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Convection in rotating magnetic nanofluids
Applied Mathematics and Computation 219 (2013) 6284–6296
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Convection in rotating magnetic nanofluids Amit Mahajan ⇑, Monika Arora Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176 215, India
a r t i c l e
i n f o
Keywords: Magnetic nanofluids Convection Rotation Microgravity
a b s t r a c t Effect of rotation for convective instability in a thin layer of a magnetic nanofluid is examined within the frame work of linear theory. The model used incorporates the effect of Brownian diffusion, thermophoresis and magnetophoresis. The Eigen value problem is solved by employing the Chebyshev Pseudospectral method and the results are discussed for all the three boundary conditions: free–free, rigid-free and rigid–rigid for water and ester based magnetic nanofluids. The effect of magnetic field, rotation and modified particle density increment has been analyzed on the onset of convection. It is seen that magnetic mechanism predominates over the buoyancy mechanism in fluid layers about 1 mm thick. In the microgravity environment, the magnetic nanofluid is more resilient to convection and, in general, for all boundary conditions requires higher temperature gradient for the threshold of convection. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Nanofluids are engineered colloids made of a base fluid and nano particles (1–100 nm). The term nanofluid was coined by Choi [1]. The characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. [2]. A comprehensive survey of convective transport in nanofluids was made by Buongiorno [3] who concluded that in the absence of turbulent effects it is the Brownian diffusion and the thermophoresis that will be important. Magnetic nanofluids (ferrofluids) represent a special category of smart nano materials, consisting of stable dispersions of magnetic nano particles in different liquid carriers. The most stable magnetic nanofluids are known among those based on non- polar solvents, in which case the presence of a single layer of surfactant on the surface of magnetic nano particles is enough to avoid particle agglomeration. In the absence of applied field, the particles in magnetic nanofluid are randomly oriented and the fluid has no net magnetization. The fluid behaves similar to the carrier fluid. However, when placed in a strong magnetic field, these fluids flow toward regions of magnetic field and preserve their liquid character as long as the magnetic field is present. This characteristic has found several applications of these fluids including their use in sealing of hard disc drives, rotating X-ray tubes and rotating vacuum feedthroughs where reliable sealing at low friction is required [4,5]. Magnetic fluids are also used in cooling and damping of loud speakers, in shock absorbers and in jet printing of magnetic inks [6–8]. Recent investigations are also finding magnetic nanofluids useful in biomedical applications, such as drug targeting, as radio isotopes targeted by magnetic guidance and as a contrast agent for magnetic resonance imaging scans [9,10]. The continuum description of the magnetic fluids, termed as ferrohydrodynamics, has been in existence since the work of Neuringer and Rosensweig [11]. Finlayson [12] studied the convective instability of a magnetic fluid for a fluid layer heated from below in the presence of a uniform vertical magnetic field. Further work in this direction is done by a number of authors [13–20].
⇑ Corresponding author. E-mail addresses: [email protected] (A. Mahajan), [email protected] (M. Arora). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.12.012
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Nomenclature Latin symbols b subscript; basic state cf nanofluid specific heat (J/kg K) cp nanoparticle specific heat (J/kg K) D thickness of the nanofluid layer (m) Brownian diffusion coefficient (m2/s) DB DH magnetophoretic diffusion coefficient (m2/s) DT thermophoretic diffusion coefficient (m2/s) f subscript; fluid g = (0, 0, g) acceleration due to gravity (m/s2) H = (0, 0, H) magnetic field (T) hp specific enthalpy of nanoparticles (J/kg) mass flux for nanoparticles (kg/m2 s) jp ^ k unit vector in the z-direction k1 thermal conductivity (W/m K) kB Boltzmann’s constant (J/K) magnetic coefficients Km; Kp M magnetization (Amp/m) Ms magnetic saturation P the reduced pressure (Pa) pf the fluid pressure (Pa) p0 the perturbation in pressure (Pa) q velocity of the nanofluid (m/s) q0 ¼ ðu; v ; wÞ the perturbation in velocity (0, 0, 0) (m/s) T time (s) T temperature (K) Th constant average temperature at the bottom surface z = 0 (K) constant average temperature at the upper surface z = d (K) Tc Greek symbols a coefficient of thermal expansion (1/K) aL Langevin parameter b a uniform temperature gradient (K/m) j thermal diffusivity (m2/s) l viscosity of nanofluid (kg/ms) l0 magnetic permeability of vacuum (H/m) qf fluid density (kg/m3) qp particle density (kg/m3) h the perturbation in temperature T (K) / nanoparticle volume fraction v tangent magnetization susceptibility v2 chord magnetization susceptibility X ¼ ð0; 0; XÞ angular velocity (rad/s) r gradient operator (1/m) Non-dimensional parameters Le Lewis number M1 ; M01 ; M3 ; M 03 magnetic parameters NA ; N 0A modified diffusivity ratios NB modified particle density increment Ng magnetic thermal Rayleigh number Pr Prandtl number Ra thermal Rayleigh number Rn concentration Rayleigh number TA Taylor number
Study of nanofluids under the effect of rotation exhibits special characteristics. So investigating the effects of rotation on thermal convective instability is scientifically and technologically important. Gupta and Gupta [21] studied thermal convective instability in a rotating layer of ferrofluid heated uniformly from below. Venkatasubramanian and Kaloni [22] have
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discussed the effect of rotation on thermo convective instability of a layer of ferrofluid confined between stress-free, rigidparamagnetic and rigid-ferromagnetic boundaries. Thermal convection in a rotating layer of a magnetic fluid is discussed by Auernhammer and Brand [23]. The weakly non-linear instability of a rotating ferromagnetic fluid layer heated from below is discussed by Kaloni and Lou [24]. Sunil and Mahajan [25] have performed non-linear stability analysis for rotating magnetized ferrofluid heated from below. The Bénard problem for a nanofluid was studied by Tzou [26,27], Nield and Kuznetsov [28] on the basis of transport equations of Buongiorno [3]. Thermal instability in a rotating nanofluid layer is investigated by Yadav et al. [29] and a weak nonlinear analysis is performed by Bhadauria and Agarwal [30]. Shliomis [31] and Shliomis and Smorodin [32] have studied the convective instability of magnetized ferrofluids by treating the fluids as binary mixtures. These authors consider the influence of concentration gradients due to magnetophoresis and Soret effects. Effect of magnetic field on nanofluid convection in different geometries is studied in [33–35]. Some other aspects of magnetic nanofluids are studied in [36–38]. In the present work we examine the onset of convection in a horizontal layer of magnetic nanofluid heated from below and rotating about a vertical axis in the presence of a uniform magnetic field for rigid–rigid, rigid-free and stress-free boundary conditions. Momentum equation involving the Coriolis term and incorporating the effect of Brownian motion along with thermophoresis and magnetophoresis has been considered. The Eigen value problem is solved by the Chebyshev Pseudospectral method for all boundary conditions in gravitational as well as microgravity environment for water and ester based magnetic nanofluids and results are depicted graphically. The effect of magnetic field, rotation, particle concentration and the modified particle density increment is observed at the onset of convection. 2. Conservation equations for magnetic nanofluid Consider an infinite horizontal layer of incompressible magnetic nanofluid heated from below. The fluid is assumed to occupy the layer z 2 ð0; dÞ with gravity acting in negative z-direction and the magnetic field, H ¼ Hext 0 k acts outside the layer (see Fig. 1). The whole system is assumed to rotate with angular velocity X ¼ ð0; 0; XÞ along the vertical axis, which is taken as z-axis. Under the Boussinesq approximation the equations of continuity and momentum (Buongiorno [3]; Finlayson [12]; Sunil and Mahajan [20,25]) are
r q ¼ 0;
qf
ð1Þ
h i @q þ qf q rq ¼ rp þ lr2 q þ l0 M rH qp / þ qf ð1 /Þf1 aðT T c Þg gk þ 2qf ðq XÞ; @t
ð2Þ
where qf ; qp ; q; t; l; l0 ; M and / are the fluid density, nanoparticle density, velocity, time, viscosity, magnetic permeability of vacuum, magnetization and nanoparticle volume fraction, respectively. The effect of rotation contributes two terms: (a) cenq trifugal force 12 gradjX rj2 and (b) Coriolis acceleration 2ðq XÞ. In (2), p ¼ pf 2f jX rj2 is the reduced pressure. The conservation equation for the nanoparticles in the absence of chemical reactions (Buongiorno [3]; Shliomis [31]) is
@/ 1 þ q r/ ¼ r jp ; @t qp
ð3Þ
where jp is the matter flux for the nanoparticles, given by the sum of ‘‘diffusiophoresis’’, ‘‘thermophoresis’’ and ‘‘magnetophoresis’’ as
Fig. 1. Geometric configuration of the problem.
A. Mahajan, M. Arora / Applied Mathematics and Computation 219 (2013) 6284–6296
jp ¼ jpB þ jpT þ jpM ¼ qp DB r/ qp DT
rT T
þ qp DH rH:
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ð4Þ
Here, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, DH is the magnetophoretic diffusion coefficient. Using Eq. (4), the equation for nanoparticles become
@/ rT þ q r/ ¼ r DB r/ þ DT DH rH : @t T
ð5Þ
The thermal energy equation for a nanofluid can be written as
ðqcÞf
@T þ q rT ¼ r V þ hp r jp ; @t
where cf is the nanofluid specific heat, T is nanofluid temperature, hp is the specific enthalpy of the nanoparticle material and V is the energy flux, relative to a frame moving with the magnetic nanofluid velocity q, given by
V ¼ k1 rT þ hp jp ; where k1 is the nanofluid thermal conductivity. Substituting, we get
ðqcÞf
@T rT rT þ q rT ¼ r ðk1 rTÞ þ qp cp DB rT r/ þ DT DH rT rH : @t T
ð6Þ
In deriving this equation use has been made of vector identity and rhp ¼ cp rT, where cp is the nanoparticle specific heat of the material constituting the nanoparticles. Maxwell’s equations in the magnetostatic limit are
r B ¼ 0; r H ¼ 0;
B ¼ l0 ðM þ HÞ:
ð7Þ
At equilibrium, magnetization is aligned with the stationary magnetic field and is a function of magnetic field, particle concentration and temperature. It is assumed to be described by Langevin formula [31,38]
Meq ¼
H H M s /LðaL Þ ¼ M eq ðH; /; TÞ; H H
LðaL Þ ¼ cothðaL Þ
1
aL
;
aL ¼
mH : kB T
ð8Þ
To obtain the solution of the quiescent state we first linearize magnetization equation M eq in the same manner as Finlayson [12] did
Meq ðH; /; TÞ ¼ M0 þ vðH H0 Þ K m ðT T h Þ þ K p ð/ /0 Þ;
ð9Þ
where v is tangent magnetic susceptibility; K m , K p are the magnetic coefficients. The tangent and chord magnetic susceptibility v; v2 can be estimated by the Langevin formula (8) for a different Langevin parameter aL as Rosensweig [6]:
8 Ms m ; 1; v ¼ 3k > > BTh > < Ms m 0 mH0 aL ¼ ¼ ’ 1; v ¼ kB T h L ðaL Þ; kB T h > > > : 1; v ¼ Ms kB2T h ; mH 0
v2 ¼ v v2 ¼ MT hs LðaL Þ
v2 ¼ MH0s 1 a1L
ð10Þ
:
We assume that the temperature and the volumetric fraction of particles are constant on the boundaries. Thus the boundary conditions are
w ¼ 0;
T ¼ Th;
/ ¼ /0 at z ¼ 0;
ð11Þ
w ¼ 0;
T ¼ Tc;
/ ¼ /1 at z ¼ d;
ð12Þ
2
with @w ¼ 0 on rigid surface and @@zw2 ¼ 0 on stress-free surface. The magnetic boundary conditions are that the normal com@z ponent of magnetic induction and tangential component of magnetic field are continuous across the boundary. c ðd zÞ þ T c is a good approximation to the quiescent state solution. According to Nield and Kuznetsov [28], T b ¼ T h T d Following this, the quiescent state solution of Eqs. (1)–(9) is
qb ¼ 0;
Tb ¼
¼ M0 þ
Th Tc ðd zÞ þ T c ; d
pb ¼ pb ðzÞ;
K m ðT h T c Þz K p ð/0 /1 Þz ; ð1 þ vÞd ð1 þ vÞd
Hb ¼ H0
/b ¼
ð/0 /1 Þz þ /0 ; d
Mb
K m ðT h T c Þz K p ð/0 /1 Þz þ : ð1 þ vÞd ð1 þ vÞd
ð13Þ
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We note that K m ¼ vTH0 is in general functions of magnetic field and temperature and K p ¼ v/H00 is function of magnetic field and h particle concentration. 3. Linear analysis To study the linear stability of the quiescent state, we now take infinitesimally small perturbations as
q ¼ qb þ q0 ;
p ¼ pb þ p0 ;
T ¼ T b þ h;
/ ¼ /b þ /0 ;
M ¼ Mb þ M0 ;
H ¼ Hb þ H0 :
ð14Þ
This gives the following set of linearized perturbation equations (dropping primes)
r q ¼ 0; qf
ð15Þ
@q K m ðT h T c Þz K p ð/0 /1 Þz @ rw ¼ rp þ l0 M 0 þ @t ð1 þ vÞd ð1 þ vÞd @z q agðT h T c Þ K m ðT h T c Þz K p ð/0 /1 Þz @w þ l0 v K m h þ K p / k þ lr2 q ðqp qf Þg/k f ðd zÞ/k þ ð1 þ vÞd ð1 þ vÞd @z d / /1 þ ð1 /0 Þ þ ð 0 Þz qf aghk þ 2qf ðq XÞ; d
" # ðqcÞp @h ðT h T c Þ DB ðT h T c Þ @/ DH ðT h T c Þ @ 2 w 2 w ¼ jr h þ þ @t d d @z d @z2 ðqcÞf ðqcÞp DB ð/0 /1 Þ 2DT ðT h T c Þ DH K m ðT h T c Þ DH K p ð/0 /1 Þ @h ; þ þ d dT c dð1 þ vÞ dð1 þ vÞ @z ðqcÞf @/ ð/0 /1 Þ DT @ r2 w w ¼ DB r2 / þ ; r2 h DH @t d @z Tc ð1 þ vÞ where
@2w @h @/ þ ð1 þ v2 Þr21 w K m þ Kp ¼ 0; @z2 @z @z
ð16Þ
ð17Þ
ð18Þ
ð19Þ
j ¼ ðqkcÞ1 f , v2 ¼ MH00 and we have used H ¼ rw since r H ¼ 0.
On taking vertical component of curl curl of Eq. (16) and vertical component of curl of Eq. (16) the resulting equations along with Eqs. (17)–(19), in dimensionless form are
@ r21 w 1 @ r2 w ¼ r4 w RaM3 Ras M03 þ RaM1 RaM 4 þ RaN ð1 þ N / zÞ r21 h Pr @t @z @n RaM4 Ras M 01 þ Rn þ RaN N/ ð1 zÞ r21 / T 1=2 ; A @z
ð20Þ
1 @n @w ¼ r2 n þ T A1=2 ; Pr @t @z
ð21Þ
@h NB @/ NB N0A @ 2 w NB 2NA N B NB N 0A M 1 NB N 0A M 01 @h ¼ r2 h þ w þ ; þ þ @t Le @z Le @z2 Le Le LeM 3 LeM03 @z
ð22Þ
@/ 1 NA 2 N0 @ r2 w r h A ¼ w þ r2 / þ ; @t Le Le Le @z
ð23Þ
@ 2 w ð1 þ v2 Þ 2 M @h M01 @/ þ ¼ 0; þ r1 w 1 2 @z ð1 þ vÞ M 3 @z M03 @z
ð24Þ
where we have introduced n ¼ @@xv @u as the z-component of vorticity and the following non-dimensional quantities and non@y dimensional parameters
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A. Mahajan, M. Arora / Applied Mathematics and Computation 219 (2013) 6284–6296 Table 1 Some Properties of magnetic nanofluids (Ta = 298 K), from [6,18]. Base fluid
Density of fluid qf (kg/m3)
Viscosity coefficient l kg/(m s)
Thermal conductivity k1 (W/m K)
Temperature expansion coefficient a (K1)
Heat capacity of fluid (J/(kg K))
Prandtl number Pr
Water
1180
0.007
0.59
5:2 104
3545.76
42.0683
Ester
1150
0.014
0.31
8:1 104
3238.26
146.244
Fig. 2. Neutral stability curves of Ra, the thermal Rayleigh number for (a) water based magnetic nanofluid and (b) ester based magnetic nanofluid; in gravitational environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of Langevin parameter aL for T A ¼ 100; Le ¼ 100.
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A. Mahajan, M. Arora / Applied Mathematics and Computation 219 (2013) 6284–6296 2
ðx; y; zÞ ¼ dðx ; y ; z Þ; w ¼ dH0 w ;
Ra ¼
t¼
d
j
t ;
p¼
lj 2
d
p ;
q¼
j d
q ;
/ ¼ ð/0 /1 Þ/ ;
h ¼ ðT h T c Þh ;
qf g ad3 ðT h T c Þ l v2 H20 ðT h T c Þ l0 v2 H20 ð/0 /1 Þ 0 ; M1 ¼ 0 ; M ¼ ; 1 lj qf g adð1 þ vÞ/20 qf g adð1 þ vÞT 2h
l0 vH20 l vH 2 l DT ðT h T c Þ DH H0 ; M 03 ¼ 0 0 ; Pr ¼ ; NA ¼ ; N0A ¼ ; DB T c ð/0 /1 Þ qf g adT h qf g ad/0 qf j DB ð/0 /1 Þ 3 4 ðqCÞp ð/0 /1 Þ ðqp qf Þð/0 /1 Þgd j / /1 4X2 d ; Le ¼ ; N/ ¼ 0 ; TA ¼ ; Rn ¼ ; NB ¼ DB 1 /0 m2 lj ðqCÞf qf g ad3 ð/0 /1 Þ M01 M23 Ra l0 v2 H20 ð/0 /1 Þ M3 M01 ¼ ; M ¼ ; RaN ¼ ð1 /0 ÞRa; Ng ¼ M 1 Ra; Ras ¼ 4 ¼ 2 lj qf g adð1 þ vÞ/0 T h M03 M 1 M 03 M3 ¼
ð25Þ
where Ra is the thermal Rayleigh number, Rn is concentration Rayleigh number, Le is Lewis number, Pr is Prandtl number, N A ; N 0A are modified diffusivity ratios, N B is modified particle-density increment, M 1 ; M01 ; M 3 ; M 03 are magnetic parameters, Ng ¼ M1 Ra is the magnetic thermal Rayleigh number, T A is the Taylor number.
Fig. 3. Neutral stability curves of Ra, the thermal Rayleigh number for (a) water based magnetic nanofluid and (b) ester based magnetic nanofluid; in gravitational environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of Taylor number T A for aL ¼ 2; Le ¼ 100.
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Table 2 Critical Rayleigh numbers and critical wave numbers for the onset of stationary convection for water based magnetic nanofluid in gravitational environment for aL ¼ 2 (i.e. H0 ¼ 28027:3). TA
1 10 100 200 400 600 800 1000 2000 3000 4000 5000 10000 100000 1000000
Rigid–rigid surface
Rigid-free surface
Free–free surface
kc
Rac
Ng c
kc
Rac
Ng c
kc
Rac
Ng c
2.86822 2.87202 2.90858 2.94922 3.02422 3.09264 3.15786 3.21806 3.47124 3.67013 3.83436 3.97438 4.47442 6.82907 10.95567
521.82 522.34 527.39 532.82 543.16 552.89 562.09 570.81 608.94 640.54 667.76 691.80 783.79 1305.3 2302.8
2688.9 2694.1 2746.5 2803.3 2913.2 3018.5 3119.8 3217.4 3661.6 4051.5 4403.1 4725.9 6066.2 16823.1 52363.4
2.45081 2.46198 2.57147 2.68365 2.87633 3.03314 3.16860 3.28408 3.70407 3.98892 4.20746 4.38616 4.99367 7.73984 12.12076
446.23 447.41 458.84 470.65 491.62 509.76 525.78 540.16 596.78 638.97 673.26 702.49 808.87 1368.4 2414.3
1966.2 1976.6 2078.9 2187.3 2386.6 2566.0 2729.7 2881.2 3516.8 4031.6 4476.0 4873.0 6460.6 18490.2 57557.6
2.06145 2.08630 2.31729 2.53082 2.84559 3.07424 3.25022 3.39774 3.90350 4.23073 4.47693 4.67609 5.34376 8.29924 12.86463
366.09 368.26 389.67 410.44 443.70 469.98 491.94 510.98 581.87 632.15 672.06 705.56 824.67 1419.3 2500.5
1323.4 1339.1 1499.4 1663.5 1944.0 2181.1 2389.7 2578.2 3343.3 3946.0 4460.0 4915.7 6715.6 19891.3 61739.6
Fig. 4. Neutral stability curves of Ra, the thermal Rayleigh number for (a) water based nanofluid and (b) ester based nanofluid; in gravitational environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of volumetric fraction D/ for T A ¼ 100; Le ¼ 100; aL ¼ 2.
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Equations (20)–(24) are five equations for five variables w; n; h; /; w In order to match the domain of Chebyshev pseudospectral–QZ method; we reset the present domain from ½0; 1 to ½1; 1 with coordinate transformation z to 2z 1 in equations. We now perform the standard normal mode analysis and look for solution of variables w; n; h; /; w in the form
½w; n; h; /; w ¼ ½wðzÞ; nðzÞ; hðzÞ; /ðzÞ; wðzÞ exp½rt þ iðkx x þ ky yÞ:
ð26Þ
On substituting (26) into Eqs. (20)–(24) in the new domain, we obtain
r
Pr
r Pr
4D2 k
2
2 zþ1 2 2 2 2 Þ k h w ¼ 4D2 k w þ ½RaM3 Ras M03 2k Dw ½RaM 1 RaM 4 k h RaN 1 þ N/ ð 2 1z 2 k / 2T A1=2 Dn; þ RaM 4 Ras M 01 þ Rn þ RaN N/ 2 2
n ¼ ð4D2 k Þn þ 2T A1=2 Dw;
rh ¼ 4D2 k2 h þ w
2NB 4N B N0A 2 NB 2NA NB NB N0A M 1 NB N0A M01 þ Dh; D/ þ D w2 þ 0 Le Le Le Le LeM3 LeM3
ð27Þ
ð28Þ
ð29Þ
Fig. 5. Neutral stability curves of Ra, the thermal Rayleigh number for (a) water based nanofluid and (b) ester based nanofluid; in gravitational environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of Lewis number Le for T A ¼ 100; aL ¼ 2.
A. Mahajan, M. Arora / Applied Mathematics and Computation 219 (2013) 6284–6296
r/ ¼ w þ
1 2 NA 2 2N0A 2 2 2 4D k / þ 4D k h ð4D2 k ÞDw; Le Le Le
# 2 k ð1 þ v2 Þ 2M 1 2M 0 w 4D Dh þ 0 1 D/ ¼ 0; ð1 þ vÞ M3 M3
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ð30Þ
"
2
ð31Þ
with the boundary conditions w ¼ 0; h ¼ 0; / ¼ 0 at z ¼ 1; 2ð1 þ vÞDw kw ¼ 0 at z ¼ 1; 2ð1 þ vÞDw þ kw ¼ 0 at z ¼ þ1 and n ¼ 0, Dw ¼ 0 at z ¼ 1, on rigid–rigid surface n ¼ 0, Dw ¼ 0 at z ¼ 1; Dn ¼ 0, D2 w ¼ 0 at z ¼ þ1, on rigid-free surface
Dn ¼ 0;
D2 w ¼ 0 at z ¼ 1;
on free —free surface
ð32Þ
The system of Eqs. (27)–(31) with boundary conditions (32) is solved by Chebyshev pseudospectral method [39]. We follow the procedure and algorithms of Kaloni and Lou [40]. For a given b, wave number k, H0 and with other physical parameters, we employ QZ algorithm, EIG function in MATLAB, to solve system for determining neutral stability curves and find the leading eigenvalue r ¼ rr þ iri for corresponding wave number k. Adjusting b by secant method, we get the temperature
Fig. 6. Neutral stability curves of Ra, the thermal Rayleigh number for (a) water based nanofluid and (b) ester based nanofluid; in gravitational environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of modified particle density increment N B for T A ¼ 100; Le ¼ 100; aL ¼ 2.
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gradient b when the real part rr of the leading eigenvalue r ¼ rr þ iri is zero. For the neutral stability curves, the critical temperature gradient bc with critical wave number kc can be defined as
bc ¼ minbðPr; Le; . . .Þ k
ð33Þ
The minimization of Eq. (33) is carried out by the function FMINBND of MATLAB, which is a combination of golden section and parabolic method. Rac and Ng c are calculated by (25) once bc has been determined. 4. Numerical results and discussion The numerical results are presented here for a 1 mm thick layer of water and ester based magnetic nanofluids heated from below and rotating about the vertical axis, and the values of physical quantities for water and ester based nanofluids are provided in Table 1. The calculations are based on 10 nm nanoparticles suspended in carrier fluid. In Fig. 2, the neutral stability curves for thermal Rayleigh number, Ra, are plotted for water and ester based magnetic nanofluids with rigid–rigid, rigid-free and free–free boundary conditions for different values of aL to see the effect of magnetic field. It is seen that the increase in magnetic field increases the value of critical thermal Rayleigh number. Thus, the effect of increase in magnetic field is to delay the onset of convection in water as well as ester based magnetic nanofluid. It is also seen from these figures that the onset of convection in magnetic nanofluids occur at much lower value than the value for a clear viscous fluid [41]; a similar behavior was predicted by Schwab et al. [42]. Fig. 3 is plotted for the neutral stability curves of thermal Rayleigh number Ra for different values of Taylor number T A for water and ester based fluids in different boundary conditions. It is well known that rotation introduces vorticity into the fluid
Fig. 7. Neutral stability curves of Ng, the magnetic thermal Rayleigh number for (a) water based nanofluid and (b) ester based nanofluid; in microgravity environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of thickness of fluid layer d for T A ¼ 100; Le ¼ 100; aL ¼ 2.
A. Mahajan, M. Arora / Applied Mathematics and Computation 219 (2013) 6284–6296
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in case of Newtonian fluid [41]. Then the fluid moves in the horizontal planes with higher velocities. On account of this motion, the velocity of the fluids perpendicular to the plane reduces, and hence delays the onset of convection. Such a delay in the onset of convection is also observed in magnetic nanofluids with the increase in the rate of rotation and so the Taylor number is found to have a stabilizing effect on the system. It is also observed that the cell size decreases with increase in rate of rotation. Stabilizing effect of rotation can also be seen through the Table 2. For the values of rates of rotation, the critical thermal Rayleigh number for magnetic nanofluid is reduced substantially when compared to that of clear fluid [41]. In Fig. 4 effect of increase in particle concentration near lower boundary is observed by varying the volumetric fraction D/ ¼ /0 /1 . The neutral stability curves for both water and ester based magnetic nanofluids shows that the increase in particle concentration near lower boundary will stabilize the system more. It is also observed that the ester based magnetic nanofluids are more resilient to convection than the water based magnetic nanofluids. In Fig. 5 neutral stability curves for different values of Lewis number are plotted. Increase in Lewis number is first found to have a stabilizing effect on water and ester based magnetic nanofluids while a dual nature of first stabilizing and then destabilizing with further increase in value of Lewis number is observed. It is also observed that the cell size decreases with increase in values of Lewis number. Fig. 6 is plotted for increase in modified particle density increment N B . It can be seen that the increase in particle density increment destabilizes the system because the heavier nanoparticles moving through the base fluids makes more stronger disturbances as compared with the lighter nanoparticles. The effect is observed for all boundary conditions and it is found that the effect of increase in N B is more profound in rigid–rigid boundary conditions over other boundary conditions. In Fig. 7 neutral stability curves are plotted in the microgravity environment for magnetic thermal Rayleigh number Ng for different values of thickness of fluid layer d and it is observed that the magnetic thermal Rayleigh number increases with increase in thickness of layer d. It is also seen that in the microgravity environment, the magnetic nanofluid is more resilient to convection and, in general, for all boundary conditions requires higher temperature gradient for the threshold of convection. From the figure it is also clear that a magnetic nanofluid in microgravity environment is more stable, not only to the case when gravity is present, but also to the pure viscous case [41]. 5. Conclusions In this paper, the effect of rotation on thermal convection in a magnetic nanofluid layer heated from below subject to a uniform magnetic field has been investigated. The behavior of magnetic field, rotation, particle density increment, particle concentration is analyzed numerically using Chebyshev Pseudo-spectral method for water and ester based magnetic nanofluids and the results are depicted graphically. The results shows that the increase in magnetic field, rate of rotation and particle concentration near lower boundary tends to stabilize the system while increase in the values of particle density increment tends to destabilize the system. Increase in Lewis number is found to have a dual nature of first stabilizing and then destabilizing the system. The magnetic mechanism predominates over the buoyancy mechanism in thin fluid layers of about 1 mm only. The increase in particle concentration near lower (upper) wall can enhance (reduce) the thermal Rayleigh number. It is observed that the onset of convection is lowered by a substantial amount by the application of magnetic nanoparticles in gravitational environment. However in microgravity environment, the onset of convection has been resilient and requires a higher temperature gradient for all boundary conditions. The magnetic nanofluid in microgravity environment is more stable, not only to the case when gravity is present, but also to the pure viscous case. Acknowledgments The financial assistance to Monika Arora in the form of research fellowship from Central University of Himachal Pradesh is gratefully acknowledged. The authors are indebted to the three anonymous referees for their constructive remarks which improved the work considerably. References [1] S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of NonNewtonian Flows, ASME, FED-Vol.231/MD-vol. 66, 1995, pp. 99–105. [2] H. Masuda, A. Ebata, K. Teramae, N. Hishinuma, Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles, Netsu Bussei 7 (4) (1993) 227–233. [3] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [4] I. Borbath, Z. Kasco, L. David, I. Potencz, D. Bica, O. Marinica, L. Vekas, Applications of magnetic nanofluids in rotating seals, in: Convergence of microand nano-engineering, Romanian Academy Publ. House, Bucharest, 2006, pp. 200–210. [5] T. Borbath, D. Bica, I. Potencz, L. Vekas, I. Borbath, T. Boros, Magnetic nanofluids and magnetic composite fluids in rotating seal systems, IOP: Conf. Series: Earth Env. Sci. 12 (2010) 012105 (1–10). [6] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, Cambridge, 1985. [7] B.M. Berkovsky, V.E. Medvedev, M.S. Krakov, Magnetic Fluids, Engineering Applications, Oxford University Press, Oxford, 1993. [8] B.M. Berkovsky, V.G. Bashtovoy, Magnetic Fluids and Applications Handbook, Begell House, New York, 1996. [9] Q.A. Pankhurst, J. Connolly, S.K. Jones, J. Dobson, Applications of magnetic nanoparticle in biomedicine, J. Phys. D: Appl. Phys. 36 (13) (2003) R167– R181. [10] I. Sharifi, H. Shokrollahi, S. Amiri, Ferrite-based magnetic nanofluids used in hyperthermia applications, J. Magn. Magn. Mater. 324 (6) (2012) 903–915. [11] J.L. Neuringer, R.E. Rosensweig, Ferrohydrodynamics, Phys. Fluids 7 (1964) 1927–1937. [12] B.A. Finlayson, Convective instability in ferromagnetic fluids, J. Fluid Mech. 40 (1970) 753–767. [13] K. Gotoh, M. Yamada, Thermal convection in a horizontal layer of magnetic fluids, J. Phys. Soc. Jpn 51 (1982) 3042–3048.
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[14] D.P. Lalas, S. Carmi, Thermoconvective stability of ferrofluids, Phys. Fluids 14 (1971) 436–438. [15] P.J. Stiles, M. Kagan, Thermal convective instability of a horizontal layer of ferrofluid in a strong vertical magnetic field, J. Coll. Inter. Sci. 134 (1990) 196–198. [16] A. Recktenwald, M. Lucke, Thermoconvection in magnetized ferrofluids: the influence of boundaries with finite heat conductivity, J. Magn. Magn. Mater. 188 (1998) 326–332. [17] P.J. Blennerhassett, F. Lin, P.J. Stiles, Heat transfer through strongly magnetized ferrofluid, Proc. Roy. Soc. London, Ser. A 433 (1991) 165–177. [18] P.N. Kaloni, J.X. Lou, Convective instability of magnetic fluids, Phys. Rev. E 70 (2004) 026313. [19] Sunil, A. Mahajan, A nonlinear stability analysis for magnetized ferrofluid heated from below, Proc. Roy. Soc. A 464 (2008) 83–98. [20] Sunil, P. Sharma, A. Mahajan, A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid, Appl. Math. Comp. 218 (2011) 2785– 2799. [21] M.D. Gupta, A.S. Gupta, Convective instability of a layer of a ferromagnetic fluid rotating about a vertical axis, Int. J. Eng. Sci. 17 (1979) 271–277. [22] S. Venkatasubramanian, P.N. Kaloni, Effects of rotation on the thermoconvective instability of a horizontal layer of ferrofluids, Int. J. Eng. Sci. 32 (2) (1994) 237–256. [23] G.K. Auernhammer, H.R. Brand, Thermal convection in a rotating layer of a magnetic fluid, Eur. Phys. J. B 16 (2000) 157–168. [24] P.N. Kaloni, J.X. Lou, Weakly nonlinear instability of a ferromagnetic fluid rotating about a vertical axis, J. Magn. Magn. Mater. 284 (2004) 54–68. [25] Sunil, A. Mahajan, A nonlinear stability analysis for rotating magnetized ferrofluid heated from below, Appl. Math. Comp. 204 (2008) 299–310. [26] D.Y. Tzou, Thermal instability of nanofluids in natural convection, Int. J. Heat Mass Trans. 51 (2008) 2967–2979. [27] D.Y. Tzou, Instability of nanofluids in natural convection, ASME J. Heat Trans. 130 (2008) D72401 (1–9). [28] D.A. Nield, A.V. Kuznetsov, The onset of convection in a horizontal nanofluid layer of finite depth, Eur. J. Mech. B/Fluids 29 (2009) 217–233. [29] D. Yadav, G.S. Agrawal, R. Bhargava, Thermal instability of rotating nanofluid layer, Int. J. Eng. Sci. 49 (2011) 1171–1184. [30] B.S. Bhadauria, S. Agarwal, Natural convection in a nanofluid saturated rotating porous layer: a nonlinear study, Trans. Porous Med. 87 (2011) 585–602. [31] M.I. Shliomis, Convective instability of magnetized ferrofluids: Influence of magnetophoresis and Soret effect, in: W. Kohler, S. Weigand (Eds.), Thermal Nonequilibrium Phenomenon in Fluid Mixtures, Lecture Notes in Physics, vol. 584, Springer, Berlin, 2002, pp. 355–371. [32] M.I. Shliomis, B.L. Smorodin, Convective instability of magnetized ferrofluids, J. Magn. Magn. Mater. 252 (2002) 197–202. [33] S.M. Aminossadati, A. Raisi, B. Ghasemi, Effects of magnetic field on nanofluid forced convection in a partially heated microchannel, Int. J. Non-Linear Mech. 46 (10) (2011) 1373–1382. [34] B. Ghasemi, S.M. Aminossadati, A. Raisi, Magnetic field effect on natural convection in a nanofluid-filled square enclosure, Int. J. Therm. Sci. 50 (9) (2011) 1748–1756. [35] A.H. Mahmoudi, I. Pop, M. Shahi, Effect of magnetic field on natural convection in a triangular enclosure filled with nanofluid, Inter. J. Therm. Sci. 59 (2012) 126–140. [36] R. Patel, Effective viscosity of magnetic nanofluids through capillaries, Phys. Rev. E 85 (2) (2012) 026316. [37] K. Parekh, H.S. Lee, Experimental investigation of thermal conductivity of magnetic nanofluids, in: Proceedings of the 56th DAE Solid State Physics Symposium 2011, AIP Conference Proceedings, vol. 1447, 2012, pp. 385–386. [38] M.I. Shliomis, Effective viscosity of magnetic suspensions, Sov. Phys. JETP 34 (6) (1972) 1291–1294. [39] C. Canuto, M.Y. Hussaini, A. Quateroni, T. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, 1998. [40] P.N. Kaloni, J.X. Lou, Stability of Hadley circulations in a Maxwell fluid, J. Non-Newtonian Fluid Mech. 103 (2002) 167–186. [41] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, New York, 1981. [42] L. Schwab, U. Hildebrandt, K. Stierstadt, Magnetic Bénard convection, J. Magn. Magn. Mater. 39 (1983) 113–114.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Convection in rotating magnetic nanofluids Amit Mahajan ⇑, Monika Arora Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176 215, India
a r t i c l e
i n f o
Keywords: Magnetic nanofluids Convection Rotation Microgravity
a b s t r a c t Effect of rotation for convective instability in a thin layer of a magnetic nanofluid is examined within the frame work of linear theory. The model used incorporates the effect of Brownian diffusion, thermophoresis and magnetophoresis. The Eigen value problem is solved by employing the Chebyshev Pseudospectral method and the results are discussed for all the three boundary conditions: free–free, rigid-free and rigid–rigid for water and ester based magnetic nanofluids. The effect of magnetic field, rotation and modified particle density increment has been analyzed on the onset of convection. It is seen that magnetic mechanism predominates over the buoyancy mechanism in fluid layers about 1 mm thick. In the microgravity environment, the magnetic nanofluid is more resilient to convection and, in general, for all boundary conditions requires higher temperature gradient for the threshold of convection. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Nanofluids are engineered colloids made of a base fluid and nano particles (1–100 nm). The term nanofluid was coined by Choi [1]. The characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. [2]. A comprehensive survey of convective transport in nanofluids was made by Buongiorno [3] who concluded that in the absence of turbulent effects it is the Brownian diffusion and the thermophoresis that will be important. Magnetic nanofluids (ferrofluids) represent a special category of smart nano materials, consisting of stable dispersions of magnetic nano particles in different liquid carriers. The most stable magnetic nanofluids are known among those based on non- polar solvents, in which case the presence of a single layer of surfactant on the surface of magnetic nano particles is enough to avoid particle agglomeration. In the absence of applied field, the particles in magnetic nanofluid are randomly oriented and the fluid has no net magnetization. The fluid behaves similar to the carrier fluid. However, when placed in a strong magnetic field, these fluids flow toward regions of magnetic field and preserve their liquid character as long as the magnetic field is present. This characteristic has found several applications of these fluids including their use in sealing of hard disc drives, rotating X-ray tubes and rotating vacuum feedthroughs where reliable sealing at low friction is required [4,5]. Magnetic fluids are also used in cooling and damping of loud speakers, in shock absorbers and in jet printing of magnetic inks [6–8]. Recent investigations are also finding magnetic nanofluids useful in biomedical applications, such as drug targeting, as radio isotopes targeted by magnetic guidance and as a contrast agent for magnetic resonance imaging scans [9,10]. The continuum description of the magnetic fluids, termed as ferrohydrodynamics, has been in existence since the work of Neuringer and Rosensweig [11]. Finlayson [12] studied the convective instability of a magnetic fluid for a fluid layer heated from below in the presence of a uniform vertical magnetic field. Further work in this direction is done by a number of authors [13–20].
⇑ Corresponding author. E-mail addresses: [email protected] (A. Mahajan), [email protected] (M. Arora). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.12.012
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Nomenclature Latin symbols b subscript; basic state cf nanofluid specific heat (J/kg K) cp nanoparticle specific heat (J/kg K) D thickness of the nanofluid layer (m) Brownian diffusion coefficient (m2/s) DB DH magnetophoretic diffusion coefficient (m2/s) DT thermophoretic diffusion coefficient (m2/s) f subscript; fluid g = (0, 0, g) acceleration due to gravity (m/s2) H = (0, 0, H) magnetic field (T) hp specific enthalpy of nanoparticles (J/kg) mass flux for nanoparticles (kg/m2 s) jp ^ k unit vector in the z-direction k1 thermal conductivity (W/m K) kB Boltzmann’s constant (J/K) magnetic coefficients Km; Kp M magnetization (Amp/m) Ms magnetic saturation P the reduced pressure (Pa) pf the fluid pressure (Pa) p0 the perturbation in pressure (Pa) q velocity of the nanofluid (m/s) q0 ¼ ðu; v ; wÞ the perturbation in velocity (0, 0, 0) (m/s) T time (s) T temperature (K) Th constant average temperature at the bottom surface z = 0 (K) constant average temperature at the upper surface z = d (K) Tc Greek symbols a coefficient of thermal expansion (1/K) aL Langevin parameter b a uniform temperature gradient (K/m) j thermal diffusivity (m2/s) l viscosity of nanofluid (kg/ms) l0 magnetic permeability of vacuum (H/m) qf fluid density (kg/m3) qp particle density (kg/m3) h the perturbation in temperature T (K) / nanoparticle volume fraction v tangent magnetization susceptibility v2 chord magnetization susceptibility X ¼ ð0; 0; XÞ angular velocity (rad/s) r gradient operator (1/m) Non-dimensional parameters Le Lewis number M1 ; M01 ; M3 ; M 03 magnetic parameters NA ; N 0A modified diffusivity ratios NB modified particle density increment Ng magnetic thermal Rayleigh number Pr Prandtl number Ra thermal Rayleigh number Rn concentration Rayleigh number TA Taylor number
Study of nanofluids under the effect of rotation exhibits special characteristics. So investigating the effects of rotation on thermal convective instability is scientifically and technologically important. Gupta and Gupta [21] studied thermal convective instability in a rotating layer of ferrofluid heated uniformly from below. Venkatasubramanian and Kaloni [22] have
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discussed the effect of rotation on thermo convective instability of a layer of ferrofluid confined between stress-free, rigidparamagnetic and rigid-ferromagnetic boundaries. Thermal convection in a rotating layer of a magnetic fluid is discussed by Auernhammer and Brand [23]. The weakly non-linear instability of a rotating ferromagnetic fluid layer heated from below is discussed by Kaloni and Lou [24]. Sunil and Mahajan [25] have performed non-linear stability analysis for rotating magnetized ferrofluid heated from below. The Bénard problem for a nanofluid was studied by Tzou [26,27], Nield and Kuznetsov [28] on the basis of transport equations of Buongiorno [3]. Thermal instability in a rotating nanofluid layer is investigated by Yadav et al. [29] and a weak nonlinear analysis is performed by Bhadauria and Agarwal [30]. Shliomis [31] and Shliomis and Smorodin [32] have studied the convective instability of magnetized ferrofluids by treating the fluids as binary mixtures. These authors consider the influence of concentration gradients due to magnetophoresis and Soret effects. Effect of magnetic field on nanofluid convection in different geometries is studied in [33–35]. Some other aspects of magnetic nanofluids are studied in [36–38]. In the present work we examine the onset of convection in a horizontal layer of magnetic nanofluid heated from below and rotating about a vertical axis in the presence of a uniform magnetic field for rigid–rigid, rigid-free and stress-free boundary conditions. Momentum equation involving the Coriolis term and incorporating the effect of Brownian motion along with thermophoresis and magnetophoresis has been considered. The Eigen value problem is solved by the Chebyshev Pseudospectral method for all boundary conditions in gravitational as well as microgravity environment for water and ester based magnetic nanofluids and results are depicted graphically. The effect of magnetic field, rotation, particle concentration and the modified particle density increment is observed at the onset of convection. 2. Conservation equations for magnetic nanofluid Consider an infinite horizontal layer of incompressible magnetic nanofluid heated from below. The fluid is assumed to occupy the layer z 2 ð0; dÞ with gravity acting in negative z-direction and the magnetic field, H ¼ Hext 0 k acts outside the layer (see Fig. 1). The whole system is assumed to rotate with angular velocity X ¼ ð0; 0; XÞ along the vertical axis, which is taken as z-axis. Under the Boussinesq approximation the equations of continuity and momentum (Buongiorno [3]; Finlayson [12]; Sunil and Mahajan [20,25]) are
r q ¼ 0;
qf
ð1Þ
h i @q þ qf q rq ¼ rp þ lr2 q þ l0 M rH qp / þ qf ð1 /Þf1 aðT T c Þg gk þ 2qf ðq XÞ; @t
ð2Þ
where qf ; qp ; q; t; l; l0 ; M and / are the fluid density, nanoparticle density, velocity, time, viscosity, magnetic permeability of vacuum, magnetization and nanoparticle volume fraction, respectively. The effect of rotation contributes two terms: (a) cenq trifugal force 12 gradjX rj2 and (b) Coriolis acceleration 2ðq XÞ. In (2), p ¼ pf 2f jX rj2 is the reduced pressure. The conservation equation for the nanoparticles in the absence of chemical reactions (Buongiorno [3]; Shliomis [31]) is
@/ 1 þ q r/ ¼ r jp ; @t qp
ð3Þ
where jp is the matter flux for the nanoparticles, given by the sum of ‘‘diffusiophoresis’’, ‘‘thermophoresis’’ and ‘‘magnetophoresis’’ as
Fig. 1. Geometric configuration of the problem.
A. Mahajan, M. Arora / Applied Mathematics and Computation 219 (2013) 6284–6296
jp ¼ jpB þ jpT þ jpM ¼ qp DB r/ qp DT
rT T
þ qp DH rH:
6287
ð4Þ
Here, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, DH is the magnetophoretic diffusion coefficient. Using Eq. (4), the equation for nanoparticles become
@/ rT þ q r/ ¼ r DB r/ þ DT DH rH : @t T
ð5Þ
The thermal energy equation for a nanofluid can be written as
ðqcÞf
@T þ q rT ¼ r V þ hp r jp ; @t
where cf is the nanofluid specific heat, T is nanofluid temperature, hp is the specific enthalpy of the nanoparticle material and V is the energy flux, relative to a frame moving with the magnetic nanofluid velocity q, given by
V ¼ k1 rT þ hp jp ; where k1 is the nanofluid thermal conductivity. Substituting, we get
ðqcÞf
@T rT rT þ q rT ¼ r ðk1 rTÞ þ qp cp DB rT r/ þ DT DH rT rH : @t T
ð6Þ
In deriving this equation use has been made of vector identity and rhp ¼ cp rT, where cp is the nanoparticle specific heat of the material constituting the nanoparticles. Maxwell’s equations in the magnetostatic limit are
r B ¼ 0; r H ¼ 0;
B ¼ l0 ðM þ HÞ:
ð7Þ
At equilibrium, magnetization is aligned with the stationary magnetic field and is a function of magnetic field, particle concentration and temperature. It is assumed to be described by Langevin formula [31,38]
Meq ¼
H H M s /LðaL Þ ¼ M eq ðH; /; TÞ; H H
LðaL Þ ¼ cothðaL Þ
1
aL
;
aL ¼
mH : kB T
ð8Þ
To obtain the solution of the quiescent state we first linearize magnetization equation M eq in the same manner as Finlayson [12] did
Meq ðH; /; TÞ ¼ M0 þ vðH H0 Þ K m ðT T h Þ þ K p ð/ /0 Þ;
ð9Þ
where v is tangent magnetic susceptibility; K m , K p are the magnetic coefficients. The tangent and chord magnetic susceptibility v; v2 can be estimated by the Langevin formula (8) for a different Langevin parameter aL as Rosensweig [6]:
8 Ms m ; 1; v ¼ 3k > > BTh > < Ms m 0 mH0 aL ¼ ¼ ’ 1; v ¼ kB T h L ðaL Þ; kB T h > > > : 1; v ¼ Ms kB2T h ; mH 0
v2 ¼ v v2 ¼ MT hs LðaL Þ
v2 ¼ MH0s 1 a1L
ð10Þ
:
We assume that the temperature and the volumetric fraction of particles are constant on the boundaries. Thus the boundary conditions are
w ¼ 0;
T ¼ Th;
/ ¼ /0 at z ¼ 0;
ð11Þ
w ¼ 0;
T ¼ Tc;
/ ¼ /1 at z ¼ d;
ð12Þ
2
with @w ¼ 0 on rigid surface and @@zw2 ¼ 0 on stress-free surface. The magnetic boundary conditions are that the normal com@z ponent of magnetic induction and tangential component of magnetic field are continuous across the boundary. c ðd zÞ þ T c is a good approximation to the quiescent state solution. According to Nield and Kuznetsov [28], T b ¼ T h T d Following this, the quiescent state solution of Eqs. (1)–(9) is
qb ¼ 0;
Tb ¼
¼ M0 þ
Th Tc ðd zÞ þ T c ; d
pb ¼ pb ðzÞ;
K m ðT h T c Þz K p ð/0 /1 Þz ; ð1 þ vÞd ð1 þ vÞd
Hb ¼ H0
/b ¼
ð/0 /1 Þz þ /0 ; d
Mb
K m ðT h T c Þz K p ð/0 /1 Þz þ : ð1 þ vÞd ð1 þ vÞd
ð13Þ
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We note that K m ¼ vTH0 is in general functions of magnetic field and temperature and K p ¼ v/H00 is function of magnetic field and h particle concentration. 3. Linear analysis To study the linear stability of the quiescent state, we now take infinitesimally small perturbations as
q ¼ qb þ q0 ;
p ¼ pb þ p0 ;
T ¼ T b þ h;
/ ¼ /b þ /0 ;
M ¼ Mb þ M0 ;
H ¼ Hb þ H0 :
ð14Þ
This gives the following set of linearized perturbation equations (dropping primes)
r q ¼ 0; qf
ð15Þ
@q K m ðT h T c Þz K p ð/0 /1 Þz @ rw ¼ rp þ l0 M 0 þ @t ð1 þ vÞd ð1 þ vÞd @z q agðT h T c Þ K m ðT h T c Þz K p ð/0 /1 Þz @w þ l0 v K m h þ K p / k þ lr2 q ðqp qf Þg/k f ðd zÞ/k þ ð1 þ vÞd ð1 þ vÞd @z d / /1 þ ð1 /0 Þ þ ð 0 Þz qf aghk þ 2qf ðq XÞ; d
" # ðqcÞp @h ðT h T c Þ DB ðT h T c Þ @/ DH ðT h T c Þ @ 2 w 2 w ¼ jr h þ þ @t d d @z d @z2 ðqcÞf ðqcÞp DB ð/0 /1 Þ 2DT ðT h T c Þ DH K m ðT h T c Þ DH K p ð/0 /1 Þ @h ; þ þ d dT c dð1 þ vÞ dð1 þ vÞ @z ðqcÞf @/ ð/0 /1 Þ DT @ r2 w w ¼ DB r2 / þ ; r2 h DH @t d @z Tc ð1 þ vÞ where
@2w @h @/ þ ð1 þ v2 Þr21 w K m þ Kp ¼ 0; @z2 @z @z
ð16Þ
ð17Þ
ð18Þ
ð19Þ
j ¼ ðqkcÞ1 f , v2 ¼ MH00 and we have used H ¼ rw since r H ¼ 0.
On taking vertical component of curl curl of Eq. (16) and vertical component of curl of Eq. (16) the resulting equations along with Eqs. (17)–(19), in dimensionless form are
@ r21 w 1 @ r2 w ¼ r4 w RaM3 Ras M03 þ RaM1 RaM 4 þ RaN ð1 þ N / zÞ r21 h Pr @t @z @n RaM4 Ras M 01 þ Rn þ RaN N/ ð1 zÞ r21 / T 1=2 ; A @z
ð20Þ
1 @n @w ¼ r2 n þ T A1=2 ; Pr @t @z
ð21Þ
@h NB @/ NB N0A @ 2 w NB 2NA N B NB N 0A M 1 NB N 0A M 01 @h ¼ r2 h þ w þ ; þ þ @t Le @z Le @z2 Le Le LeM 3 LeM03 @z
ð22Þ
@/ 1 NA 2 N0 @ r2 w r h A ¼ w þ r2 / þ ; @t Le Le Le @z
ð23Þ
@ 2 w ð1 þ v2 Þ 2 M @h M01 @/ þ ¼ 0; þ r1 w 1 2 @z ð1 þ vÞ M 3 @z M03 @z
ð24Þ
where we have introduced n ¼ @@xv @u as the z-component of vorticity and the following non-dimensional quantities and non@y dimensional parameters
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A. Mahajan, M. Arora / Applied Mathematics and Computation 219 (2013) 6284–6296 Table 1 Some Properties of magnetic nanofluids (Ta = 298 K), from [6,18]. Base fluid
Density of fluid qf (kg/m3)
Viscosity coefficient l kg/(m s)
Thermal conductivity k1 (W/m K)
Temperature expansion coefficient a (K1)
Heat capacity of fluid (J/(kg K))
Prandtl number Pr
Water
1180
0.007
0.59
5:2 104
3545.76
42.0683
Ester
1150
0.014
0.31
8:1 104
3238.26
146.244
Fig. 2. Neutral stability curves of Ra, the thermal Rayleigh number for (a) water based magnetic nanofluid and (b) ester based magnetic nanofluid; in gravitational environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of Langevin parameter aL for T A ¼ 100; Le ¼ 100.
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A. Mahajan, M. Arora / Applied Mathematics and Computation 219 (2013) 6284–6296 2
ðx; y; zÞ ¼ dðx ; y ; z Þ; w ¼ dH0 w ;
Ra ¼
t¼
d
j
t ;
p¼
lj 2
d
p ;
q¼
j d
q ;
/ ¼ ð/0 /1 Þ/ ;
h ¼ ðT h T c Þh ;
qf g ad3 ðT h T c Þ l v2 H20 ðT h T c Þ l0 v2 H20 ð/0 /1 Þ 0 ; M1 ¼ 0 ; M ¼ ; 1 lj qf g adð1 þ vÞ/20 qf g adð1 þ vÞT 2h
l0 vH20 l vH 2 l DT ðT h T c Þ DH H0 ; M 03 ¼ 0 0 ; Pr ¼ ; NA ¼ ; N0A ¼ ; DB T c ð/0 /1 Þ qf g adT h qf g ad/0 qf j DB ð/0 /1 Þ 3 4 ðqCÞp ð/0 /1 Þ ðqp qf Þð/0 /1 Þgd j / /1 4X2 d ; Le ¼ ; N/ ¼ 0 ; TA ¼ ; Rn ¼ ; NB ¼ DB 1 /0 m2 lj ðqCÞf qf g ad3 ð/0 /1 Þ M01 M23 Ra l0 v2 H20 ð/0 /1 Þ M3 M01 ¼ ; M ¼ ; RaN ¼ ð1 /0 ÞRa; Ng ¼ M 1 Ra; Ras ¼ 4 ¼ 2 lj qf g adð1 þ vÞ/0 T h M03 M 1 M 03 M3 ¼
ð25Þ
where Ra is the thermal Rayleigh number, Rn is concentration Rayleigh number, Le is Lewis number, Pr is Prandtl number, N A ; N 0A are modified diffusivity ratios, N B is modified particle-density increment, M 1 ; M01 ; M 3 ; M 03 are magnetic parameters, Ng ¼ M1 Ra is the magnetic thermal Rayleigh number, T A is the Taylor number.
Fig. 3. Neutral stability curves of Ra, the thermal Rayleigh number for (a) water based magnetic nanofluid and (b) ester based magnetic nanofluid; in gravitational environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of Taylor number T A for aL ¼ 2; Le ¼ 100.
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Table 2 Critical Rayleigh numbers and critical wave numbers for the onset of stationary convection for water based magnetic nanofluid in gravitational environment for aL ¼ 2 (i.e. H0 ¼ 28027:3). TA
1 10 100 200 400 600 800 1000 2000 3000 4000 5000 10000 100000 1000000
Rigid–rigid surface
Rigid-free surface
Free–free surface
kc
Rac
Ng c
kc
Rac
Ng c
kc
Rac
Ng c
2.86822 2.87202 2.90858 2.94922 3.02422 3.09264 3.15786 3.21806 3.47124 3.67013 3.83436 3.97438 4.47442 6.82907 10.95567
521.82 522.34 527.39 532.82 543.16 552.89 562.09 570.81 608.94 640.54 667.76 691.80 783.79 1305.3 2302.8
2688.9 2694.1 2746.5 2803.3 2913.2 3018.5 3119.8 3217.4 3661.6 4051.5 4403.1 4725.9 6066.2 16823.1 52363.4
2.45081 2.46198 2.57147 2.68365 2.87633 3.03314 3.16860 3.28408 3.70407 3.98892 4.20746 4.38616 4.99367 7.73984 12.12076
446.23 447.41 458.84 470.65 491.62 509.76 525.78 540.16 596.78 638.97 673.26 702.49 808.87 1368.4 2414.3
1966.2 1976.6 2078.9 2187.3 2386.6 2566.0 2729.7 2881.2 3516.8 4031.6 4476.0 4873.0 6460.6 18490.2 57557.6
2.06145 2.08630 2.31729 2.53082 2.84559 3.07424 3.25022 3.39774 3.90350 4.23073 4.47693 4.67609 5.34376 8.29924 12.86463
366.09 368.26 389.67 410.44 443.70 469.98 491.94 510.98 581.87 632.15 672.06 705.56 824.67 1419.3 2500.5
1323.4 1339.1 1499.4 1663.5 1944.0 2181.1 2389.7 2578.2 3343.3 3946.0 4460.0 4915.7 6715.6 19891.3 61739.6
Fig. 4. Neutral stability curves of Ra, the thermal Rayleigh number for (a) water based nanofluid and (b) ester based nanofluid; in gravitational environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of volumetric fraction D/ for T A ¼ 100; Le ¼ 100; aL ¼ 2.
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Equations (20)–(24) are five equations for five variables w; n; h; /; w In order to match the domain of Chebyshev pseudospectral–QZ method; we reset the present domain from ½0; 1 to ½1; 1 with coordinate transformation z to 2z 1 in equations. We now perform the standard normal mode analysis and look for solution of variables w; n; h; /; w in the form
½w; n; h; /; w ¼ ½wðzÞ; nðzÞ; hðzÞ; /ðzÞ; wðzÞ exp½rt þ iðkx x þ ky yÞ:
ð26Þ
On substituting (26) into Eqs. (20)–(24) in the new domain, we obtain
r
Pr
r Pr
4D2 k
2
2 zþ1 2 2 2 2 Þ k h w ¼ 4D2 k w þ ½RaM3 Ras M03 2k Dw ½RaM 1 RaM 4 k h RaN 1 þ N/ ð 2 1z 2 k / 2T A1=2 Dn; þ RaM 4 Ras M 01 þ Rn þ RaN N/ 2 2
n ¼ ð4D2 k Þn þ 2T A1=2 Dw;
rh ¼ 4D2 k2 h þ w
2NB 4N B N0A 2 NB 2NA NB NB N0A M 1 NB N0A M01 þ Dh; D/ þ D w2 þ 0 Le Le Le Le LeM3 LeM3
ð27Þ
ð28Þ
ð29Þ
Fig. 5. Neutral stability curves of Ra, the thermal Rayleigh number for (a) water based nanofluid and (b) ester based nanofluid; in gravitational environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of Lewis number Le for T A ¼ 100; aL ¼ 2.
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r/ ¼ w þ
1 2 NA 2 2N0A 2 2 2 4D k / þ 4D k h ð4D2 k ÞDw; Le Le Le
# 2 k ð1 þ v2 Þ 2M 1 2M 0 w 4D Dh þ 0 1 D/ ¼ 0; ð1 þ vÞ M3 M3
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ð30Þ
"
2
ð31Þ
with the boundary conditions w ¼ 0; h ¼ 0; / ¼ 0 at z ¼ 1; 2ð1 þ vÞDw kw ¼ 0 at z ¼ 1; 2ð1 þ vÞDw þ kw ¼ 0 at z ¼ þ1 and n ¼ 0, Dw ¼ 0 at z ¼ 1, on rigid–rigid surface n ¼ 0, Dw ¼ 0 at z ¼ 1; Dn ¼ 0, D2 w ¼ 0 at z ¼ þ1, on rigid-free surface
Dn ¼ 0;
D2 w ¼ 0 at z ¼ 1;
on free —free surface
ð32Þ
The system of Eqs. (27)–(31) with boundary conditions (32) is solved by Chebyshev pseudospectral method [39]. We follow the procedure and algorithms of Kaloni and Lou [40]. For a given b, wave number k, H0 and with other physical parameters, we employ QZ algorithm, EIG function in MATLAB, to solve system for determining neutral stability curves and find the leading eigenvalue r ¼ rr þ iri for corresponding wave number k. Adjusting b by secant method, we get the temperature
Fig. 6. Neutral stability curves of Ra, the thermal Rayleigh number for (a) water based nanofluid and (b) ester based nanofluid; in gravitational environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of modified particle density increment N B for T A ¼ 100; Le ¼ 100; aL ¼ 2.
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gradient b when the real part rr of the leading eigenvalue r ¼ rr þ iri is zero. For the neutral stability curves, the critical temperature gradient bc with critical wave number kc can be defined as
bc ¼ minbðPr; Le; . . .Þ k
ð33Þ
The minimization of Eq. (33) is carried out by the function FMINBND of MATLAB, which is a combination of golden section and parabolic method. Rac and Ng c are calculated by (25) once bc has been determined. 4. Numerical results and discussion The numerical results are presented here for a 1 mm thick layer of water and ester based magnetic nanofluids heated from below and rotating about the vertical axis, and the values of physical quantities for water and ester based nanofluids are provided in Table 1. The calculations are based on 10 nm nanoparticles suspended in carrier fluid. In Fig. 2, the neutral stability curves for thermal Rayleigh number, Ra, are plotted for water and ester based magnetic nanofluids with rigid–rigid, rigid-free and free–free boundary conditions for different values of aL to see the effect of magnetic field. It is seen that the increase in magnetic field increases the value of critical thermal Rayleigh number. Thus, the effect of increase in magnetic field is to delay the onset of convection in water as well as ester based magnetic nanofluid. It is also seen from these figures that the onset of convection in magnetic nanofluids occur at much lower value than the value for a clear viscous fluid [41]; a similar behavior was predicted by Schwab et al. [42]. Fig. 3 is plotted for the neutral stability curves of thermal Rayleigh number Ra for different values of Taylor number T A for water and ester based fluids in different boundary conditions. It is well known that rotation introduces vorticity into the fluid
Fig. 7. Neutral stability curves of Ng, the magnetic thermal Rayleigh number for (a) water based nanofluid and (b) ester based nanofluid; in microgravity environment for (1) rigid–rigid (2) rigid-free and (3) free–free boundary conditions for different values of thickness of fluid layer d for T A ¼ 100; Le ¼ 100; aL ¼ 2.
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in case of Newtonian fluid [41]. Then the fluid moves in the horizontal planes with higher velocities. On account of this motion, the velocity of the fluids perpendicular to the plane reduces, and hence delays the onset of convection. Such a delay in the onset of convection is also observed in magnetic nanofluids with the increase in the rate of rotation and so the Taylor number is found to have a stabilizing effect on the system. It is also observed that the cell size decreases with increase in rate of rotation. Stabilizing effect of rotation can also be seen through the Table 2. For the values of rates of rotation, the critical thermal Rayleigh number for magnetic nanofluid is reduced substantially when compared to that of clear fluid [41]. In Fig. 4 effect of increase in particle concentration near lower boundary is observed by varying the volumetric fraction D/ ¼ /0 /1 . The neutral stability curves for both water and ester based magnetic nanofluids shows that the increase in particle concentration near lower boundary will stabilize the system more. It is also observed that the ester based magnetic nanofluids are more resilient to convection than the water based magnetic nanofluids. In Fig. 5 neutral stability curves for different values of Lewis number are plotted. Increase in Lewis number is first found to have a stabilizing effect on water and ester based magnetic nanofluids while a dual nature of first stabilizing and then destabilizing with further increase in value of Lewis number is observed. It is also observed that the cell size decreases with increase in values of Lewis number. Fig. 6 is plotted for increase in modified particle density increment N B . It can be seen that the increase in particle density increment destabilizes the system because the heavier nanoparticles moving through the base fluids makes more stronger disturbances as compared with the lighter nanoparticles. The effect is observed for all boundary conditions and it is found that the effect of increase in N B is more profound in rigid–rigid boundary conditions over other boundary conditions. In Fig. 7 neutral stability curves are plotted in the microgravity environment for magnetic thermal Rayleigh number Ng for different values of thickness of fluid layer d and it is observed that the magnetic thermal Rayleigh number increases with increase in thickness of layer d. It is also seen that in the microgravity environment, the magnetic nanofluid is more resilient to convection and, in general, for all boundary conditions requires higher temperature gradient for the threshold of convection. From the figure it is also clear that a magnetic nanofluid in microgravity environment is more stable, not only to the case when gravity is present, but also to the pure viscous case [41]. 5. Conclusions In this paper, the effect of rotation on thermal convection in a magnetic nanofluid layer heated from below subject to a uniform magnetic field has been investigated. The behavior of magnetic field, rotation, particle density increment, particle concentration is analyzed numerically using Chebyshev Pseudo-spectral method for water and ester based magnetic nanofluids and the results are depicted graphically. The results shows that the increase in magnetic field, rate of rotation and particle concentration near lower boundary tends to stabilize the system while increase in the values of particle density increment tends to destabilize the system. Increase in Lewis number is found to have a dual nature of first stabilizing and then destabilizing the system. The magnetic mechanism predominates over the buoyancy mechanism in thin fluid layers of about 1 mm only. The increase in particle concentration near lower (upper) wall can enhance (reduce) the thermal Rayleigh number. It is observed that the onset of convection is lowered by a substantial amount by the application of magnetic nanoparticles in gravitational environment. However in microgravity environment, the onset of convection has been resilient and requires a higher temperature gradient for all boundary conditions. The magnetic nanofluid in microgravity environment is more stable, not only to the case when gravity is present, but also to the pure viscous case. Acknowledgments The financial assistance to Monika Arora in the form of research fellowship from Central University of Himachal Pradesh is gratefully acknowledged. The authors are indebted to the three anonymous referees for their constructive remarks which improved the work considerably. References [1] S. 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