Electrohydrodynamic nanofluid flow and heat transfer

Journal of Molecular Liquids 216 (2016) 583–589

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Electrohydrodynamic nanofluid flow and heat transfer between two plates Houman B. Rokni a, Dhafer M. Alsaad b,⁎, P. Valipour c a b c

Department of Mechanical and Materials Engineering, Tennessee Technological University, Cookeville, TN 38505, USA School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85287, USA Department of Textile and Apparel, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

a r t i c l e

i n f o

Article history: Received 5 July 2015 Received in revised form 14 January 2016 Accepted 23 January 2016 Available online 10 February 2016 Keywords: Electrohydrodynamic Magnetohydrodynamic Nanofluid Rotating system

a b s t r a c t In this paper electro-magneto-hydrodynamics effects on nanofluid flow and heat transfer characteristics in a rotating system are studied. The fourth-order Runge–Kutta method is used in order to solve the governing equations. Effects of electric parameter, magnetic parameter, Reynolds number and rotation parameter on the magnitude of the skin friction coefficient and rate of heat transfer have been considered. Results show that the Nusselt number increases with an increase of the magnetic parameter, electric parameter and Reynolds number but it decreases with an increase of the rotation parameter. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The study of magnetic field effects has important applications in physics, chemistry and engineering. Industrial equipment, such as magnetohydrodynamic (MHD) generators, pumps, bearings and boundary layer control is affected by the interaction between the electrically conducting fluid and a magnetic field. The work of many investigators has been studied in relation to these applications. One of the basic and important problems in this area is the hydromagnetic behavior of boundary layers along fixed or moving surfaces in the presence of a transverse magnetic field. MHD boundary layers are observed in various technical systems employing liquid metal and plasma flow transverse of magnetic fields [1]. Magnetohydrodynamics treats the phenomena that arise in fluid dynamics from the interaction of an electrically conducting fluid with the electromagnetic field [2]. The study of an electrically conducting fluid flow under a transversely applied magnetic field has become the basis of many scientific and engineering applications [3]. Sheikholeslami and Ellahi [4] studied three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid. They found that thermal boundary layer thickness increases with an increase of Lorentz forces. Onyejekwe [5] has analyzed the problem of unsteady magnetohydrodynamic flow of an electrically conducting, laminar, incompressible, Newtonian, viscous fluid through a tube exposed to screen electrodes placed centrally in the flow field and at the two ends of the tube. The governing differential equations which comprise such effects as the electric field, the magnetic and flow fields are ⁎ Corresponding author. E-mail addresses: [email protected] (H.B. Rokni), [email protected] (D.M. Alsaad).

http://dx.doi.org/10.1016/j.molliq.2016.01.073 0167-7322/© 2016 Elsevier B.V. All rights reserved.

non-dimensionalized and solved numerically. Force convective heat transfer of magnetic nanofluid in a lid driven semi-annulus enclosure has been investigated by Sheikholeslami et al. [6]. Their results showed that the Nusselt number has a direct relationship with the Reynolds number and nanoparticle volume fraction while it has a reverse relationship with the Hartmann number. Magnetohydrodynamic squeezing flow of nanofluid over a porous stretching surface was investigated by Hayat et al. [7]. Sheikholeslami et al. [8] simulated electric field effect on nanofluid flow and heat transfer. They proved that the effect of the electric field on heat transfer is more pronounced at low Reynolds number. Mahmoudi et al. [9] investigated the natural convection in a square enclosure filled with a water–Al2O3 nanofluid in the presence of a magnetic field and uniform heat generation/absorption. They observed that adding nanoparticle reduces the entropy generation. The nanoparticle effect is more intense for high Hartmann number. Recently several authors investigated about the effect of magnetic and electric fields on flow style [10–28]. From an energy saving perspective, improvement of heat transfer performance in systems is a necessary subject. Low thermal conductivity of conventional heat transfer fluids such as water and oils is a primary limitation in enhancing the performance and the compactness of systems. Solids typically have a higher thermal conductivity than liquids. For example, copper (Cu) has a thermal conductivity 700 times greater than water and 3000 times greater than engine oil. An innovative and novel technique to enhance heat transfer is to use solid particles in the base fluid (i.e. nanofluids) in the range of sizes 10–50 nm. Sheikholeslami and Abelman [29] used two phase simulation of nanofluid flow and heat transfer in an annulus in the presence of an axial magnetic field. The MHD steady flow of viscous nanofluid flow over a stretching sheet was studied by Sandeep et al. [30]. They

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H.B. Rokni et al. / Journal of Molecular Liquids 216 (2016) 583–589

Nomenclature A1,A2,A3,A4,A5 Dimensionless constants B0 Magnetic field Cp Specific heat at constant pressure ~ Cf;C Skin friction coefficients f Electric parameter Eh f(η), g(η) Similarity functions h Distance between the plates k Thermal conductivity Kr Rotation parameter Nu Nusselt number P* Modified fluid pressure Pr Prandtl number qw Heat flux at the lower plate R Reynolds number u, v, w Velocity components along x, y, z axes, respectively uw Velocity of the stretching surface Greek symbols α Thermal diffusivity η Dimensionless variable θ Dimensionless temperature ρ Density ϕ Nanoparticle volume fraction σ Electrical conductivity μ Dynamic viscosity υ Kinematic viscosity τw Skin friction or shear stress along the stretching surface Ω Constant rotation velocity

Cartesian coordinate system (x,y,z) is considered as follows: the x-axis is along the plate, the y-axis is perpendicular to it and the z-axis is normal to the x–y plane (see Fig. 1). The origin is located at the lower plate, and the plates are located at y = 0 and y = h. The lower plate is being stretched by two equal opposite forces so that the position of the point (0,0,0) remains unchanged. A uniform magnetic flux with densityB0 acts along the y-axis. Also an electrical field (E = E0xy) acts along the z-axis. Under these assumptions, the Navier–Stokes and energy equations are: ∂u ∂v ∂w þ þ ¼0 ∂x ∂y ∂z

ð1Þ

! 2 2 ∂u ∂u ∂p ∂ u ∂ u þ þ μ nf ρnf u þν þ 2Ωw ¼ − ∂x ∂x ∂y ∂x2 ∂y2 −σ nf B0 ðuB0 þ EÞ;

ð2Þ

! 2 2 ∂v ∂p ∂ v ∂ v ρnf u þ þ μ nf ¼− ∂y ∂y ∂x2 ∂y2

ð3Þ

Subscripts ∞ Condition at infinity nf Nanofluid f Base fluid s Nano-solid-particles

observed that the heat and mass transfer rate in Oldroyd-B nanofluid is significantly high compared with the Jeffery and Maxwell nanofluids. Sheikholeslami et al. [31] investigated the effect of a variable magnetic field on force convection heat transfer. Their results indicated that the effects of Kelvin forces are more pronounced for high Reynolds number. The turbulent forced convection of nanofluids in shallow cavity heated from sides with uniform temperature was analyzed by Abdellahoum et al. [32]. Their obtained results show that the heat transfer is enhanced with the studied parameters for all models of thermal conductivity. Squeezing unsteady nanofluid flow and heat transfer have been studied by Sheikholeslami et al. [33].They showed that for the case in which two plates are moving together, the Nusselt number increases with an increase of nanoparticle volume fraction and Eckert number while it decreases with growth of the squeeze number. Recently several authors investigated about nanofluid flow and heat transfer [34–68]. The objective of the present paper is to study nanofluid flow and heat transfer between two horizontal parallel plates in a rotating system in the presence of both electrical field and magnetic field. The reduced ordinary differential equations are solved numerically using the fourth-order Runge–Kutta method. The effects of the parameters governing the problem are discussed.

2. Problem statement Consider steady flow of nanofluid between two horizontal parallel plates when the fluid and the plates rotate together with a constant angular velocity Ω around the axis which is normal to the plates. A

Fig. 1. Schematic theme of the problem geometry.

Table 1 Thermo-physical properties of water and nanoparticles.

Water Cu

ρ (kg/m3)

Cp (j/kg ⋅ k)

k (W/m × k)

β × 105 (K−1)

σ (Ω × m)−1

997.1 8933

4179 385

0.613 401

21 1.67

0.05 5.96 × 107


585

! 2 2 ∂w ∂w ∂ w ∂ w ρnf u þ 2 −σ nf B0 2 w; þν −2Ωu ¼ μ nf ∂x ∂y ∂x2 ∂y

ð4Þ

! 2 2 2 ∂T ∂T ∂T ∂ T ∂ T ∂ T þ þ ρ C p nf u þv þw ¼ knf ∂x ∂y ∂z ∂x2 ∂y2 ∂z2 0 " 1 # 2 2 2 ∂u ∂v ∂w B2 C þ þ B C ∂x ∂y ∂z C þμ nf B B 2 2 2C @ ∂v ∂v ∂w ∂u A þ þ þ þ ∂x ∂z ∂x ∂z

ð5Þ

where u, v and w denote the fluid velocity components along the x, y and z directions respectively, p*is the modified fluid pressure, T is the temperature and the physical meanings of the other quantities are mentioned in the nomenclature. The absence of ∂ p * /∂ z in Eq. (4) implies that there is a net cross-flow along the z-axis. The corresponding boundary conditions of Eqs. (1)–(5) are:

Fig. 2. Comparison of the temperature profiles between the present work and [69] for different values of Pr when λ=0.5, M=1, R=0.5 and Kr=0.5.

u ¼ a x; v ¼ 0; w ¼ 0; T ¼ T h at y ¼ 0 u ¼ 0; v ¼ 0; w ¼ 0; T ¼ T o at y ¼ h:

Fig. 3. Effect of electric parameter on velocity and temperature profiles when Kr=10, R=1, Ec=0.01, M=2 and ϕ=0.04.

ð6Þ

586


Fig. 4. Effect of magnetic parameter on velocity and temperature profiles when Kr=10, R=1, Ec=0.01, M=2 and ϕ=0.04. Kr=10, R=10, Ec=0.01, ϕ=0.04.

The effective density (ρnf), the effective dynamic viscosity (μnf), the effective heat capacity (ρCp)nf, the effective thermal conductivity (knf) and the electrical conductivity (σnf) of the nanofluid are defined as [36]

Therefore, the governing momentum and energy equations for this problem are given in dimensionless form by: iv

f −R ρnf ¼ ð1−ϕÞρ f þ ϕρs ;

μ nf ¼

μf ð1−φÞ2:5

knf ks þ 2 k f −2ϕ k f −ks ; ¼ kf ks þ 2 k f þ 2ϕ k f −ks

σ nf σf

;

ρ C p nf ¼ ð1−ϕÞ ρC p f þ ϕ ρC p s

σs 3 −1 ϕ σf ¼1þ σs σs þ2 − −1 ϕ σf σf

ð7Þ

y ; h

w ¼ ax g ðηÞ;

0

u ¼ axf ðηÞ;

ν ¼ −ah f ðηÞ;

T−T o θðηÞ ¼ T h −T o

where the prime denotes differentiation with respect to η.

A1 0 A1 0 M f g−f g 0 þ 2Kr f −A5 g ¼ 0 A2 A2 A2

θ″ þ Pr

A2 A3 A1 0 A1 0 2 R f θ þ Ec 4 f þ g 2 ¼ 0: A1 A4 A2 A3

ð9Þ ð10Þ ð11Þ

The dimensionless quantities in these equations are:

where ϕ is the solid volume fraction of the nanoparticles.

η¼

g ″ −R

A1 0 ″ A1 0 M ″ ‴ f f −f f −2 Kr g −A5 f þ Eh ¼ 0 A2 A2 A2

2 2 2 μ f ρC p f σ f B0 2 h ah Ωh ;M ¼ ; Kr ¼ ; Pr ¼ ; R¼ υf ρf υf υf ρf kf 2

ð8Þ

ρ f a2 h ; ρC p f ðθ0 −θh Þ Eh ¼ E0 =ðB0 ahÞ ρC p nf μ nf ρnf k σ ; A4 ¼ nf ; A5 ¼ nf ; A2 ¼ ; A3 ¼ A1 ¼ ρf μf kf σf ρC p f

Ec ¼

ð12Þ


587

Fig. 5. Effect of rotation parameter on velocity and temperature profiles when R=1, Ec=0.01, Eh =2, M=2 and ϕ=0.04.

where R , M , Kr , Pr , Ec and Eh are the Reynolds number, magnetic parameter, rotation parameter, Prandtl number (Pr), Eckert number (Ec) and electric parameter. The boundary conditions are: 0

f ¼ 0; f ¼ 1; g ¼ 0; θ ¼ 1 0 f ¼ 0; f ¼ 0; g ¼ 0; θ ¼ 0

at η ¼ 0 at η ¼ 1:

ð13Þ

3. Numerical method In order to solve the transformed ordinary differential Eqs. (9)–(11) along with the boundary conditions (13) numerically, we have used the fourth-order Runge–Kutta method. This method is sometimes known as RK4 and is a reasonably simple and robust scheme featuring a shooting technique. The Runge–Kutta method numerically integrates ordinary differential equations using a trial step at the midpoint of an interval to cancel out lower-order error terms. The fourth-order formula is:

The physical quantity of interest in this problem is the skin friction coefficient Cf along the stretching wall, which is defined as

k1 ¼ h f ðxn ; yn Þ

ð16Þ

A2 ″

C f ¼

f ð0Þ

: A1

1 1 k2 ¼ h f xn þ h; yn þ k1 2 2

ð17Þ

1 1 k3 ¼ h f xn þ h; yn þ k2 2 2

ð18Þ

k4 ¼ h f ðxn þ h; yn þ k3 Þ

ð19Þ

ð14Þ

The Nusselt number at the lower plate is defined as

knf

θ0 ð0Þ

Nu ¼

kf

ð15Þ

588


Fig. 6. Effects of Reynolds number and magnetic parameter on skin friction coefficient and Nusselt number when Kr=10, Ec=0.01, Eh =2, M=2, and ϕ=0.04.

1 1 1 1 5 ynþ1 ¼ yn þ k1 þ k2 þ k3 þ k4 þ O h : 6 3 3 6

ð20Þ

4. Results and discussion The electro-magneto-hydrodynamics effect on nanofluid flow and heat transfer characteristics in a rotating system is studied (Fig. 1). Thermo physical properties of water and nanoparticles are shown in Table 1. The comparison between the current result and those reported by Mehmood and Ali [69], shows an excellent agreement (Fig. 2). The effect of the electric parameter on velocity and temperature profiles is shown in Fig. 3. Velocity boundary layer thickness increases with an increase of the electric parameter. As the electric parameter increases the thermal boundary layer thickness decreases. Fig. 4 shows the effect of the magnetic parameter on velocity and temperature profiles. The presence of the magnetic field decreases the momentum boundary layer thickness due to the Lorentz force. The thermal boundary layer thickness decreases with increasing magnetic parameter. The skin friction coefficient decreases with augmentation of the magnetic parameter

and electric parameter. The Nusselt number enhances with an increase of the magnetic parameter while it decreases with an increase of the electric parameter. The effect of the rotation parameter on velocity and temperature profiles is shown in Fig. 5. As the rotation parameter increases, the Coriolis force increases which results in an increase in rotational velocity. So the skin friction coefficient increases with an increase of the rotation parameter. Also increasing the rotation parameter leads to a decrease in the Nusselt number due to an increase in the thermal boundary layer thickness. The effects of the Reynolds number on velocity and temperature profiles are shown in Fig. 6. Both velocity and temperature profiles decrease as Re increases and in turn increasing the Reynolds number leads to an increase in the magnitude of the skin friction coefficient and Nusselt number. 5. Conclusions In the present study combined effects of electrohydrodynamic and magnetohydrodynamic on nanofluid hydrothermal behavior in a rotating system is considered. The effects of the electric parameter, magnetic


parameter, Reynolds number and rotation parameter on the magnitude of the skin friction coefficient and rate of heat transfer have been examined. The skin friction coefficient increases as the rotation parameter and Reynolds number increase but it decreases as the magnetic parameter and electric parameter increase. The Nusselt number has a direct relationship with the magnetic parameter, electric parameter and Reynolds number but it has a reverse relationship with the rotation parameter. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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