Journal of Applied Fluid Mechanics, Vol. 9, No. 3, pp. 1081-1088, 2016. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645.
Electrothermo Convection in a Porous Medium Saturated by Nanofluid R. Chand1†, G. C. Rana2 and D. Yadav3 1 Department of Mathematics, Government College Nurpur, Himachal Pradesh, 176202, INDIA Department of Mathematics, Government College Hamirpur, Himachal Pradesh, 177033, INDIA 3 School of Mechanical Engineering, Yonsei University Seoul, South Korea
2
†Corresponding Author Email:
[email protected] (Received April 2, 2015; accepted July 1, 2015)
ABSTRACT Thermal instability in a horizontal layer of nanofluid with vertical AC electric field in a porous medium is investigated. The flux of volume fraction of nanoparticles is taken to be zero on the isothermal boundaries and the eigenvalue problem is solved using the Galerkin method. Darcy model is used for the momentum equation. The model used for nanofluid incorporates the effect of Brownian diffusion and thermophoresis. Linear stability theory based upon normal mode technique is employed to find the expressions for Rayleigh number for stationary and oscillatory convection. Graphs have been plotted to study the effects of Lewis number, modified diffusivity ratio, concentration Rayleigh number, AC electric Rayleigh number and porosity on stationary convection.
Keywords: Nanofluid; AC electric Rayleigh number; Brownian motion; Galerkin method; Porosity. NOMENCLATURE a dimensionless resultant wave number d thickness of fluid layer D differential operator DB Brownian diffusion coefficient DT thermophoretic diffusion coefficient g acceleration due to gravity km thermal conductivity Le Lewis number n growth rate of disturbances NA modified diffusivity ratio modified particle -density increment NB p pressure q Darcy velocity vector Ra thermal Rayleigh number Rac critical Rayleigh number Rn concentration Rayleigh number t time T temperature T0 temperature at z = 0 T1 temperature at z = d (u, v, w) Darcy velocity components (x, y, z) space co-ordinates α coefficient of the thermal expansion μ viscosity ε porosity
ρ ρ0 ρp
density of the nanofluid density of nanofluid at z = 0
ρf
density of base fluid heat capacity of fluid in porous medium
(ρc)m
density of nanoparticles
ω σ
heat capacity of nanoparticles volume fraction of the nanoparticles reference volume fraction of the nanoparticles at z = 0 thermal diffusivity dimensionless frequency of oscillation thermal capacity ratio
2H
horizontal Laplacian operator
2
Laplacian operator
(ρc)p φ φ0
κ
Superscripts ' non - dimensional variables '' perturbed quantities Subscripts c critical s stationary convection p particle b basic state f fluid
R. Chand et al. / JAFM, Vol. 9, No. 3, pp. 1081-1088, 2016.
1.
fluids.
INTRODUCTION
When a small amount of nano-sized particles are added to the base fluid, the thermal conductivity of the fluid enhances and such a fluid is called nanofluid which was first coined by Choi (1995). Nanofluids have unique properties that make them potentially useful in many applications of heat transfer and thus nanofluids considered to be the next-generation heat transfer fluids. Recent developments in the study of heat transfer using nanofluids can be reported by Wong and Leon (2010), Yu and Xie (2012), Taylor et al. (2013). Thermal convection in nanofluids in a porous medium is an important phenomenon due to its wide ranges of applications in geophysics, food processing, oil reservoir modeling, petroleum industry, bio-mechanics, building of thermal insulations and nuclear reactors. The detailed study of thermal convection in a layer of nanofluid in porous medium based upon Buongiorno (2006) model has been given by Kuznetsov and Nield (210a, b, c), Nield and Kuznetsov (2009, 2010, 2011), Chand (2013), Chand and Rana (2012a, b, c), Rana et al. (2014a, b), Yadav and Kim (2015), Chand et al. (2015) and Yadav and Lee (2015). In all the above studies boundary condition on volume fraction of nanoparticle is physically not realistic as it is difficult to control the nanoparticle volume fraction on the boundaries and suggested the normal flux of volume fraction of nanoparticles is zero on the boundaries as an alternative boundary condition which is physically more realistic. Nield and Kuznetsov (2014), Chand and Rana (2014a, b), Chand et al. (2014) pointed out that this type of boundary condition on volume fraction of nanoparticles is physically not realistic as it is difficult to control the nanoparticle volume fraction on the boundaries and suggested the normal flux of volume fraction of nanoparticles is zero on the boundaries as an alternative boundary condition which is physically more realistic. Under the circumstances, it is desirable to investigate convective instability problems by utilizing these boundary conditions to get meaningful insight in to the problems. Natural convection under AC/DC electric field of electrically enhanced heat transfer in fluids and possible practical applications has been reviewed by Jones (1978) and Chen et al. (2003). Stiles et al. (1993) studied the problem of convective heat transfer through polarized dielectric liquids. They observed that the convection pattern established by the electric field is quite similar to the familiar Bénard cells in normal convection. Recently Shivakumara et al. (2012) studied the effect of velocity and temperature boundaries conditions on electro-thermal convection in a rotating dielectric fluid and found that AC electric field is to enhance the heat transfer and to hasten the onset of convection. Several studies have been carried out to assess the effect of AC and DC electric fields on natural convection due to the fact that many problems of practical importance involve dielectric
The nano dielectric fluid may be used in an electrical apparatus and other electrical equipment such as distribution transformers, regulating transformers, shunt reactors, converter transformers, instrument transformers and power transformers. The nano dielectric fluid used herein exhibits an increased thermal conductivity as compared to the insulating liquids commonly used without the nanoparticles. A high thermal conductivity of the nanoparticles is often desirable for the nano dielectric fluids. Although EHD instability has been extensively investigated in Newtonian and non-Newtonian fluid dielectric fluid layer, but no effort has been made to study the EHD instability in a layer of nanofluid. So keeping in view the importance of thermal convection of dielectric (when fluid layer is subjected to a uniform vertical AC/DC electric field) nanofluids, an attempt has been made to study the electro thermo convection in a horizontal layer of nanofluid in a porous medium.
2.
MATHEMATICAL FORMULATIONS PROBLEM
OF
THE
Consider an infinite horizontal layer of nanofluid of thickness‘d’ bounded by planes z = 0 and z = d. Fluid layer is heated from below in a porous medium whose medium permeability is k1 and porosity is ε. Nanoparticles are being suspended in the nanofluid using either surfactant or surface charge technology, preventing the agglomeration and deposition of these on the porous matrix. Layer of fluid is subjected to a uniform vertical AC electric field applied across the layer; lower surface is grounded and upper surface is kept at an alternating potential whose root mean square is 1. A Cartesian coordinate system (x, y, z) is chosen with the origin at the bottom of the fluid layer and the z- axis normal to the fluid layer in the gravitational field g (0,0,-g). The normal component of the nanoparticles flux has to vanish at an impermeable boundary and the temperature T is taken to be T0 at z = 0 and T1 at z = d, (T0 > T1) as shown in Fig. 1. The reference scale for temperature and nanoparticles fraction is taken to be T1 and φ0 respectively. According to the works of Chandrasekhar (1981), Nield and Kuznetsov (2014) and Shivakumara et al. (2012), the relevant equations under the Oberbeck- Boussinesq approximation in a porous medium are q 0,
(1)
0 p φρp 1 φρf01 αT T0 g
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1 E E 2
k1
q ,
(2)
R. Chand et al. a / JAFM, Voll. 9, No. 3, pp. 1081-1088, 201 16.
Heated from below ation of the prooblem. Fig. 1. Physiical Configura
ρcm T qT km2Tρcp DBφT t
DT TT T0
w 0, T T1, DB
DT T 0 at z d. z T1 z
(3)
φ 1 D q φ DB2φ T 2T , t ε T0
(4)
(9)
2.1 2 Steady State S and its S Solutions
where q is the t Darcy veloocity of fluid, p is the pressure, ρ0 is the density of nanofluid at lower layer, ρp is the t density of nanoparticles, φ is the volume fracttion of the nanoparticles, n T is the temperature, α is coefficcient of the thermal expansion, g is accelerationn due to graviity, k1 is medium perm meability of fluuid, ε is the poorosity of porous mediuum and μ is thee viscosity, (ρcc)m is the heat capacityy of fluid in porous p medium m, (ρc)p is the heat capacity of nanoparrticles, km is thee thermal B conductivity of the fluid, DB is the Brownian diffusion cooefficient, D is the therm moporetic T diffusion coeefficient of the nanoparticles, E is the root mean squuare value of thhe electric fieldd and is the dielectric constant.
The T steady statee is given as
Since there iss no free charge, the relevant Maxwell equations are
Also A we have
E 0,
(5)
E 0 .
(6)
In view of eqquation (5), E can c be expressedd as
E ,
where is the root meann square valuue of the electric potenntial. The dielectric constant is assumed a to bee a linear function of teemperature in thhe form
We assume that the tempeerature is consstant and o the boundariies. Thus nanoparticless flux is zero on boundary connditions (Channdrasekhar 19881, Nield and Kuznetsoov 2014) are
DT T 0 at z 0 z T1 z
(10)
The T solution off steady state is Ts T0 -
and
D ΔT ΔT z, s 0 T z, d D B T1d
T ˆ , s 0 1 z k d Es
, E0 kˆ T 1 z d
where w subscriptt s denote the stteady state.
s z
E0 d
T
T ˆ llog1 k , d
1
T
d is the root mean log 1 T square s value off the electric fieeld at z = 0. E0
2.2 2 Perturbaation Equatioons Let L the initial steady s state as described by eq quation be b slightly pertturbed so that the perturbed state is given g by q q,T TsT, p ps p,s , E Es E, s ,
(8)
where γ(>0) is i the thermal expansion e coeffficient of dielectric connstant and is asssumed to be sm mall.
w 0, T T0 , DB
E Es z, s z.
where w
(7)
0 1 T T0 0 ,
q qs 0, p p(z), T Ts (z) ( , s z,
(11)
where w the prim me denote the perturbed quaantities. Substituting S the Eq. (11) into the Eqs. (1)) – (9), linearizing l by neglecting thee products of primed quantities, q the obtained pertuurbations equattions to be b converted into to non-ddimensional fo orm by introducing i thee following dim mensionless vaariables as a follows x, y, z (x, y, z) , d
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u , v, w (u , v, w ) d, κ
R. Chand et al. / JAFM, Vol. 9, No. 3, pp. 1081-1088, 2016.
κ
t
t , p
k1
p, T
T
,
ΔT μκ d 2 km , where κ the thermal ρc f E 0 ΔTd diffusivity of the fluid is, σ
ρc p m ρc p f
is the
thermal capacity ratio. The linearized perturbation equations in nondimensional form q 0,
(12)
0 p - q RaTeˆ z Rnφeˆ z ,
(13)
N T φ 2N A N B T T , w 2T B Le z z Le z t (14) 1 φ 1 1 N w 2φ A 2T (15) t Le Le
(16)
Where
κ ρ gαTk1d is the Lewis number, Ra 0 DB μκ is the thermal Rayleigh number, ρp ρ φ1 - φ0 gk1d is the nanoparticle Rn μκ Le
electric
Rayleigh
2 E 02 T 2 d 2 μκ
number, NA
modified diffusivity ratio, NB
DT T is DBT1φ0
ρcpφ0 ρcf
the
is the
Non- dimensional boundary conditions are given as 2w z
2
T 0, N z z
A
T z
0
at z = 0 and z = 1.
D
Ra 2HT
- Rn 2H
2
D
. z (18)
(23)
D
is the
The boundary conditions of the problem in view of normal mode analysis are W 0, D2W 0,Θ 0,D 0,D NAD 0 at z 0,1 (24)
4.
LINEAR STABILITY ANALYSIS
For the present formulation, we have considered the case of free -free boundaries for which system of Eqs. (20) - (23) together with the boundary conditions (24) constitute a linear eigenvalue problem with variable coefficient for the growth rate of disturbance of the system. The resulting eigenvalue problem is solved numerically by the Galerkin method of first order (N = 1), which gives the expression for Rayleigh number Ra as 1 a2
2
Le 2 a2 ε nLe 2 a2 σ
n
a2 2 a2 n
NA 2 a2
a2 2
a2
Re
Rn. (25)
For neutral stability, the real part of n is zero. Hence on putting n = iω, ( where ω is real and is dimensionless frequency) in Eq. (25), we have
where
NORMAL MODES ANALYSIS
a 2 D,
Ra 1 i 2 , 3.
d and a k x2 k y2 dz dimensionless the resultant wave number.
(17)
Re 2H T
a 2 W a 2 RaΘ a 2 Rn a 2 Re - D 0, (20) 2NANB N 2 NA W D DD - a 2 n Θ B D 0, Le Le Le (21) WNA N A 2 2 1 2 2 n D a Θ D a 0, Le Le (22)
Ra
Now eliminating the pressure from Eq. (13) by using the identity curl curl = grad div– 2 together with Eq. (12), we obtain z-component of the momentum equation as
w
Using Eq. (19), Eqs. (18), (14) - (16) become
is the AC
modified particle-density increment.
2
where kx and ky are wave numbers in x and y directions respectively, while ‘n’ is the growth rate of disturbances.
Rayleigh number, Re
w
(19)
where
[Dashes ( '' ) are dropped for simplicity]
w,T,, W(z),Θ(z),z, zexpikx x iky y nt,
2
T . z
2
and assuming that the perturbed quantities are of the form
Analyzing the disturbances into the normal modes
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(26)
R. Chand et al. / JAFM, Vol. 9, No. 3, pp. 1081-1088, 2016.
Δ
2
a2
1
Re
2
2
a
2
a2
N
2 2
a
a2 2
A
2 2
a2
found to 4π2, so the corresponding critical Rayleigh Le number given by Ra s 4 2 N A Rn. (32)
Le ω2 ε σε
ωLe σ
2
.
(27)
This is same result which was derived by Nield and Kuznetsov (2010).
Rn
One recognizes that in the absence of nanoparticles (Rn = Le = NA = 0) and AC vertical electric field (Re = 0), one recovers the well- known results that the critical Rayleigh-Darcy number is equal to 4π2.
and
a 2
Δ2
Le Le Le a2 NA ε Rn.
2
2
a2
2
a
ωLe σ
2 2
2
(28) Since Ra is a physical quantity, so it must be real. Hence, it follow from the Eq. (26) that either ω = 0 (exchange of stability, steady state) or Δ2 = 0 (ω ≠ 0 over stability or oscillatory onset).
4.2 Oscillatory Convection
4.1 Stationary Convection For the case of stationary convection [n = ω = 0], Eq. (25) reduces to
Ras π
2
a2 a
2
2
a2
2
differentiated with respect to ‘a2’ and then equated to zero. A polynomial in a2 whose coefficients are function of parameter AC Rayleigh number Re influencing the stability is given as
- π Rea
2 2 a c2
3
2
2 2 c
2 6 a c2 8 0. (30)
The above equation is solved for various value of AC Rayleigh number Re and critical value ac is obtained and then corresponding critical Rayleigh number is obtained. In the absence of electrical field (Re = 0), Eq. (29) reduces to
Ra s π
2
a2 a
2
2
Le NA Rn .
The minimum of first term of right- hand side of
ac
2 2
Since ω2 0 for the occurrence oscillatory convection, but the values of the parameters considered in the range 10 2 Ra 10 5 (thermal 1 N A 10 (modified Rayleigh number), diffusivity ratio), 10 2 Le 10 4 (Lewis number), 10 1 Rn 10 (nanoparticles Rayleigh number), 0 .1 5 (capacity ratio), 0.1 1 (porosity parameter) (Chand and Rana 2012a) and (AC electric Rayleigh number) 10 Re 104 (Shivakumara et al. 2012), the value of ω2 in Eq. (33) is found to be negative, which imply that oscillatory convection is not possible for the problem. 5.
RESULT AND DISCUSSION
An expression for the stationary Rayleigh number, which characterize the stability of the system are obtained for free-free boundary conditions. The computations are carried out for different values of parameters considered in the range 10 2 Ra 10 5 (thermal Rayleigh number), 1 N A 10 (modified diffusivity ratio), 10 2 Le 10 4 (Lewis number),
(31)
This result agrees with the result obtained by Nield and Kuznetsov (2012). Eq. (31) is attained at
2
a 2 1 1 Le 2 a 2 N A Rn - . ε Le Le ε σ (33)
To find the critical value of Ra s , Eq. (29) is
2 4 c
For oscillatory convection ω ≠ 0, we must have Δ2 = 0, thus Eq. (28) gives an expression for frequency of oscillations as
Le Re N A Rn. π a (29) 2
It is observed that stationary Rayleigh number Ra s is function of the Lewis number Le, the modified diffusivity ratio NA, the nanoparticles Rayleigh Rn, porosity ε and AC electric Rayleigh number Re but independent of modified particledensity increment NB. Thus the instability is purely a phenomenon due to buoyancy coupled with the conservation of nanoparticles.
a
Thus presence of the nanoparticles lowers the value of the critical Rayleigh number by usually by substantial amount. Also parameter NB does not appear in the Eq. (29), thus instability is purely phenomena due to buoyancy coupled with conservation of nanoparticles. Thus average contribution of nanoparticles flux in the thermal energy equation is zero with one-term Galerkin approximation.
and minimum value
10 1 Rn 10 (nanoparticles Rayleigh number), 0 .1 5 (capacity ratio), 0 .1 1 (porosity parameter) [Chand and Rana (2012a)] and 10 Re 104 (AC electric Rayleigh number) [Shivakumara et al. (2012)] to find the effects of various parameters on the stationary convection.
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R. Chaand et al. / JAF FM, Vol. 9, No. 3, pp. 1081-10088, 2016. Stabiliity curves for Lewis numbeer Le, modifieed diffusiivity ratio NA, AC electric Rayleigh R numbeer Re, naanoparticles Rayyleigh number Rn and porositty param meter are shown in figures 2-6.
Fig. 4 shows s the variiation of stationary Rayleigh numberss (Ra)s with wave numberr for different value off modified diffuusivity ratio andd fixed value off other paarameters and it is found tthat stationary Rayliegh h number decreeases with an iincrease in the value modified m diffusivvity ratio, indiccating that the modified d diffusivity raatio destabilze the stationary convectiion. This is good g agrement of the result obtained d by Chand andd Rana (2014).
Fig. 2. Variation of the t thermal Raayleigh numbeer (Ra))s with wave nu umber for diffferent value off Lewis number.
o the thermal Rayleigh Fig.. 4. Variation of numbeer (Ra)s with wave w number ffor different value v of modiffied diffusivity ratio. Fig. 5 shows s the variiation of stationary Rayleigh numberss (Ra)s with wave numberr for different values of o nanoparticlee Rayleigh num mber and it is found th hat stationary Rayliegh num mber decreases with an n incease in the value of nanoparticles Rayleigh h number. Thiis is good agrrement of the result ob btained by Channd and Rana (20014).
Fig. 3. Variation of the t thermal Raayleigh numbeer n for diffferent value off (Ra)s with wave number AC electriic Rayleigh num mber. v of thee stationary Raayleigh numberrs The variation (Ra)s with wave number havee been plotteed graphiically.
Fig. 2 shows the vaariation of statiionary Rayleig gh numbeers (Ra)s with h wave numbeer for differen nt value of Lewis num mber Le and fixed value of o nd it is found that stationarry other parameters an Raylieegh number decreases as the values of o Lewiss number incrreases, indicatting that Lewis numbeer destabilize the stationaary convection n. This is i good agrem ment of the ressult obtained by b Chand d and Rana (20 014). Fig. 3 shows the vaariation of stationary Rayleig gh numbeers (Ra)s with h wave numbeer for differen nt value of AC electrric Rayleigh number n Re an nd fixed value v of other parameters and d it is found that station nary Rayliegh number deccreases with an a increase in the vallue of AC ellectric Rayleig gh he AC electriic numbeer Re, indicaating that th Rayleiigh number Re R destabilze the stationarry convecction. This is good agremen nt of the resu ult obtain ned by Shivakum mara et al. (2012).
Fig. 5. Variation of the Rayleigh nu umber (Ra)s witth wave numbeer for differentt value of nanoparticle Rayleigh num mber. Fig. 6 reepresent the varriation of stationary Rayleigh numberss (Ra)s with wave number for different value off porosity paraameter and it is found that stationarry Rayliegh number n increaases with an increase in the value of porosity paarameter, thus porosity y stabilize the stationary conveection. This is good agrrement of the result r obtained by Chand and Rana (20 014).
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R. Chand et al. a / JAFM, Voll. 9, No. 3, pp. 1081-1088, 201 16.
layer satu urating a Darrcy-Brinkman porous medium. Int. Journal of Heat and d Mass Transfer 55, 5 5417-5424 4. Chand, C R. annd G. C. Ranna (2012). Thermal T instability y of Rivlin n-Ericksen ellasticoviscous nanofluid n saturrated by a porous medium. J. J Fluids Eng. 134(12), 12120 03. Chand, C R. annd G. C. Raana (2014). Thermal T instabilityy in a Brinkm man porous medium m saturated by nanofluid with no nanop particle flux on booundaries. Speccial Topics & Reviews R in Porouss Media: An IInternational Journal J 5(4), 277--286.
Fig. 6. Variiation of the Raayleigh numbeer (Ra)s with wave nu umber for diffferent value of porosity parameeter.
Chand, C R. annd G. C. Ranna (2014). Magneto M convectionn in a layer off nanofluid in porous medium-A A more realisticc approach. Jou urnal of Nanofluidds 4, 196-202.
7. CO ONCLUSION NS Thermal insstability in a horizontal layer l of nanofluid witth vertical AC electric e field in a porous medium is stuudied using lineear instability theory t by employing a model for nannofluid that incorporates the effects off Brownian mottion and thermoophoresis. The flux of volume fractiion of nanopaarticles is taken to be zero z on the isothermal boundaries and the eigenvaluue problem is soolved using the Galerkin residual methhod. The main conclusions of o present analysis are as a follows (i)
(ii)
The innstability purely phenomenon due to buoyanncy coupled with w the conserrvation of nanopaarticle and iss independentt of the contribbution of Brownian B mottion and thermoophoresis. Oscillaatory convectioon is not possibble for the probleem. (iii)
Porosity param meter stabilizes the
mber, AC stationary coonvection whille Lewis num electric field, modified diffusivity raatio and n destabilize the concentrationn Rayleigh number stationary connvection. ACKNO OWLEDGMEN NT The authors are grateful to the reviewers for their c comments annd suggestioons for valuable improvementt of the paper. REFER RENCES Buongiorno, J. (2006). Convective C trannsport in nanofluiids. ASME Jouurnal of Heat Transfer 128, 2400-250. Chand, R. (22013). Thermall instability off rotating nanoflu uid. Int. J. Ap ppl. Math and Mech. 9(3), 70 0-90. Chand, R. and a G. C. Ranna (2012a). Oscillating O convection of nanofluuid in porous medium. Transp Porous P Med 955, 269-284. Chand, R. annd G. C. Rana (2012). ( On thee onset of thermall convection in rotating nanofluid n
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