stepha (213) analyzed the problem o chemical reaction and mass transer eects on the lo o a micropolar luid past a continuously moving porous plate ith variable viscosity. In recent years, the study o heat transer in the presence o nanoluids has attracted enormous interest rom researchers, due to its potential or a high rate o heat transer enhancement properties. A nanoluid reers to the suspension o nanometer sized particles in a base luid. These luids have very high thermal conductivity and the competence to meet challenges, such as stability, sedimentation, erosion and additional pressure drop. The term nanoluid as introduced by Choi (1995). Buongiorno (26) noted that the nanoparticles absolute velocity can be vieed as the sum o the base luid s velocity and a relative velocity. Karthikeyan et al. (28) ound that the eective thermal conductivity o a nanoluid increases ith increasing volume raction. The present paper has considered the problem, using the nanoluid model proposed by Tiari and Das (27), hich as successully utilized by many authors, such as Abu- Nada (28), Muthamilselvan (21), Oztop and Abu-Nada (28), Yacob et al. (211) and Loganathan et al. (213). An excellent collection o papers on nanoluids can be ound in the book by Das et al. (27), and the revie papers by Trisaksri and Wongises (27), Wang and Mujumdar (27), and Kakac and Pramuanjaroennkij (29). The study o magnetohydrodynamic lo and heat transer has received considerable attention in recent years due to its ide variety o applications in engineering and technology, such as MHD generators, plasma studies, nuclear reactors and geothermal energy extractions. Bala Anki Reddy (212) investigated the MHD and radiation eects on the unsteady lo along an exponentially accelerated isothermal vertical plate, ith uniorm mass diusion in the presence o a heat source. The magnetic nanoluid aids in improving the extremely good long term stability or applications, such as mechanical seals, sensors, biomedicine, and many others. The nanoluid containing magnetic nanoparticles also acts as a super-paramagnetic luid, hich, in an alternating electromagnetic ield, absorbs energy and produces a controllable hyperthermia, to be used in cancer treatment. Radiation can be quite signiicant at high operating temperatures. Radiative heat transer becomes more important or the design o the pertinent equipment. Due to the signiicance o magnetic and radiation eects, Loganathan and Vimala (213) analyzed the inluence o magnetic and radiation eects on orced convection boundary layer lo o a nanoluid past an exponentially stretching sheet. Futhermore, the eect o stratiication is also one o the important aspects that has to be considered in the study o heat transer. Stratiication o luids occurs due to the temperature variations or concentration dierences, or the presence o dierent luids o dierent densities. This has great importance in many engineering processes, or applications such as solar collectors, polymer production, drying processes, reactors and metallurgy, by hydromagnetic methods. Rosmila et al. (212) used Lie group transormation, to study the problem o the magnetic eect on the natural convective lo o a nanoluid, over a linearly porous stretching sheet in the presence o thermal stratiication. The process o suction also has its importance in many engineering activities, such as in the design o thrust bearing and radial diusers, and thermal oil recovery. Due to numerous applications and the importance o the actors discussed above, our objective is to study, the inluence o combined eects o magnetic, suction and radiation on the boundary layer lo o nanoluids past an exponentially stretching sheet embedded in a thermally stratiied medium. The present analysis inds its applications in the rotating seals, aerodynamic sensors, nano -structured magneto rheological luids or semi active vibration dampers and biomedical applications. In the present study, the inluence o the solid volume raction, thermal radiation, suction and thermal stratiication, and magnetic eects, is investigated and presented in detail. Four dierent types o nanoluids, namely, Cuater, Ag-ater, Al 2 O 3 -ater and TiO 2 -ater are studied. The obtained results are compared ith the relevant results in the existing literature, and are ound to be in good agreement. 2. MATHEMATICAL FORMULATION 86

3 Consider the steady laminar to-dimensional lo o an electrically conducting nanoluid over an exponentially stretching sheet in the presence o x L u U U e v V ( x ), T T ( x )at y = o u v, T T as y (4) here u and v are the velocity components in the x and y directions, T is the local temperature o the nanoluid, L is the characteristic length o the plate. U is the velocity parameter o the stretching surace, x2l qr is the radiation heat lux andv ( x ) V oe is the constant parameter ith or suction and or injection. Fig.1. Physical model and coordinate system radiation. The x axis is taken along the stretching sheet in the direction o the motion, and the y axis is normal to it. There are no pressure gradients in the x and y directions. p as the momentum boundary y layer is very thin. p as U is constant or the x lo over a lat surace at zero incidence. (The application o Bernoulli s equation reveals that pressure is constant in the x direction in the ree stream). The sheet stretches at a velocity o and a given temperature distributiont, and is embedded in a thermally stratiied medium, through a quiescent ambient luid o variable temperature T here T T. A uniorm magnetic ield B is applied normal to the sheet as shon in Fig. 1. The induced magnetic ield is assumed to be small compared to the applied magnetic ield, and is neglected (the magnetic Reynolds number is small). The luid is a ater based nanoluid containing our dierent types o nanoparticles, namely, Cu, Ag, Al 2 O 3 and TiO 2. It is also assumed that the base luid and the suspended nanoparticles are in thermal equilibrium, and no slip occurs beteen them. Under the above assumptions, and the olloing model equations proposed by Tiari and Das (27), the basic continuity, momentum and energy equation or this problem can be ritten as ollos: u v x y u u 2 u B 2 u v u x y n y 2 n U u v k 2 T T 1 T qr n 2 x y C p y y n. The boundary conditions o Eqs. (1)-(3) are taken to be (1) (2) (3) x (2 L) The magnetic ield B(x) is o the orm B B e here B is the constant magnetic ield. It is assumed that T T be x 2L, T T ce x 2L here T is the parameter o the temperature distribution in the stretching surace, b and c are constants. The thermo physical properties o the nanoluids, namely, viscosity n, density n, thermal diusivity n, and heat capacitance ( Cp) n are given by (Brinkman, 1952; Xuan et al., 2). n 2.5, (1 ) n (1 ) s, k n n, ( C p ) n ( C p ) n (1 )( C p ) ( C p ) s. Here, k n is the thermal conductivity o the nanoluid, is the solid volume raction o the nanoluid, is the density o the luid raction, s is the density o the solid raction, is the viscosity o the luid raction, k is the thermal conductivity o the luid raction, k s is the thermal conductivity o the solid volume raction. (C p ) is the speciic heat parameter o the luid raction and ( C ) is the speciic heat parameter o the solid raction. p s (5) The eective thermal conductivity o the nanoluid is approximated by the Maxell-Garnetts model k k n ( ks 2k ) 2 ( k ks ) ( k 2k ) ( k k ) s s (6) Using the Rosseland approximation (Brester, 1992) or the thermal radiation, the radiative heat lux is simpliied as: 87

4 4 q r 3k T 4 y here is the Stean-Boltzmann constant and k is the mean absorption coeicient. 4 Hence, e express T as a linear unction o temperature T using the Taylor series about a ree stream temperature T and neglecting the higher order terms, e get: T 4 4T 3 T 3T 4 By using Eqs. (7) and (8), the energy equation becomes 2 3 T T T 16 T 2 T u v x y n y C k y p n (7) (8) (9) Introducing the olloing transormations (Sajid and Hayat, 28) 12 U x 2L e y, 2 L 12 x 2L 2 U L e ( ), T T ( ). T T (1) The stream unction satisies Eq. (1) automatically ith u = and v = -. (11) y x On substituting the transormations (1), Eqs. (2) and (3) are reduced to the olloing set o ordinary dierential equations: (1 ( / ))( " 2 ' ) (1 ) s M ' (12) 4 1 k (1 N ) n (1 3 R Pr k (( C ) / ( C ) ) p s p ( ) ' St (13) Subject to the transormed boundary conditions are:, 1, 1 St at =,, as. (14) here the primes denote dierentiation, ith respect to 2 B 2 L the similarity variable η, M o is the U o magnetic parameter, Pr is the Prandtl number,, 4 T 3 N is the radiation parameter, and R k k n St c b is the stratiication parameter. St implies a stably stratiied environment, hile St corresponds to an unstratiied environment. The skin riction coeicient C number Nu x, are deined as C and local Nusselt xq and Nu U 2 x k ( T T ) (15) here is the skin riction or the shear stress and q is the heat lux rom the plate; these are given by u n y y T and q k. n y y (16) Substituting the transormations (1) into (15) and (16), e obtain 1 2Re C (), (1 ) k x Re Nu n () x x (17) k 2L here Re x x L xu e is the local Reynolds number. 2. NUMERICAL METHOD The system o non-linear ordinary dierential Eqs. (12) and (13) along ith boundary conditions (14) orms a to-point boundary value problem. A total o ive initial conditions are required to solve these equations. Hence, to initial conditions ƒ() and () hich are not prescribed are to be obtained, using the numerical shooting technique by utilizing the Natchtsheim-Siget iteration technique (Alan Adams and Rogers, 1973) and then solved numerically, using the ourth order Runge- Kutta integration scheme. Here, various groups o parameters M, St,, ф, N R and Pr are considered in dierent phases. We have chosen a step size Δɳ =.1, to satisy the convergence criterion o 1-4 in all o dierent phases mentioned above. 3. RESULTS AND DISCUSSION In order to obtain a clear insight into the physical problem, the numerical results are displayed ith the help o graphical and tabular illustrations. In the current investigation, dierent types o nanoluids, 88

5 namely, Cu-ater, Ag-ater Al 2 O 3 -ater and TiO 2 - ater have been studied ith the Prandtl number Pr = 6.2 and the range o particles raction ф is Table 1 Thermophysical properties o the luid and nanoparticles (Oztop and Abu-Nada, 28). C P (J/kg K) K (W/m K) (kg/m 3 ) Water Copper ( Cu) Silver (Ag) Alumina (Al 2 O 3 ) Titanium oxide (TiO 2) the temperature o the luid increases ith the increase in the volume raction o the nanoparticles. This agrees ith the physical behavior, in that, hen the addition o copper increases the thermal Table 2 Comparison o the results o the temperature gradient -() hen St =, M =, =, N R = and = Pr Magyari and Keller (1999) El.Aziz (29) Bidin and Nazar (29) Present results Fig. 2. Velocity proiles o Cu-ater nanoluid or dierent values o volume raction parameter ф. analyzed, or the values beteen and.2. The eect o the thermal stratiication level is considered or a range St 1. The thermophysical properties o the base luid (ater) and the nanoparticles chosen, are listed in Table 1. The inluence o the physical parameters, namely, volume raction ф, suction parameter, magnetic parameter M, stratiication parameter St and the radiation parameter N R in the lo and heat transer analysis, are depicted in Figs In order to test the accuracy o the present results, the values o the temperature gradient '() or M = St = = ф = N R = are compared ith those o the previously published results, as shon in Table 2, and are ound to be in good agreement. Figures 2 and 3 display the eect o the volume raction on the velocity and temperature proiles, or the Cu-ater nanoluid. In the presence o a uniorm strength o the magnetic ield, it is observed that the velocity o the luid decreases, hereas Fig. 3. Temperature proiles o Cu-ater nanoluid ith dierent values o ф. conductivity increases, and then the thermal boundary layer thickness increases. The values o the magnetic parameter are considered as.5, 1 and 1.5 in the present analysis. Figures 4 and 5 exhibit that an increase in the magnetic parameter decreases the velocity proiles, hile increasing the temperature proiles. This clearly indicates that the transverse magnetic ield opposes the lo. This is because, as the magnetic ield increases the Lorentz orce also increases, hich then produces more resistance to the motion o the lo. On the other hand, as the Lorentz orce increases, the luid exhibits aresistance to this orce, by increasing the riction beteen its layers, thereby resulting in an increase in the temperature. Figures 6 and 7 depict the eects o the suction parameter on the velocity and temperature proiles, respectively. It is observed that on increasing value o, the velocity o the luid decreases, and the 89

6 temperature o the luid increases. This is due to the act that as increases, the resistance o the luid increases, and has a tendency to slo don the velocity o the luid, and to increase the temperature proiles. Fig. 4. Velocity proiles o Cu-ater nanoluid or dierent values o magnetic parameter M Fig. 5. Temperature proiles o Cu-ater nanoluid ith dierent values o M. Fig. 6. Velocity proiles o Cu-ater nanoluid or dierent values o suction parameter Fig. 7. Temperature proiles o Cu-ater nanoluid ith dierent values o ƒ Fig. 8. Temperature proiles o Cu-ater nanoluid ith dierent values o N R Fig. 9. Temperature proiles o Cu-ater nanoluid ith dierent values o stratiication parameter. 9

7 o N R ill cause no change in the velocity proile. Fig. 1. Volume raction eects o Cu-ater nanoluid on temperature in stratiied and unstratiied medium. Fig. 11. Velocity proiles o dierent types o nanoparticles hen N R = 2, =.2, M = 1, St =.8 and ф =.1. Table 3 Values o () and () or various parameters ф, St hen M = N R = 1 and =.2 Pr ф () St.8 St () () Fig. 12. Temperature proiles o dierent types o nanoparticles hen NR = 2, =.2, M = 1, St=.8 and ф =.1. This is because o the combined eects o the strength o thermal stratiication, and the presence o the nanoparticle volume raction. Figure 8 shos the inluence o the radiation parameter on the temperature proile. When the radiation parameter increases rom to 2, the thermal boundary layer thickness is increased. This agrees ith the physical behavior that, hen the radiation parameter N R increases, the mean absorption coeicient k ill decrease, hich in turn, increases the divergence o the radiative heat lux. Hence, the rate o radiative heat transerred to the nanoluid ill be increased, so that the nanoluid temperature ill be increased. Since the Eq. (12) is uncoupled rom the temperature equation, the values Figure 9 shos the inluence o the stratiication parameter on the temperature proile. It is observed that the temperature o the Cu-ater nanoluid is decreased signiicantly, ith an increase o the thermal stratiication parameter. From Fig.1 it is noted, that the volume raction is more inluential in the stratiied medium, than in the unstratiied medium. Besides, a negative temperature proile has been observed; this is because or a high thermal stratiication, the temperature at the all is reduced to a value loer than that o the ree stream. It is ound that the eects o the radiation parameter, volume raction, and suction are the same on the temperature proiles, in contrast to the eects o the thermal stratiication parameter. Figures 11 and 12 illustrate the velocity and temperature proiles o dierent nanoparticles, namely, Cu, Ag, Al 2 O 3 and TiO 2. It can be seen, that on using dierent kinds o nanoluids, the momentum and the thermal boundary layer thickness 91

8 change, hich means that the additions o nanoparticles in the base luid, are capable o changing the velocity and temperature proiles in the boundary layer. Table 3 displays the numerical results o () and () or various values o ф and St hen M = N R = 1, =.2 and Pr = 6.2. From Table 3, it is seen that, ith the increase in ф the magnitudes o "() increases, and '() decreases or dierent kinds o nanoluids, hich in turn, increases the skin riction; hereas it decreases the heat transer rate ith the volume raction ф, but increases ith the stratiication parameter St. 5. CONCLUSIONS The present study gives the numerical solutions or the eects o MHD, suction and thermal radiation in the to-dimensional boundary layer lo o nanoluids past an exponentially stretching sheet, embedded in a thermally stratiied medium. The shooting technique, along ith the ourth order Rung-Kutta integration scheme is employed, to obtain the solution o the governing equations. The obtained results are compared ith the knon results available in the literature, and the comparison shos good agreement. The results rom the present study can be summarized as ollos: An increase in the magnetic parameter M, volume raction parameter ф, and the suction parameter decreases the thickness o the velocity boundary layers. An increase in the stratiication parameter, leads to a decrease in the nanoluid s temperature, hereas the opposite is true ith increasing ф, M, and N R. Besides, a negative temperature is observed due to the inluence o thermal stratiication. The addition o nanoparticles in the base luid is capable o changing the velocity and temperature proiles in the boundary layer. Cu-ater nanoluid shos a thicker thermal boundary layer than the other nanoluids. On using dierent kinds o nanoluids, the momentum and the thermal boundary layer thickness change, hich means that the addition o nanoparticles in the base luid are important in the cooling and heating processes. The magnitude o the skin riction coeicient increases, ith increasing values o the volume raction parameter ϕ. The heat transer rate decreases ith increasing values o ф, hile an opposite eect is observed or the stratiication parameter St. REFERENCES Abu-Nada, E. (28). Application o nanoluids or heat transer enhancement o separated lo encountered in a backard acing step. Int. J. Heat Fluid Flo, 29, Alan Adams, J. and Rogers, D.F. (1973). Computeraided heat transer analysis. Mc Gra_Hill book company, Ne York. Bala Anki Reddy, P., Bhaskar Reddy.N., & Suneetha, S. (212), Radiation eects on MHD Flo past an exponentially accelerated isothermal vertical plate ith uniorm mass diusion in the presence o heat source. Journal o Applied Fluid Mechanics, 5, Bidin, B. and Nazar, R. (29), Numerical solution o the boundary layer lo over an exponentially stretching sheet ith thermal radiation. Euro J. Sci. Res,. 33, Brester, M.Q. (1992). Thermal radiative transer properties, Wiley, Ne york. Brinkman, H.C. (1952). The Viscosity o concentrated suspensions and solution. J. Chem. Phys. 2, Buongiorno, J. (26). Convective transport in nanoluids. Transactions o the ASME. 128, Choi, S.U.S. (1995). Enhancing thermal conductivity o luids ith nanoparticles. Developments and Applications o Non-Netonian Flos. ASME FED. 231/MD 66, Crane, L.J. (197). Flo past a stretching plate. Z. Agne. Math. Phys. 21, Das, S.K., Choi, S.U.S., Yu, W. and Pradeer, T. (27), Nanoluids: Science and Technology. Wiley, Ne Jersey. El-Aziz, M.A. (29). Viscous dissipation eect on mixed convection lo o a micropolar luid over an exponentially stretching sheet. Can. J. Phys. 87, Gupta, P.S., Gupta, A.S. (1977). Heat and mass transer on a stretching sheet ith suction or bloing. Can.J. Chem. Eng. 55, Kakac, S., and Pramuanjaroenkij, A. (29). Revie o convective heat transer enhancement ith 92

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