Mhd Slip Flow Between Parallel Plates Heated with a Constant Heat Flux

MHD SLIP FLOW BETWEEN PARALLEL PLATES HEATED WITH A CONSTANT HEAT FLUX

Ayşegül ÖZTÜRK Trakya University, Department of Mechanical Engineering, 22180 Edirne, Turkey, [email protected]

(Geliş Tarihi: 12. 04. 2011, Kabul Tarihi: 06. 07. 2011)

Abstract: The steady fully developed laminar flow and heat transfer of an electrically conducting viscous fluid between two parallel plates heated with a constant heat flux is investigated analytically in the presence of a transverse magnetic field. The momentum and energy equations are solved under the first order boundary conditions of slip velocity and temperature jump. The effects of the Knudsen number, Brinkman number and Hartmann number on the velocity and temperature distribution and heat transfer characteristics are discussed. Keywords: Parallel plate, MHD, slip-flow, viscous dissipation

SABİT ISI AKISINA MARUZ PARALEL PLAKALAR ARASINDAN MHD KAYMA AKIŞI

Özet: Sabit ısı akısına maruz iki paralel plaka arasından, elektrik iletkenliği olan bir viskoz akışkanın, daimi, tam gelişmiş laminer akışı ve ısı transferi akışa dik bir manyetik alanın varlığında analitik olarak incelenmiştir. Momentum ve enerji denklemleri birinci mertebe kayma hızı ve sıcaklık sıçraması sınır şartları altında çözülmüştür. Knudsen sayısı, Brinkman sayısı ve Hartmann sayısının hız, sıcaklık dağılımı ve ısı transferi karakteristikleri üzerindeki etkisi tartışılmıştır. Anahtar Kelimeler: Paralel plaka, MHD, kayma akışı, viskoz yayılım

NOMENCLATURE Greek symbols B0 magnetic field strength [Tesla] 2 Br Brinkman number [ ]  thermal diffusivity of the fluid [m /s]

Brq modified Brinkman number [ ]  specific heat ratio cp specific heat at constant pressure [J /kg K]  molecular mean free path [m] Fv tangential momentum accommodation  dynamic viscosity [Pa s] coefficient  density [kg/m3] Ft thermal accommodation coefficient 2  electrical conductivity of the fluid [1/(  m)] h convective heat transfer coefficient [W/ m K] 2  kinematic viscosity [m /s] H half distance between plates [m]  dimensionless temperature [ ] k thermal conductivity [W/ m K] modified dimensionless temperature Kn Knudsen number [ ] [ ] M Hartmann number [ ] Nu Nusselt number [=h(2H)/k] dimensionless temperature [ ] p pressure [Pa] modified dimensionless temperature

P dimensionless pressure [ ]

Pr Prandtl number [ ] -2 q heat flux [W/m ] Subscripts T temperature [K] u velocity component in the x-direction [ m/s] c centerline U dimensionless velocity component in the X- m mean direction s fluid properties at the wall x dimensional axial coordinate [m] w wall X dimensionless axial coordinate y dimensional normal coordinate [m] Y dimensionless normal coordinate

INTRODUCTION and presented predictions for the dissipation in terms of mean flow velocity in pressure-driven and gravity- The researchers have recently focused their attention on driven Poiseuille flows. Viscous dissipation effects in flow and heat transfer in microchannels because of microtubes and microchannels were studied by (Koo several application areas such as electronic industry, and Kleinstreuer, 2003, 2004) using dimensional microfabrication technology and biomedical analysis and experimentally validated computer engineering. Flow and heat transfer characteristics in simulations. Their results showed that ignoring viscous microscale exhibit major differences compared to dissipation could affect accurate flow simulations and macroscale. For example, Navier-Stokes and energy measurements in microconduits. Laminar slip-flow equations with no slip and no temperature jump are no forced convection in rectangular microchannels was longer valid if the characteristic length scale is in studied analytically by (Yu and Ameel, 2001) by micron range. In micron range, gas no longer reaches applying a modified generalized integral transform the velocity or the temperature of the surface and technique and found that increasing temperature jump therefore a slip condition for the velocity and a jump for reduces heat transfer and thermal entrance length and temperature have to be adopted. Another parameter that Nusselt number shows a decrease with increasing aspect should be taken into account in flows in micron range is ratio. The wall effect on heat transfer characteristics was viscous dissipation. As a result of shear stress, viscous investigated by (Li et al., 2000) for laminar flow dissipation behaves as an energy source and therefore through microchannels. Their investigation showed that affects the temperature distribution. the change in thermal conductivity of gas in wall adjacent layer has a significant influence on the heat There are four different flow regimes depending on the transfer when passage size is small. (Aydın and Avcı, value of Knudsen number, which is defined as the ratio 2007) investigated forced convective heat transfer in a of the gas mean free path to the characteristic dimension micro channel heated with a constant heat flux or a in the flow field. constant wall temperature. They determined the interactive effects of the Brinkman number and the Continuum flow Kn <10-3 Knudsen number on the Nusselt numbers analytically. Slip Flow 10-3< Kn <10-1 Heat and fluid flow in a rectangular microchannel filled Transition flow 10-1< Kn <10 with a porous medium was analyzed by (Hooman, Free molecular flow 10< Kn 2008). The results showed that as the permeability is decreased, velocity distribution tends to be more The Knudsen number is a measure of rarefaction and as uniform and therefore Nusselt number increases. Kn increases, rarefaction effects become more (Shams et al., 2009) studied slip flow in rhombus important, and eventually continuum approach breaks microchannels and concluded that aspect ratio and down. Knudsen number have important effect on Poiseuille number and Nusselt number. The results also showed A great deal of research have been undertaken to study that Reynolds number has also considerable influence the fundamentals of the flow and heat transfer in on Nu number at low Re numbers. (Darbandi et al. , microchannels. (Beskok et al., 1996) proposed models 2008) presented a theoretical approach to predict the and a computational strategy to simulate gas micro flow temperature field in micro-Poiseuille channel flow with in the slip flow region for Knudsen number less than constant temperature. Their formulations predict the 0.3. (Arkilic et al., 1997) made an analytic and temperature field in the channel readily compared to the experimental investigation into gaseous flow with slight past analytical solutions. Heat transfer characteristics of rarefaction through long microchannels. Their results gaseous flows in microchannel with negative heat flux showed that the analytic solution for the streamwise were investigated by (Hong and Asako, 2007) and a mass flow corresponds well with the experimental correlation was proposed for the prediction of the wall results. Convective heat transfer for steady laminar temperature. Flow and heat transfer characteristics in hydrodynamically developed flow in circular micro-Couette flow was investigated by (Xue and Shu, microtubes and rectangular microchannels were 2003). Their results showed that the solution of Burnett investigated by (Tunc and Bayazitoglu, 2001, 2002). equations is superior to that of the Navier-Stokes They found that the temperature jump effect diminishes equations at relatively high Kn numbers in the slip flow as the Prandtl number increases. (Hadjiconstantinou and regime. Viscous dissipation effect on heat transfer Simek, 2002) studied the constant wall temperature characteristics of rectangular microchannels was studied convective heat transfer characteristics of a model by (Aynur et al. , 2006). It was found that the viscous gaseous flow in two-dimensional micro and dissipation is negligible for gas flows in microchannels nanochannel under hydrodynamically and thermally since the contribution of this effect on Nu number is fully developed conditions and concluded that the about 1%. (Yu and Ameel, 2001) found a universal Nusselt number decreases monotonically with Nusselt number for laminar slip flow heat transfer at the increasing Knudsen number both in slip flow and entrance of a conduit. They also obtained that the Nu transition regime. The results also showed that axial number expression is valid for both isothermal and heat conduction increases the Nusselt number in slip isoflux thermal boundary conditions and for any conduit flow regime.(Hadjiconstantinou, 2003) studied the geometries. (Wang et al., 2008a, 2008b) used inverse effect of viscous dissipation in small scale gaseous flow

12 temperature sampling method to deal with diatomic gaseous flow and heat transfer in a microchannel. Their results showed that this model can accurately simulate In this study, MHD slip flow between two parallel the flow under uniform heat flux boundary condition. plates heated with a constant heat flux was studied and They also concluded that gaseous rarefaction and the effect of rarefaction, viscous dissipation and compressibility increase with the increase of the wall magnetic field on the flow and heat transfer was heat flux. (Shams et al., 2009) performed a numerical revealed. simulation for incompressible and compressible fluid flows through microchannels in slip flow regime and ANALYSIS found that the effect of compressibility ise more noticeable when relative roughness increases. (Zhang et In this study, slip flow and heat transfer between al., 2010) studied the effect of viscous heating on heat parallel plates heated with a constant heat flux is transfer performance in microchannel slip flow region investigated analytically in the presence of a transverse and found that the viscous heating causes severely magnetic field. The geometry and the coordinate system distortions on the temperature profile. (Aydın and Avcı, are shown in Figure 1.The flow is assumed to be steady, 2006) performed an analysis on forced convection heat laminar, incompressible and fully developed and fluid transfer in a micropipe in slip flow and analytically properties to be constant. The magnetic Reynolds determined the Nusselt number as a function of number is assumed very small and the induced magnetic Brinkman number and Knudsen number. The Knudsen field due to motion of the electrically conducting fluid is number was shown to decrease the Nusselt number for neglected. Hall effect and Joule heating are also low values of the Brinkman number. Laminar forced neglected. The coordinate system is located at the center convection slip-flow in a micro-annulus between two of the microchannel. The x-axis is taken along the concentric cylinders was studied by (Avcı and Aydın, centerline of the microchannel and y-axis is taken 2008). The results showed that an increase at Kn normal to it. u is the dimensional velocity component in number decreases the Nu due to the temperature jump at the x-direction. Under these conditions, the governing the wall. Furthermore, Nusselt number decreases with equations including the viscous dissipation effect are as increasing Brinkman number for the hot wall and follows: increases with increasing Brinkman number for the cold B0 , qw wall. They also found that Br is more effective on Nu for lower values of Kn than for higher values of Kn. As is seen from the literature given above, there are many studies on slip flow in microchannels. In these studies, y various parameters such as geometry and boundary 2H u conditions have been into consideration to reveal the x effect of these parameters on flow and heat transfer. But there is only a little number of studies on the effect of magnetic field on the flow. In one of these studies,(Soundalgekar, 1966) studied MHD Couette flow of a rarefied gas and found that velocity profile is B0 , qw affected considerably by magnetic field. (Soundalgekar, Figure 1. Geometry and the coordinate system. 1967) also investigated MHD slip flow and heat transfer in a channel with walls having different temperatures 2 d u 1 dp B2 and found that Nusselt number decreases with an 0 2 u increase in the slip parameter and temperature jump dy d coefficient. (Makinde and Chinyoka, 2010) analyzed T 2T du 2 u MHD transient flow and heat transfer of a dusty fluid 2 y cp dy numerically in a channel with Navier slip condition and found that an increase in the Hartmann number causes a Here c T , are the dynamic decrease in the velocity and therefore in the dissipation. p viscosity, pressure, magnetic field strength, thermal A new approximate solution for the velocity profile of diffusivity, kinematic viscosity, specific heat, steady incompressible MHD flow in a rectangular temperature and electrical conductivity of the fluid, microchannel was proposed by (Kabbani et al., 2008). respectively. In a slip flow regime, due to the Their results showed that the proposed solutions agree rarefaction effect, flow will slip and the temperature better with existing experimental and computational will jump at the wall. First order velocity-slip and reports than previous approximations. (Smolentsev, temperature-jump boundary conditions are defined as 2009) studied MHD duct flows under hydrodynamic (Gad-el-Hak, 2006): slip condition and concluded that the Hartmann layers still exhibit in the duct flows with the slip. (Srinivas and u Muthuraj, 2010) examined the problem of MHD flow in u v (3 s y a vertical wavy porous space in the presence of a v temperature-dependent heat source with slip flow t 2 T T T boundary condition. They concluded that cross velocity s t 1 Pr y profiles exhibit oscillatory character.

13 where u is the slip velocity, the molecular mean free T dT dT s s 1 path, Fv the tangential momentum accommodation d d coefficient, Ts the temperature of the gas at the wall, Tw the wall temperature, Ft thermal accommodation Dimensionless temperature can be defined as: coefficient, the specific heat ratio and Pr the Prandtl number. F and F are dependent on the gas and material T T v t s 1 of the surface and their values near unity are typical for Ts Tc engineering applications (particularly for air) (Zhang, 2007). Therefore, they are taken as unity. Dimensionless energy equation then takes the following form: Velocity Profile 2 d 2 Dimensionless variables are defined as: a Br 1 2 d

y u u pH s H2u dT s P ( where a m s and Br is the Brinkman H H um um um Ts Tc d number, which shows the effect of viscous dissipation where um is the mean velocity and is defined as and it is defined as:

H 2 1 um um udy Br 1 2H H k Ts Tc

The dimensionless momentum equation in the x- Dimensionless thermal boundary conditions therefore direction and associated boundary conditions for take the following form: hydrodynamically fully-developed MHD Poiseuille flow take the following form: d 1 0 1 d 0 d2 dP 0 at 1 M2 2 d The solution of Eq. (17) under thermal boundary d 0 conditions given in Eqs. (19-20) are: d 0

d s 2Kn Ts T d 1 Ts Tc where M is the Hartmann number defined as: Br 2 1 Br 2 21 2 3 2 1 M 1 1 2 2 2 2 B H 3 cosh M KnM sinh M 22 0 10 2 M M M2 cosh 2M 2M2 1 23

The analytical solution of Eq.(7) subjected to the 1 1 boundary conditions given in Eqs.(8-9) is obtained as cosh M cosh M M2 1 2 M 2 follows: 1 KnM2 sinh M 1 2 M 1 2 M 2 cosh 2M cosh 2M M cosh M cosh M 2KnMsinh M 1 1 2 sinh M 1 2KnM2 Mcosh M 1 M 1 1 1 2 2

We can also use the modified Brinkman number defined Temperature Distribution as:

The constant heat flux is assumed at the wall. u2 Br m 2 T H k 1 y y H In this case, temperature distribution takes the following

form: The temperature gradient for the fluid is equal to the T T temperature gradient of the wall for the constant heat s Br 1 Br 2 H 2 2 flux case: 2 2 2 k

14 M3 12M2 12 Kn2M KnM2 M sinh 2M 2M 2 1 2 2 2 M 3 2 cosh M M 2 1 1 2 2 M cosh M cosh M M 1 2 1 M 2 sinh 2M 2KnM 12KnM2 1 KnM2 sinh M 2 1 2 12M 2 2KnMsinh M 3 M cosh 2M cosh 2M M2 2M2 2 1 30 sinh 3M KnM2 sinh M KnM 13 M Eqs. (21) and (27) are in terms of T . These equations KnM2 cosh 3M 2KnM2 M2 s 32 can be transformed into the equations in terms of Tw 1 using the following conversion formulas: 2 cosh M KnM2 M 33M2 3

M Ts T Kn 2 31 cosh 2M 2M 1 0 T T 1 Pr s c 1

1 2 2 Ts T Kn 1 cosh M M 2 KnM sinh M 32 2M H 1 Pr 1 1 1 k M

Therefore, Eqs. (21) and (27) become as follows:

T T Br 1 Br 1 2 2 Ts Tc 2 3 2 ( 2

(33 3 1 M 2 2 2M 2M cosh 2M Kn M 10 cosh M M 2 1 2M 1 1 2 sinh M 1 KnM2 3 12Kn M sinh 2M KnM M 2M 2 2 KnM2 1 Kn2M 12Kn M2 3 M 2 11 2Msinh 2M cosh 2M 2M 1 3

M2 M Mcosh 3M Kn 1 cosh M 3Kn and 1 1 3 1 M2 2 sinh 3M 12KnM2

(3 2 sinh M 32KnM 12KnM

The forced convective heat transfer coefficient is For fully developed flows, mean fluid temperature is defined as: usually used instead of the centerline temperature in defining the Nusselt number. The mean temperature is defined as: h T Tm

uTdA Therefore Nusselt number can be given as: T 3 m udA d 2 The dimensionless mean temperature in terms of d 1 Brinkman and modified Brinkman number can be u m written as : 2

T T m m 12 13 1 m Br 2 1 Br 3 Ts Tc 2 2 3 2 2 After the necessary substitution, the Nusselt number is obtained as (3

1 2 2 12 cosh 2M Kn M KnM M 2 M2

2 1 For u Br 2 1 1 Br 2 2 3 2 2 12 13 1 lim Br 1 Br M 0 2 3 2 2 3 2 2 (55)

As is shown, temperature distribution for MHD slip

flow approaches asymptotically to that for the ordinary

slip flow (Aydın and Avcı, 200 .

M 1 1 sinh M cosh M RESULTS AND DISCUSSION 2 M 1 KnM2sinh M In this study, hydrodynamically and thermally fully M 2 2 developed MHD slip flow between parallel plates 1 M 2Msinh 2M cosh 2M 2M 1 ( 0 heated with a constant heat flux is studied analytically. The results are obtained for the Knudsen numbers Using the modified Brinkman number, the Nusselt ranging from 0 to 0.1, for the Brinkman number ranging number can also be written as follows: from -0.1 to 0.1 and for the Hartmann number ranging from 0 to 2. Air is taken as the working fluid with

Pr=0.71. The dimensionless axial velocity profile for ( 1 various values of Knudsen number and Hartmann

number are shown in Figure 2. As it can be seen from Asymptotic Behavior of the Velocity and the figure, slip velocity of fluid on the microchannel Temperature Distribution wall increases as the Knudsen number increases. Furthermore, higher slip is associated with the lower The asymptotic behavior of the flow can be obtained by core velocities, as expected. As shown in Figure 2, using the Taylor series expansions. presence of a transverse magnetic field leads to a decrease at velocity with an increase of the velocity gradients near the walls. 2 M 1.0 1 2 Kn 1 3 M=0 2 M=1 2 M2 M=2 1 Kn 1 0.5

For Kn Y 0.0 3 1 2 Kn lim ( 3 M 0 2 1 Kn Kn=0.10, 0.05, 0 -0.5 As is seen, the velocity distribution for MHD slip flow Pr=0.71 approaches asymptotically to that for the ordinary slip flow. If we follow the same procedure, equation for -1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 temperature distribution can be written as: U Figure 2. Fully developed velocity profiles for various values M2 M2 M 3 of Hartmann number. Br Kn 1 120 2 The dimensionless temperature distributions are shown

2 in Figure 3 for different values of the governing M2 M2 M M2 parameters. As was stated before, velocity takes lower 2 Kn 1 1 10 120 2 values with increasing of Kn and M numbers. Therefore, temperature takes higher values.

M2 M M2 2 Kn 1 2 2 Variation of the Nusselt number with the Knudsen 120 2 number is shown in Figures 4 and 5 for various values of the Hartmann number, Brinkman number and M2 M modified Brinkman number. As the Knudsen number 2 Kn 1 2 1 120 increases, temperature jump at the microchannel walls increases due to the rarefaction effect. Temperature

jump at the wall reduces the Nusselt number. The percentage decrease in the Nusselt number in the slip

16 flow range taken into consideration was given in Table 4.8 Br=- 0.1 M=0 1. As it can be observed, decrease in the Nusselt number Br=- 0.05 M=1 4.4 Br= 0 M=2 is more pronounced for negative values of the Brinkman Br=0.05 number, particularly for higher values of Hartmann Br=0.1 number. 4.0

1.0 3.6 Nu M=0

3.2 0.8 Pr=0.71 Pr=0.71

Kn=0.1 2.8 0.6 Kn=0.05 2.4

Y Kn=0 0.00 0.02 0.04 0.06 0.08 0.10 Kn 0.4 Figure 4. Variation of the Nu number with Kn number for various values of Br and M. Br=-0.1 0.2 Br=-0.05 Br=0 5.5 Br=0.05 M=0 Brq=- 0.1 M=1 Br=0.1 Br =- 0.05 5.0 q 0.0 M=2 Brq= 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Br =0.05 ~ q  Br =0.1 4.5 q (a)

4.0 1.0 Nu M=1

Pr=0.71 3.5 0.8 Pr=0.71 3.0

Kn=0.1 0.6 2.5

Y Kn=0.05 0.00 0.02 0.04 0.06 0.08 0.10 Kn Kn=0 0.4 Figure 5. Variation of the Nu number with Kn number for various values of Brq and M. Br=-0.1 Br=-0.05 0.2 Br=0 Table 1. Percentage decrease in Nusselt number between Br=0.05 Kn=0 and Kn=0.1 (slip flow range). Br=0.1 0.0 Br M=0 M=1 M=2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 % % % ~  -0.1 40.16 40.58 41.77 (b) -0.05 37.94 38.27 39.20

1.0 0 35.48 35.68 36.89 M=2 0.05 32.72 32.82 33.03 29.62 29.54 29.25 Pr=0.71 0.1 0.8 Positive values of the Brinkman number mean that the fluid is being heated by a hot wall, while the opposite is 0.6 Kn=0.1 true for negative values of it. For the cold wall case,

Y Kn=0.05 viscous dissipation increases the temperature difference

0.4 Kn=0 between the wall and the fluid. Therefore, the Nusselt number increases with an increase of the Brinkman Br=-0.1 number in the negative direction. On the other hand, Br=-0.05 0.2 viscous dissipation decreases the temperature difference Br=0 Br=0.05 between the wall and the fluid for the hot wall case. Br=0.1 Therefore, the Nusselt number takes lower values with 0.0 increasing Brinkman number. Because of the increase in 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 ~ the velocity gradient on the microchannel walls, Nusselt  (c) number increases with an increase of the Hartmann Figure 3. Variation of the temperature profiles for a) M=0, b) number. A comparison between Figures 4 and 5 shows M=1 and c) M=2. that the behavior of the Nusselt number versus the Knudsen number for different Brq is similar to for different Br. The variation of the Nusselt number with

17 Brinkman number and modified Brinkman number is shown in Figures 6 and 7 for various values of Knudsen In order to validate the analysis, the results of this study number and Hartmann number. for M=0 were compared with the results available in the literature (see Table 3). As it can be observed from 1000 Table 3, there is a good agreement between the results. Kn= 0

Kn= 0.05 Table 3. The fully developed Nusselt numbers for M=0 Kn= 0.10 (Pr=0.71). 100 Hadjiconstantinou, Aydın and Kn Present (2003) Avcı, (2007) Br 0 0.01 0 0.01 0 0.01 10 0 4.118 4.077 4.118 4.086 4.118 4.077

logNu 0.05 3.271 3.258 3.271 3.258 3.271 3.258 0.1 2.657 2.652 2.657 2.651 2.657 2.652

M=0 1 M=1 CONCLUSION M=2 In this study, the effect of transverse magnetic field in Pr=0.71 slip flow in a microchannel between two parallel plates 0.1 heated with a constant heat flux is studied analytically. -20 -15 -10 -5 0 5 10 15 20 25 30 It was found that the velocity and temperature field Br Figure 6. Variation of the Nu number with Br for various were strongly affected by the presence of magnetic field values of Kn and M. and the Kn number. As the Knudsen number increases, Nusselt number decreases because of the increasing 100 temperature jump at the channel walls. The Nusselt Kn=0 number increases with an increase of the Brinkman Kn=0.5 Kn=1 number in the negative direction for the cold wall case and decreases with an increase of the Brinkman number 10 for the hot wall case. The Nusselt number takes higher values with an increase of the Hartmann number.

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Ayşegül ÖZTÜRK received her BSc degree in Mechanical Engineering Department of Trakya University in 1984 and received her MSc and PhD from Heat Process division of Institute of Natural Science of Yıldız Technical University in 1989 and 1995 respectively. She is an associate professor of thermodynamic division of Department of Mechanical Engineering at the Trakya University, Edirne. Her research interests are fluid mechanics and heat transfer.