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The onset of Brinkman ferroconvection in an anisotropic porous medium
International Journal of Engineering Science 49 (2011) 497–508
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The onset of Brinkman ferroconvection in an anisotropic porous medium C.E. Nanjundappa a,⇑, I.S. Shivakumara b,1, Jinho Lee b,⇑, M. Ravisha c a b c
Department of Mathematics, Dr. Ambedkar Institute of Technology, Bangalore 560 056, India School of Mechanical Engineering, Yonsei University, Seoul 120-749, South Korea Department of Mathematics, Smt. Rukmuni Shedthi Memorial National Government First Grade College, Barkur 576210, Udup District, India
a r t i c l e
i n f o
Article history: Received 27 April 2010 Accepted 29 December 2010 Available online 5 March 2011 Keywords: Ferroconvection Anisotropic porous medium Paramagnetic Ferromagnetic
a b s t r a c t Onset of ferroconvection in an anisotropic porous layer heated from below is investigated theoretically using modified Brinkman extended-Darcy equation with fluid viscosity different from effective viscosity. The isothermal bounding surfaces of a porous layer are considered to be either free or rigid-paramagnetic/ferromagnetic. The eigenvalue problem is solved exactly for free boundaries, while for realistic rigid-paramagnetic or rigidferromagnetic boundaries the critical stability parameters are obtained numerically using the Galerkin method. It is seen that the stability of the system depends on the nature of boundaries and rigid-paramagnetic boundaries are found to be preferred to the ferromagnetic ones as well as free boundaries in controlling ferroconvection in an anisotropic porous layer. It is observed that increase in the value of thermal anisotropy parameter and viscosity ratio is to delay the onset of ferroconvection, while increase in the value of mechanical anisotropy parameter and magnetic number is to hasten the onset of ferroconvection. Moreover, increasing the value of thermal anisotropy parameter and decreasing the value of mechanical anisotropy parameter is to narrow the convection cells. Ó 2011 Published by Elsevier Ltd.
1. Introduction Magnetic nanofluids are stable colloidal suspensions of single-domain nano-scale magnetic particles with typical dimensions of about 3–10 nm dispersed in non-conducting carrier liquids like water, kerosene, ester, hydrocarbons, etc. These fluids are also referred to as ferrofluids. Since 1960s, when these fluids were initially synthetized, their technological applications have been step up over the years. These fluids are found to have numerous applications, for example in loud speakers, rotatory exclusion seals, bearings, dampers, shock absorbers, medicine drug targeting, and in many other thermal transport applications. A detailed introduction to ferrofluids along with their diverse applications is well documented in the books by Rosensweig (1985), Bashtovoy and Berkovsky (1996) and Rerkovsky et al. (1993). Thermo-gravitational convection in a layer of ferrofluid in the presence of a uniform magnetic field, known as ferroconvection, is analogous to classical Benard convection and has received due attention in the literature because of promising potential in heat transfer applications. Finlayson (1970) was the first to study this problem theoretically. Thermoconvective instability of ferrofluids without considering buoyancy effects has been investigated by Lalas and Carmi (1971), whereas Shliomis (1974) has analyzed the linear relation for magnetized perturbed quantities at the limit of instability. A similar analysis but with the ferrofluid confined between ferromagnetic plates has been carried out by Gotoh and Yamada (1982) using linear stability analysis. Schwab, Hilderbrandt, and Stierstadt (1987) have conducted ⇑ Corresponding authors. Tel.: +91 080 3211506 (C.E. Nanjundappa). 1
E-mail address: [email protected] (C.E. Nanjundappa). Permanent address: UGC-CAS in Fluid Mechanics, Department of Mathematics, Bangalore University, Bangalore 560 001, India.
0020-7225/$ - see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.ijengsci.2010.12.014
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experiments and showed that their results are in good agreement with theoretical predictions. Stiles and Kagan (1990) have extended the problem to allow for the dependence of effective shear viscosity on temperature and colloidal concentration, while Stiles, Lin, and Blennerhassett (1992) have analyzed linear and weakly nonlinear thermoconvective instability in a thin layer of ferrofluid subjected to a weak uniform external magnetic field in the vertical direction. The effect of rotation on linear and weakly nonlinear thermal convection in a magnetic fluid layer is investigated by Auernhammer and Brand (2000), while the effect of different forms of basic temperature gradients on the onset of ferroconvection driven by combined surface tension and buoyancy forces has been discussed by Shivakumara, Rudraiah, and Nanjundappa (2002) with the sole motto of understanding control of ferroconvection. Kaloni and Lou (2004) have theoretically investigated the convective instability problem in a thin horizontal layer of magnetic fluid heated from below under alternating magnetic field by considering the quasi stationary model with internal rotation and vortex viscosity. In his review article, Odenbach (2004) has focused on recent developments in the field of rheological investigations of ferrofluids and their importance for the general treatment of ferrofluids. The influence of magnetic field on heat and mass transport in ferrofluids has been discussed by Volker, Blums, and Odenbach (2007). Nanjundappa and Shivakumara (2008) have investigated a variety of velocity and temperature boundary conditions on the onset of ferroconvection in an initially quiescent ferrofluid layer. Thermal convection of ferrofluids saturating a porous medium has also attracted considerable attention in the literature owing to its importance in controlled emplacement of liquids or treatment of chemicals, and emplacement of geophysically imageable liquids into particular zones for subsequent imaging, etc. Rosensweig, Zahn, and Volger (1978) have studied experimentally the penetration of ferrofluids in the Heleshaw cell. The stability of the magnetic fluid penetration through a porous medium in high uniform magnetic field oblique to the interface is studied by Zhan and Rosensweig (1980). Thermal convective instability in a layer of ferrofluid saturating a porous medium in the presence of a vertical magnetic field is studied by Vaidyanathan, Sekar, and Balasubramanian (1991). Their analysis is limited to free-free boundaries and to the case of effective viscosity equal to fluid viscosity. Qin and Chadam (1995) have carried out the non-linear stability analysis of ferroconvection in a porous layer by including the inertial effects to accommodate high velocity. Sekar, Vaidynathan, and Ramathan (1996) have investigated ferroconvection in an anisotropic porous medium for stress-free boundaries by considering anisotropy only in the permeability of the porous medium. The laboratory scale experimental results of the behaviour of ferrofluids in porous media consisting of sands and sediments are presented in detail by Borglin, Mordis, and Oldenburg (2000). Ramathan and Suresh (2004) have analyzed the effects of magnetic field dependent viscosity and mechanical anisotropy of porous medium on ferroconvection. Recently, Shivakumara, Nanjundappa, and Ravisha (2008, 2009) have investigated in detail the onset of thermomagnetic convection in a ferrofluid saturated porous medium for various types of velocity and temperature boundary conditions. The onset of buoyancy-driven convection in a porous layer heated from below using the Brinkman–Lapwood extended Darcy equation with fluid viscosity different effective viscosity is investigated by Nanjundappa, Shivakumara, and Ravisha (2010). Many technological applications of practical importance involve porous materials of high porosity and for such a porous medium Givler and Altobelli (1994) have demonstrated that the effective viscosity is about ten times the fluid viscosity. Under the circumstances, a theoretical study which is general enough to yield accurate results for ferroconvection in porous media is of fundamental and practical interest. In many practical situations, it is reasonable to expect the porous materials to be anisotropic in their mechanical and thermal properties. Anisotropy is generally a consequence of preferential orientation or asymmetric geometry of porous matrix or fibers and is in fact encountered in numerous systems in industry and nature. Anisotropy can also be a characteristic of artificial porous materials like pelletting used in chemical engineering process and fiber material used in insulating purposes. Copious literature is available on thermal convection in a layer of an anisotropic porous medium because the anisotropy significantly alters the convective instability of the problem. For the literature on this problem see, for example, the survey in Nield and Bejan (2006). The main objective of the paper is, therefore, to obtain the criterion for the onset of ferroconvection in a ferrofluid saturated anisotropic Brinkman porous layer heated from below in the presence of a uniform vertical magnetic field. Keeping in mind the realistic situation, the boundaries are considered to be rigid and are either paramagnetic or ferromagnetic. The opportunity is also being used to review the results on free boundaries in the present context. The resulting eigenvalue problem is solved exactly for free–free boundaries and for the case of rigid–rigid boundaries the problem is solved numerically using a Galerkin technique. To achieve the above objectives, the paper is organized as under. Section 2 is devoted to mathematical formulation. The method of solution is discussed in Section 3. In Section 4, the numerical results presented are discussed and some important conclusions follow in Section 5.
2. Mathematical formulation We consider an initially quiescent ferrofluid saturated horizontal anisotropic porous layer of depth d in the presence of a ^ uniform applied magnetic field Ho acting in the vertical direction. A Cartesian coordinate system (x, y, z) is used with ð^i; ^j; kÞ being the standard basis and the gravity acts in the negative vertical z-direction. The lower and upper boundaries of the porous layer are maintained at constant but different temperatures T0 and Tl(
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499
The governing equations are as follows: The continuity equation
r ~ q¼0
ð1Þ
where, ~ q ¼ ðu; v ; wÞ is the velocity vector. The momentum equation
q0 @~ q ~~ ~ f r2~ q þ r ðH ¼ rp þ q0 ½1 at ðT T 0 Þ~ g lf k1 ~ qþl BÞ e @t
ð2Þ
~ is the magnetic field, ~ where, p is the pressure, H B is the magnetic induction, T is the temperature, q0 is the reference density, ~ ~ f is the effective viscosity, at is the thermal expansion coefg is the acceleration due to gravity, lf is the dynamic viscosity, l ficient, e is the porosity of the porous medium, r2 = @ 2/@x2 + @ 2/oy2 + @ 2/@z2 is the Laplacian operator and k is the permeabil 1 1 ^ ^ k, where kh and kv are the ity tensor. The horizontal isotropy is assumed and hence we have k1 ¼ kh ð^i^i þ ^j^jÞ þ kv k permeability in the horizontal and vertical directions respectively. The energy equation
2
~ ~ @M e4q0 C V;H l0 H @T
3
! V;H
~ 5 DT þ ð1 eÞðq0 CÞ @T þ l T @ M S 0 Dt @t @T
! V;H
~ DH ¼ r ðj rTÞ Dt
ð3Þ
~ is the magnetization, l0 is the where, C is the specific heat, CV,H is the specific heat at constant volume and magnetic field, M ^k, ^ magnetic permeability of vacuum and j is the effective thermal diffusivity tensor which is given by j ¼ jh ð^i^i þ ^j^jÞ þ jv k while jh and jv are the effective thermal diffusivity in the horizontal and vertical directions respectively. The Maxwell equations in the magnetostatic limit are:
~ ¼ 0 or H ~ ¼ ru r ~ B ¼ 0r H
ð4a;bÞ
~ and H ~ are related by where, u is the magnetic potential. Further, ~ B, M
~ ~ þ HÞ ~ B ¼ l0 ð M
ð5Þ
We assume that the magnetization is aligned with the magnetic field, but allow a dependence on the magnitude of magnetic field as well as the temperature in the form
~ ~ ¼ H MðH; T Þ M H
ð6Þ
The magnetic equation of state is linearized about H0 and T0 to become
M ¼ M 0 þ vðH H0 Þ KðT T 0 Þ
ð7Þ
where, v ¼ ð@M=@HÞH0 ;T 0 is the magnetic susceptibility of the ferrofluid, K ¼ ð@M=@TÞH0 ;T 0 is called the pyromagnetic co-effi~ and M ¼ jMj. ~ cient, M0 = M(H0, T0), H ¼ jHj The basic state is quiescent and is given by
~ qb ¼ 0 1 l M0 jb l j2 b2 2 pb ðzÞ ¼ p0 q0 gz q0 at gbz2 0 z 0 z 2 1þv 2ð1 þ vÞ2 T b ðzÞ ¼ T 0 bz Kbz Hb ðzÞ ¼ H0 1þv Kbz M b ðzÞ ¼ M0 þ 1þv
ð8Þ
where the subscript b denotes the basic state and b = DT/d is the temperature gradient. To study the linear stability of the above steady state solution, the variables are perturbed in the form
~ q ¼~ q0 ;
p ¼ pb ðzÞ þ p0 ;
T ¼ T b ðzÞ þ T 0 ;
~¼H ~b ðzÞ þ H ~0 ; H
~ ¼M ~ b ðzÞ þ M ~0 M
ð9Þ
where the primed quantities are perturbed ones and are assumed to be small. Substituting Eq. (9) into Eqs. (6) and (7) and using Eq. (5), we get (after ignoring the primes):
M0 Hx ; Hx þ M x ¼ 1 þ H0 Hz þ Mz ¼ ð1 þ vÞHz KT
Hy þ M y ¼
M0 Hy 1þ H0
ð10Þ ð11Þ
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where (Hx, Hy, Hz) and (Mx, My, Mz) are the components of perturbed magnetic field and magnetization, respectively. In obtaining the above equations, it is assumed that Kbd (1 + v)H0. After substituting Eq. (9) into Eq. (2), linearizing, eliminating the pressure term by operating curl twice, and using Eqs. ~0 ¼ r/0 , the z-component of the resulting equation can be obtained as (after ignoring the (10) and (11) together withH primes):
q0
l l @2w @ @ 2 l0 K 2 b 2 ~ f r2 r2 w þ f r2h w þ f l ¼ l0 Kb rh u þ r T þ q0 at g r2h T 2 @t @z 1þv h kv kh @z
ð12Þ
where r2h ¼ @ 2 =@x2 þ @ 2 =@y2 is the horizontal Laplacian operator. As before, using Eq. (9) in Eq. (3) and linearizing the equation, we get (after ignoring the primes)
ðq0 CÞ1
# " @T @ @u l T0K2 @2T ¼ ðq0 CÞ2 0 bw þ jh r2h T þ jv 2 l0 T 0 K @t @t @z @z 1þv
ð13Þ
where (q0C)1 = eq0CV,H + el0H0K + (1 e)(q0C)s and (q0C)2 = eq0CV,H + el0H0K. ~0 ¼ ru0 , Eqs. (4a,b) may be written as (after Finally, after substituting Eq. (9) and using Eqs. (10) and (11) together with H ignoring the primes)
M0 @2u @T ¼0 1þ r2h u þ ð1 þ vÞ 2 K @z H0 @z
ð14Þ
Assume the normal mode expansion of the dependent variables in the form
fw; T; ug ¼ fWðzÞ; HðzÞ; UðzÞg exp½ið‘x þ myÞ þ rt
ð15Þ
where ‘ and m are wave numbers in the x and y directions, respectively and r is the growth rate. Substituting Eq. (15) into Eqs. (12)–(14) and non-dimensionalizing the variables by setting
z z ¼ ; d
W ¼
d W; mA
t ¼
m d
t; 2
2
r ¼
d
m
r; H ¼
jv bv d
H;
U ¼
ð1 þ vÞjv 2
Kbv d
U
ð16Þ
where v = lf/q0 is the kinematic viscosity and A = (q0C)1/(q0C)2 is the ratio of heat capacities, we obtain (after ignoring the asterisks)
1 KðD2 a2 Þ2 Da1 D2 a2 rðD2 a2 Þ W ¼ a2 R½M1 DU ð1 þ M 1 ÞH n
ð17Þ
ðD2 ga2 PrrÞH PrM 2 rDU ¼ ð1 M 2 AÞW
ð18Þ
2
2
ðD a M3 ÞU DH ¼ 0
ð19Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here D = d/dz is the differential operator, a ¼ ‘2 þ m2 is the overall horizontal wave number, R = atgbd4/mjvA is the thermal Rayleigh number, M1 = l0K2b/(1 + v)atq0g is the magnetic number, M2 = l0T0K2/(1 + v)(q0C)1 is the magnetic parameter, ~ f =lf is the M3 = (1 + M0/H0)/(1 + v) is the measure of nonlinearity of magnetization, Pr = m/j is the Prandtl number, K ¼ l ratio of viscosities, n = kh/kv is the anisotropic permeability parameter, g = jh/jv is the anisotropic effective thermal diffusivity parameter and Da = kv/d2 is the Darcy number. The typical value of M2 for magnetic fluids with different carrier liquids turns out to be of the order of 106 and hence its effect is neglected as compared to unity. The above equations are solved subject to the following three different types of boundary conditions: (i) Lower and upper boundaries are free with large magnetic susceptibility
W ¼ 0 ¼ D2 W;
H ¼ 0;
DU ¼ 0 at z ¼ 0; 1
ð20Þ
(ii) Lower and upper boundaries are rigid-paramagnetic
W ¼ 0 ¼ DW;
H ¼ 0 at z ¼ 0; 1
ð1 þ vÞDU aU ¼ 0 at z ¼ 0 ð1 þ vÞDU þ aU ¼ 0 at z ¼ 1
ð21aÞ ð21bÞ ð21cÞ
(iii) Lower and upper boundaries are rigid-ferromagnetic
W ¼ 0 ¼ DW;
H ¼ 0;
U ¼ 0 at z ¼ 0; 1
ð22Þ
3. Method of solution Since the principle of exchange of stability is valid irrespective of the boundary conditions considered (see Appendix A), we take r = 0 in Eqs. (17)–(19) and obtain
C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508
1 KðD2 a2 Þ2 Da1 D2 a2 W ¼ a2 R½M 1 DU ð1 þ M 1 ÞH n
501
ð23Þ
ðD2 ga2 ÞH ¼ W
ð24Þ
ðD2 a2 M3 ÞU DH ¼ 0
ð25Þ
Eqs. (23)–(25) together with different types of boundary conditions (20) or (21) or (22) constitute an eigenvalue problem with R as an eigenvalue. The resulting eigenvalue problem is solved exactly for stress free boundaries. While for realistic rigid–rigid paramagnetic/ferromagnetic boundaries the critical eigenvalues are obtained numerically using the Galerkin method. 3.1. Free–free boundaries We assume the solution satisfying the boundary conditions (20) in the form
W ¼ A0 sin pz;
H ¼ B0 sin pz and U ¼ ðC 0 =pÞ cos pz
ð26Þ
where A0, B0 and C0 are constants. Substituting Eq. (26) into Eqs. (23)–(25), and eliminatingA0, B0 and C0 from the equations, we obtain an expression for the Rayleigh number in the form
R¼
fKd4 þ Da1 ðp2 =n þ a2 Þgðp2 þ ga2 Þðp2 þ M 3 a2 Þ a2 fp2 þ M3 ð1 þ M 1 Þa2 g
ð27Þ
where d2 = p2 + a2. For an isotropic case (i.e., n = 1 = g), the above equation coincides with the one obtained by Shivakumara et al. (2008). To find the critical value of R (i.e., Rc) with respect to the wave number a, Eq. (27) is differentiated with respect to a2 and equated to zero. A polynomial in a2c whose coefficients are functions of the parameters influencing the instability is obtained as follows:
5 4 3 2
b1 a2c þ b2 a2c þ b3 a2c þ b4 a2c b5 a2c ¼ 0
ð28Þ
where,
b1 ¼ 2M 23 Kgnð1 þ M 1 Þ b2 ¼ M 3 n½3p2 Kg þ ð1 þ M1 Þfp2 KðM3 þ g þ 2M 3 gÞ þ M3 gDa1 g b3 ¼ 2p2 n½M3 gDa1 þ Kp2 ðg þ M 3 þ 2M3 gÞ h n o n oi b4 ¼ p4 p2 Kn 1 þ 2g M23 M 1 M3 ð2 þ M3 þ gÞ Da1 ð1 þ M 1 ÞM23 gn þ M 1 M 3 ðg þ nÞ b5 ¼ 2M 3 ð1 þ M 1 Þp6 ðp2 Kn þ Da1 Þ Eq. (29) is solved numerically for various values of M1, M3, K, n, g and Da, to obtain the critical value a2c . Using this a2c in Eq. (27), the critical Rayleigh number Rc is obtained, above which the ferroconvection in an anisotropic porous medium sets in. When M3 = 0 (i.e., non-magnetic case), Eq. (27) reduces to
R¼
fKd4 þ Da1 ðp2 =n þ a2 Þgðp2 þ ga2 Þ a2
ð29Þ
This is the result for the ordinary viscous fluid saturating an anisotropic porous medium. For very large M1, we obtain the results for the magnetic mechanism alone operating in the absence of buoyancy effects. The corresponding magnetic Rayleigh number Rm can be expressed as follows:
Rm ¼ RM1 ¼
fKd4 þ Da1 ðp2 =n þ a2 Þgðp2 þ ga2 Þðp2 þ M 3 a2 Þ M 3 a4
ð30Þ
We note that Rmattains its critical value Rmc at a = ac, where ac satisfies the equation
4 3
c1 a2c þ c2 a2c c3 a2c c4 ¼ 0 Here,
c1 ¼ 2M 3 ngK c2 ¼ Kp2 ðgn þ nM 3 þ 2M3 gnÞ þ M 3 gnDa1 c3 ¼ Kp6 nð2 þ g þ M 3 Þ þ p4 ðg þ n þ M3 ÞDa1 c4 ¼ 2p6 ðKp2 n þ Da1 Þ:
ð31Þ
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3.2. Rigid–rigid paramagnetic boundaries For this case, it is difficult to obtain an analytical solution in the closed form to the eigenvalue problem as in the case of free–free boundaries and we have to resort to numerical methods. To extract the critical stability parameters, the Galerkin method is employed and accordingly the dependent variables are written in series of basis functions in the form
W¼
n X
Ai W i ðzÞ;
HðzÞ ¼
n X
i¼1
Bi Hi ðzÞ and UðzÞ ¼
i¼1
n X
C i Ui ðzÞ
ð32Þ
i¼1
where the trial functions Wi(z), Hi(z) and Ui (z) will be generally chosen in such a way that they satisfy the respective boundary conditions, while Ai, Bi and Ci are constants. Substituting Eq. (32) into Eqs. (23)–(25), multiplying Eq. (23) by Wj(z), Eq. (24) by Hj(z) and Eq. (25) by Uj(z); performing the integration by parts with respect to z between z = 0 and z = 1 and using the boundary conditions (21), we obtain the following system of linear homogeneous algebraic equations:
C ji Ai þ Dji Bi þ Eji C i ¼ 0
ð33Þ
F ji Ai þ Gji Bi ¼ 0
ð34Þ
Hji Bi þ Iji C i ¼ 0
ð35Þ
The coefficients Cji Iji involve the inner products of the basis functions and are given by
! Da1 hDW j DW i i þ a2 ða2 K þ Da1 ÞhW j W i i C ji ¼ KhD W j D W i i þ 2a K þ n 2
2
2
Dji ¼ a2 Rð1 þ M 1 ÞhW j Hi i;
Eji ¼ a2 RM1 hW j DUi i;
F ji ¼ hHj W i i
2
Gji ¼ hDHj DHi i þ ga hHj Hi i; Hji ¼ hDUj Hi i a Iji ¼ ½Uj ð1ÞUi ð1Þ þ Uj ð0ÞUi ð0Þ þ hDUj DUi i þ a2 M 3 hUj Ui i 1þv R1 where the inner product is defined as h i ¼ 0 ð Þdz. The above set of homogeneous algebraic equations will have a non-trivial solution if and only if
C ji F ji 0
Dji Gji Hji
Eji 0 ¼0 Iji
ð36Þ
The eigenvalue has to be extracted from the above characteristic equation. For this, we select the trial functions as
W i ¼ ziþ3 2ziþ2 þ ziþ1 ;
Hi ¼ ziþ1 zi ;
Ui ¼ zi 1=2
ð37Þ
It may be noted that the magnetic potential Ui does not satisfy the boundary conditions (21b,c) but the boundary residuals technique is used for the function Ui (see Finlayson, 1972) and the first term in Iji represents this residual term. 3.3. Rigid–rigid ferromagnetic boundaries For this case, we consider the trial functions as
W i ¼ ziþ3 2ziþ2 þ ziþ1 ;
Hi ¼ ziþ1 zi ;
Ui ¼ ziþ2 3ziþ1 þ 2zi
ð38Þ
Here, we note that Wi, Hi and Ui satisfy all the corresponding boundary conditions (22). Hence, the boundary residuals for the function Ui become zero and in this case the coefficient Iji reduces to
Iji ¼ hDUj DUi i þ a2 M3 hUj Ui i
ð39Þ
4. Results and discussion The linear stability analysis has been carried out to investigate the onset of ferroconvection in an anisotropic horizontal porous layer heated from below in the presence of a uniform vertical magnetic field using a non-Darcian model. The lower and upper isothermal boundaries are considered to be rigid-paramagnetic/ferromagnetic or free with large magnetic susceptibility. The critical stability parameters computed numerically by the Galerkin method as explained above, are found to converge by considering six terms in the series expansion of the basis functions. The numerical solution procedure has been validated first by comparing the results with those of Stiles et al. (1992) for the non-porous case (Da1 = 0, K = 1, n = 1 and g = 1). For this a new magnetic parameter Q, independent of the temperature gradient, was introduced in the form 4 Q = Rm/R2, where Q ¼ l0 K 2 Ajv lf =½ð1 þ vÞq20 g 2 a2t d . The critical thermal Rayleigh number (Rc), critical magnetic Rayleigh number (Rmc) and the corresponding wave number (ac) computed numerically for different values of Q for the non-porous
503
C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508 Table 1 Comparison of critical stability parameters with those of Stiles et al. (1992) for different values of Q and v(non-porous case). Q
v
Stiles et al. (1992)
Present values
ac
Rc
ac
Rc
0
2 4 6
3.116 3.116 3.116
1707.8 1707.8 1707.8
0 0 0
3.116 3.116 3.116
1707.9 1707.9 1707.9
0 0 0
104
2 4 6
3.181 3.186 3.188
1567.7 1572.6 1575.1
245.8 247.3 248.1
3.181 3.186 3.188
1567.8 1572.7 1575.2
245.8 247.3 248.1
103
2 4 6
3.404 3.430 3.444
1056.0 1066.7 1072.1
1115.1 1137.9 1149.4
3.404 3.430 3.444
1056.0 1066.7 1072.1
1115.1 1137.9 1149.4
102
2 4 6 2 4 6
3.637 3.685 3.710 3.777 3.837 3.867
456.9 463.0 466.1 52.2 52.9 53.3
2087.6 2143.9 2172.4 2723.1 2801.1 2840.6
3.637 3.685 3.710 3.777 3.837 3.867
456.9 463.0 466.1 52.2 52.9 53.3
2087.6 2143.9 2172.4 2723.4 2798.4 2840.8
102
2 4 6
3.793 3.854 3.884
5.29 5.36 5.40
2795.8 2876.2 2916.9
3.794 3.854 3.884
5.29 5.37 5.40
2798.4 2883.6 2916.0
1
2 4 6
3.794 3.856 3.886
0 0 0
2804.0 2884.7 2925.5
3.794 3.856 3.886
0 0 0
2804.0 2884.7 2925.8
1
Rmc
Rmc
case are compared in Table 1 with those of Stiles et al. (1992). The comparison shows an excellent agreement on the numerical results. The results obtained for different types of boundary conditions covering a wide range of various parameters are illustrated in Figs. 1–4. The variation of Rc and the corresponding ac as a function of Da1 is shown in Figs. 1a and 1b, respectively for different values of K when M3 = 1, M1 = 5, n = 2 and g = 5. From Fig. 1a, it is seen that Rc increases monotonically with Da1 indicating the effect of decrease in the vertical permeability of the porous medium is to delay the onset of ferroconvection. Also, increase in the value of viscosity ratio K is to delay the onset of ferroconvection and this may be due to the increase in viscous diffusion. Further inspection of the figure reveals that the deviation in the Rc values for different boundaries becomes predominant as the value of K increases and the nature of boundaries plays an important role in deciding the stability of the system. The rigid-paramagnetic boundaries with higher magnetic susceptibility (v = 6) are more stable followed by rigidparamagnetic boundaries with low magnetic susceptibility (v = 0), then the rigid-ferromagnetic boundaries and the least stable is free boundaries. In other words, the rigid paramagnetic boundaries have stronger effect in stabilizing the thermally unstable ferrofluid saturated anisotropic porous layer than that of ferromagnetic boundaries because of their dampening 12000 R gi Ri g d-par a am amagnetic wit w th 1+ χ = 7 R gi Ri g d-par a am amagnetic wi w th t 1+ χ = 1 R gi Ri g d-fe ferrom omagnetic
10000
Λ=5
F ee-Fr Fr Free
8000 5
Rc
3
5
6000 3 5
3
4000 3
1
1 1
1
2000
0 10
20
30
40
-1
50 Da D 60
70
80
90
100
Fig. 1a. Variation of critical Rayleigh number Rc as a function of Da1 for different values of K when M3 = 1, M1 = 5, n = 2 and g = 5.
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3.2 Λ=5
3 3.0
Rigid-paramagnetic with 1+χ = 7
1
Rigid-paramagnetic with 1+χ = 1 R gid-fe Ri f rromagnetic
2.8
F ee-Fr Fr F ee
5
ac
3
1
2.6 1 2.4
5 3
2.2 1 2.0 10
20
30
-1
50 Da 60
40
1
Fig. 1b. Variation of critical wave number ac as a function of Da
70
80
90
100
for different values of K when M3 = 1, M1 = 5, n = 2 and g = 5.
15000 Rigid-paramagnetic with 1+ χ = 7 Rigid-paramagnetic with 1+ χ = 1 Rigid-ferromagnetic Free-Free -5
12000
M1 = 10
Rc
0.1
9000
1 1 1
6000 -5
10 0.1 3000 1
2
3
4
ξ
5
6
1 7
8
9
Fig. 2a. Variation of critical Rayleigh number Rc as a function of n for different values of M1 when g = 5, Da1 = 50, K = 2 and M3 = 1.
effect due to an increase in the magnetic induction. Also, it is evident that increasing magnetic susceptibility is to delay the onset of ferroconvection. From Fig. 1b it is evident that the critical wave numbers decrease with Da1 in the case of rigidparamagnetic/ferromagnetic boundaries but an opposite trend could be seen in the case of free boundaries. Irrespective of the magnetic boundary conditions, increasing K and v is to increase ac in the case of rigid boundaries and thus their effect is to narrow the convection cells. To the contrary, increase in K is to decrease ac and thus its effect is to widen the convection cells when the boundaries are free. The ac values are higher for large magnetic susceptibility rigid-paramagnetic boundaries compared to low susceptibility boundaries and the least for rigid-ferromagnetic boundaries. The important parameters which are affecting the stability characteristics of the system are the mechanical and thermal anisotropy parameters. The variations of critical Rayleigh number Rc and the corresponding critical wave number ac as a function of mechanical anisotropy parameter n(= kh/kv) for different boundaries are presented in Figs. 2a and 2b, respectively for different values of M1whenM3 = 1, g = 5,Da1 = 50 andK = 2. From Figs. 2a and 2b, it is evident that the critical Rayleigh number Rc and the corresponding wave number acincrease with decreasingn. This may be understood as follows: let us keep the vertical permeability kv fixed and decrease the horizontal permeability kh. Physically, this means that the conduction solution in the anisotropic porous medium becomes more stable and the critical wavelength decreases as the horizontal permeability decreases. Smaller horizontal permeability inhibits horizontal motion, and the conduction solution is thus
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505
2.7
M1 = 1
2.4
1
ac
1
0.1
2.1
-5
10
Rigid-paramagnetic with 1+ χ = 7 Rigid-paramagnetic with 1+ χ = 1 Rigid-ferromagnetic Free-Free
1 1.8
0.1 -5
1.5
10 1
2
3
ξ
4
5
6
7
8
9
Fig. 2b. Variation of critical wave number ac as a function of n for different values of M1 when g = 5, Da1 = 50, K = 2 and M3 = 1.
20000 Rigid-paramagnetic with 1+ χ = 7 Rigid-paramagnetic with 1+ χ = 1 Rigid-ferromagnetic Free-Free
-5
M1 = 10
15000 0.1
Rc
1 1
10000
1 -5
10 0.1
1
5000
0 1
2
3
4
η
5
6
7
8
9
Fig. 3a. Variation of critical Rayleigh number Rc as a function of g for different values of M1 when n = 2, Da1 = 50, K = 2 and M3 = 1.
stabilized. The larger resistance to horizontal flow also leads to a shortening of the horizontal wavelength (or increase in the horizontal wave number) at the onset. Further, increasing M1is to decrease the values of Rc and to hasten the onset of ferroconvection due to an increase in the magnetic force, while opposite is the trend with respect to the critical wave number. We note that increasingM1is to reduce the size of convection cells. Figs. 3a and 3b represent the variation of Rcand ac as a function of thermal anisotropy parameter g (= jh/jv) on the stability characteristics of the system for different boundaries and also for different values of M1. The results exhibited in the figures are forM3 = 1, n = 2, Da1 = 50 and K = 2. In contrast to the effect of mechanical anisotropy parameter, it is seen that decrease in the value of g is to decrease the value of Rc, whereas the critical wave number increases with decreasing g. That is, decreasing g is to hasten the onset of ferroconvection and also to diminish the size of convection cells. This may be attributed to the fact that, as g decreases, a heated fluid parcel loses less heat in the horizontal directions, and hence retains its buoyancy better. Hence, the basic state becomes less stable which in turn augments the onset of ferroconvection, and also the wavelength is reduced. The complementary effects of buoyancy and the magnetic forces are made clear in Fig. 4 by displaying the locus of the critical Rayleigh number Rc and the critical magnetic Rayleigh Rmc for M3 = 1,Da1 = 100, n = 2, K = 2 and g = 2. We note that
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3.3 Rigid-paramagnetic with 1+ χ = 7 Rigid-paramagnetic with 1+ χ = 1 Rigid-ferromagnetic Free-Free
3.0
2.7
ac
M1 = 1
2.4
1 0.1
2.1
1
-5
10
1 1.8
0.1 -5
10 1.5 1
2
3
4
η
5
6
7
8
9
Fig. 3b. Variation of critical wave number ac as a function of g for different values of M1 when n = 2, Da1 = 50, K = 2 and M3 = 1.
10000 Rigid-paramagnetic with 1+ χ = 7 Rigid-paramagnetic with 1+ χ = 1
8000
Rigid-ferromagnetic
6000
Rc 4000
2000
0 0
5000
Rmc
10000
15000
20000
Fig. 4. Locus of Rc and Rmc for M3 = 1, g = 2, Da1 = 100, K = 2 and n = 2.
Rc is inversely proportional to Rmc. In the absence of buoyancy forces (Rc = 0), the instability sets in at higher values of Rmc indicating the system is more stable when the magnetic forces alone are present.
5. Conclusions The linear stability theory is used to investigate the onset of Brinkman ferroconvection in an anisotropic porous layer heated from below in the presence of a uniform magnetic field. The resulting eigenvalue problem is solved numerically by employing the Galerkin method when the boundaries are rigid and solved exactly when the boundaries are stress-free. From the foregoing study, the following conclusions may be drawn: (i) The system is more stabilizing against the ferroconvection if the boundaries are rigid-paramagnetic with high magnetic susceptibility and least stable if the boundaries are free. It is observed that
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507
ðRc and ac Þrigid-parmagnetic > ðRc and ac Þrigid-ferromagnetic > ðRc and ac Þfree—free (ii) The effect of increasing the value of mechanical anisotropy parameter n is to hasten, while increase in the value of thermal anisotropy parameter g is to delay the onset of ferroconvection in porous media. Thus, it is possible to control (suppress or augment) ferroconvection by adjusting the mechanical and thermal anisotropy parameters. (iii) The effect of increasing the value of magnetic number M1 is to hasten the onset of ferroconvection, while increase in the value of Da1, K and v is to delay the onset of ferroconvection. (iv) The effect of increasing g, v, Da1 and M1 as well as decrease in n is to increase the critical wave number and hence their effect is to narrow the convection cells. (v) The magnetic and buoyancy forces are complementary with each other and the system is more stabilizing when the magnetic forces alone are present.
Acknowledgements One of the authors (ISS) wishes to thank the BK 21 Program of the School of Mechanical Engineering, Yonsei University, Seoul, Korea for inviting him as a visiting Professor and also the Bangalore University for sanctioning sabbatical leave. The authors (C.E.N.) and (M.R.) wish to thank the Principals of their respective colleges for their encouragement. We thank the reviewer for useful suggestions. Appendix A Here, we analyze whether ferroconvection in an anisotropic porous layer sets in as oscillatory instability or not. Following Shivakumara et al. (2008), first we eliminate U from Eq. (17) by operating (D2 a2M3) on that equation and use Eq. (19) to obtain
1 2 D a2 r W ¼ a2 R½M 1 D2 ð1 þ M 1 ÞðD2 a2 M 3 ÞH ðD2 a2 M3 Þ KðD2 a2 Þ2 Da1 n
ðA1Þ
Multiplying Eq. (A1) by W⁄ (complex conjugate of W) and integrating from z = 0 to z = 1, we get
1 2 D a2 r W ¼ a2 RhW D2 Hi a4 Rð1 þ M1 ÞM 3 hW Hi W ðD2 a2 M 3 Þ KðD2 a2 Þ2 Da1 n
ðA2Þ
R1
where hFi ¼ 0 Fdz. The left hand side of Eq. (A2), on repeatedly applying integration by parts and simplifying using the boundary conditions on the velocity can be written as 2
3
K½D W D W DW D
4
! Da1 hKjD Wj þ Ka ð2 þ M 3 Þ þ þ r jD2 Wj2 n ! ! Da1 þ r ð1 þ M 3 Þ jDWj2 þ a4 M3 ða2 K þ Da1 þ rÞjWj2 i n
W10
þ a4 Kð1 þ 2M 3 Þ þ a2
2
3
2
ðA3Þ
where jFj2 = FF⁄ and F⁄ is the conjugate of F. From Eq. (18), after neglecting the terms involving M2, we have
W þ ððD2 ga2 Prr ÞH ¼ 0
ðA4Þ
⁄
Substituting for W from Eq. (A4) on the right hand side of Eq. (A2), integrating by parts and using the boundary conditions on H we get
hjD2 Hj2 þ ðga2 þ Prr ÞjDHj2 i a2 M 3 ð1 þ M 1 ÞhjDHj2 þ ðga2 þ Pr r ÞjHj2 i
ðA5Þ
Using Eqs. (A3) and (A5) in Eq. (A2), we obtain
* 2
3
K½D W D W DW D 2
þa
4
W10
! Da1 KjD Wj þ Ka ð2 þ M3 Þ þ þ r jD2 Wj2 n ! ! 3
2
2
Da1 Ka ð1 þ 2M 3 Þ þ þ r ð1 þ M 3 Þ jDWj2 þ a4 M 3 ðKa2 þ Da1 þ r ÞjWj2 n
+
2
¼ hjD2 Hj2 i ða2 M 3 ð1 þ M 1 Þ þ ða2 þ Prr ÞÞhjDHj2 i a2 M3 ð1 þ M 1 Þða2 þ Prr Þ < jHj2 Equating the real and imaginary parts of the above equation separately, we get respectively
ðA6Þ
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*
K½D2 W D3 W DW D4 W > 10 KjD3 Wj2 þ a2 ð2 þ M3 Þ þ þa
2
! Da1 þ r jD2 Wj2 n +
! Da1 Ka ð1 þ 2M 3 Þ þ þ r ð1 þ M 3 Þ jDWj2 þ a4 M 3 ðKa2 þ Da1 ÞjWj2 n 2
¼ ½hjD2 Hj2 i þ ða2 M3 ð1 þ M1 Þ þ ðga2 þ PrÞÞhjDHj2 i þ a2 M3 ð1 þ M1 Þðga2 þ PrÞhjHj2 i
ðA7Þ
and
KhjD2 Wj2 þ a2 ð1 þ M 3 ÞjDWj2 þ a4 M 3 jWj2 i ¼ PrhjDHj2 i Pra2 M3 ð1 þ M 1 ÞhjHj2 i
ðA8Þ
Only the imaginary part gives the information about the possibility of occurring oscillatory instability. From Eq. (A9) it is evident that oscillatory instability occurs provided
KhjD2 Wj2 þ a2 ð1 þ M3 ÞjDWj2 þ a4 M3 jWj2 i þ PrhjDHj2 þ a2 M 3 ð1 þ M1 ÞjHj2 i ¼ 0
ðA9Þ
Since the left hand side of Eq. (A9) is a positive definite, the above condition will never be satisfied. Therefore oscillatory instability is not possible for ferroconvection in an anisotropic porous medium under the influence of vertical magnetic field. This result is true irrespective of the type of velocity (rigid or free) and magnetic (paramagnetic or ferromagnetic) boundary conditions. References Auernhammer, G. K., & Brand, H. R. (2000). Thermal convection in a rotating layer of a magnetic fluid. The European Physical Journal B, 16, 157–168. Bashtovoy, V. G., & Berkovsky, B. N. (1996). Magnetic fluids and applications hand book. New York: Begell house. Borglin, S. E., Mordis, G. J., & Oldenburg, C. M. (2000). Experimental studies of the flow of ferrofluid in porous media. Transport in Porous Media, 41, 61–80. Finlayson, B. A. (1970). Convective instability of ferromagnetic fluids. Journal of Fluid Mechanics, 40(4), 753–767. Finlayson, B. A. (1972). Method of weighted residuals and variational principles. Academic Press. Givler, R. A., & Altobelli, S. A. (1994). A determination of the effective viscosity for the Brinkman–Forcheimer flow model. Journal of Fluid Mechanics, 258, 355–370. Gotoh, K., & Yamada, M. (1982). Thermal convection in a horizontal layer of magnetic fluids. Journal of the Physical Society of Japan, 51, 3042–3048. Kaloni, P. N., & Lou, J. X. (2004). Convective instability of magnetic fluids under alternating magnetic fields. Physical Review E, 71, 066311-1–066311-12. Lalas, D. P., & Carmi, S. (1971). Thermoconvective stability of ferrofluids. Physics of Fluids, 14(2), 436–437. Nanjundappa, C. E., & Shivakumara, I. S. (2008). Effect of velocity and temperature boundary conditions on convective instability in a ferrofluid layer. ASME Journal of Heat Transfer, 130, 104502-1–104502-5. Nanjundappa, C. E., Shivakumara, I. S., & Ravisha, M. (2010). The onset of buoyancy-driven convection in a ferromagnetic fluid saturated porous medium. Meccanica, 45, 213–226. Nield, D. A., & Bejan, A. (2006). Convection in porous media (3rd ed.). Springer. Odenbach, S. (2004). Recent progress in magnetic fluid research. Journal of Physics Condensed Matter, 16, R1135-50. Qin, Y., & Chadam, J. (1995). A non-linear stability problem for ferromagnetic fluids in a porous medium. Applied Mathematics Letters, 8(2), 25–29. Ramathan, A., & Suresh, G. (2004). Effects of magnetic field dependent viscosity and anisotropy of porous medium on ferroconvection. International Journal of Engineering Science, 42, 411–425. Rerkovsky, B. M., Medvedev, V. F., & Krakov, M. S. (1993). Magnetic fluids, engineering applications. New York: Oxford University Press. Rosensweig, R. E. (1985). Ferrohydrodynamics. Cambridge: Cambridge University Press. Rosensweig, R. E., Zahn, M., & Volger, T. (1978). Stabilization of fluid penetration through a porous medium using magnetizable fluids. In B. Berkovsky (Ed.), Thermomechanics of magnetic fluids (pp. 195–211). Washington DC: Hemisphere. Schwab, L., Hilderbrandt, U., & Stierstadt, K. (1987). Magnetic Benard – Convection. Journal of Magnetism and Magnetic Materials, 65, 315–319. Sekar, R., Vaidynathan, G., & Ramathan, A. (1996). Ferrocnvection in an Anisotrophic porous medium. International Journal of Engineering Science, 34(4), 399–405. Shivakumara, I. S., Nanjundappa, C. E., & Ravisha, M. (2008). Thermomagnetic convection in a magnetic nanofluid saturated porous medium. International Journal of Applied Mathematics and Engineering Science, 2(2), 157–170. Shivakumara, I. S., Nanjundappa, C. E., & Ravisha, M. (2009). Effect of boundary conditions on the onset of thermomagnetic convection in a ferrofluid saturated porous medium. ASME Journal of Heat Transfer, 131, 101003-1–101003-9. Shivakumara, I. S., Rudraiah, N., & Nanjundappa, C. E. (2002). Effect of non-uniform basic temperature gradient on Rayleigh–Benard–Marangoni convection in ferrofluids. Journal of Magnetism and Magnetic Materials, 248, 379–395. Shliomis, M. I. (1974). Magnetic fluids. Soviet Physics, Uspekhi (Engl. Trans), 17(2), 153–169. Stiles, P. J., & Kagan, M. (1990). Thermoconvective instability of a ferrofluid in a strong magnetic field. Journal of Colloid Interface Science, 134, 435–448. Stiles, P. J., Lin, F., & Blennerhassett, P. J. (1992). Heat transfer through weakly magnetized ferrofluids. Journal of Colloid Interface Science, 151, 95–101. Vaidyanathan, G., Sekar, R., & Balasubramanian, R. (1991). Ferroconvective instability of fluids saturating a porous medium. International Journal of Engineering Science, 29, 1259–1267. Volker, T., Blums, E., & Odenbach, S. (2007). Heat and mass transfer phenomena in magnetic fluids. GAMM-Mitteilungen, 30(1), 185–194. Zhan, M., & Rosensweig, R. E. (1980). Stability of magnetic fluid penetration through a porous medium with uniform magnetic field oblique to the interface. IEEE Transactions of Magnetics, 16, 275–282.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
The onset of Brinkman ferroconvection in an anisotropic porous medium C.E. Nanjundappa a,⇑, I.S. Shivakumara b,1, Jinho Lee b,⇑, M. Ravisha c a b c
Department of Mathematics, Dr. Ambedkar Institute of Technology, Bangalore 560 056, India School of Mechanical Engineering, Yonsei University, Seoul 120-749, South Korea Department of Mathematics, Smt. Rukmuni Shedthi Memorial National Government First Grade College, Barkur 576210, Udup District, India
a r t i c l e
i n f o
Article history: Received 27 April 2010 Accepted 29 December 2010 Available online 5 March 2011 Keywords: Ferroconvection Anisotropic porous medium Paramagnetic Ferromagnetic
a b s t r a c t Onset of ferroconvection in an anisotropic porous layer heated from below is investigated theoretically using modified Brinkman extended-Darcy equation with fluid viscosity different from effective viscosity. The isothermal bounding surfaces of a porous layer are considered to be either free or rigid-paramagnetic/ferromagnetic. The eigenvalue problem is solved exactly for free boundaries, while for realistic rigid-paramagnetic or rigidferromagnetic boundaries the critical stability parameters are obtained numerically using the Galerkin method. It is seen that the stability of the system depends on the nature of boundaries and rigid-paramagnetic boundaries are found to be preferred to the ferromagnetic ones as well as free boundaries in controlling ferroconvection in an anisotropic porous layer. It is observed that increase in the value of thermal anisotropy parameter and viscosity ratio is to delay the onset of ferroconvection, while increase in the value of mechanical anisotropy parameter and magnetic number is to hasten the onset of ferroconvection. Moreover, increasing the value of thermal anisotropy parameter and decreasing the value of mechanical anisotropy parameter is to narrow the convection cells. Ó 2011 Published by Elsevier Ltd.
1. Introduction Magnetic nanofluids are stable colloidal suspensions of single-domain nano-scale magnetic particles with typical dimensions of about 3–10 nm dispersed in non-conducting carrier liquids like water, kerosene, ester, hydrocarbons, etc. These fluids are also referred to as ferrofluids. Since 1960s, when these fluids were initially synthetized, their technological applications have been step up over the years. These fluids are found to have numerous applications, for example in loud speakers, rotatory exclusion seals, bearings, dampers, shock absorbers, medicine drug targeting, and in many other thermal transport applications. A detailed introduction to ferrofluids along with their diverse applications is well documented in the books by Rosensweig (1985), Bashtovoy and Berkovsky (1996) and Rerkovsky et al. (1993). Thermo-gravitational convection in a layer of ferrofluid in the presence of a uniform magnetic field, known as ferroconvection, is analogous to classical Benard convection and has received due attention in the literature because of promising potential in heat transfer applications. Finlayson (1970) was the first to study this problem theoretically. Thermoconvective instability of ferrofluids without considering buoyancy effects has been investigated by Lalas and Carmi (1971), whereas Shliomis (1974) has analyzed the linear relation for magnetized perturbed quantities at the limit of instability. A similar analysis but with the ferrofluid confined between ferromagnetic plates has been carried out by Gotoh and Yamada (1982) using linear stability analysis. Schwab, Hilderbrandt, and Stierstadt (1987) have conducted ⇑ Corresponding authors. Tel.: +91 080 3211506 (C.E. Nanjundappa). 1
E-mail address: [email protected] (C.E. Nanjundappa). Permanent address: UGC-CAS in Fluid Mechanics, Department of Mathematics, Bangalore University, Bangalore 560 001, India.
0020-7225/$ - see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.ijengsci.2010.12.014
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experiments and showed that their results are in good agreement with theoretical predictions. Stiles and Kagan (1990) have extended the problem to allow for the dependence of effective shear viscosity on temperature and colloidal concentration, while Stiles, Lin, and Blennerhassett (1992) have analyzed linear and weakly nonlinear thermoconvective instability in a thin layer of ferrofluid subjected to a weak uniform external magnetic field in the vertical direction. The effect of rotation on linear and weakly nonlinear thermal convection in a magnetic fluid layer is investigated by Auernhammer and Brand (2000), while the effect of different forms of basic temperature gradients on the onset of ferroconvection driven by combined surface tension and buoyancy forces has been discussed by Shivakumara, Rudraiah, and Nanjundappa (2002) with the sole motto of understanding control of ferroconvection. Kaloni and Lou (2004) have theoretically investigated the convective instability problem in a thin horizontal layer of magnetic fluid heated from below under alternating magnetic field by considering the quasi stationary model with internal rotation and vortex viscosity. In his review article, Odenbach (2004) has focused on recent developments in the field of rheological investigations of ferrofluids and their importance for the general treatment of ferrofluids. The influence of magnetic field on heat and mass transport in ferrofluids has been discussed by Volker, Blums, and Odenbach (2007). Nanjundappa and Shivakumara (2008) have investigated a variety of velocity and temperature boundary conditions on the onset of ferroconvection in an initially quiescent ferrofluid layer. Thermal convection of ferrofluids saturating a porous medium has also attracted considerable attention in the literature owing to its importance in controlled emplacement of liquids or treatment of chemicals, and emplacement of geophysically imageable liquids into particular zones for subsequent imaging, etc. Rosensweig, Zahn, and Volger (1978) have studied experimentally the penetration of ferrofluids in the Heleshaw cell. The stability of the magnetic fluid penetration through a porous medium in high uniform magnetic field oblique to the interface is studied by Zhan and Rosensweig (1980). Thermal convective instability in a layer of ferrofluid saturating a porous medium in the presence of a vertical magnetic field is studied by Vaidyanathan, Sekar, and Balasubramanian (1991). Their analysis is limited to free-free boundaries and to the case of effective viscosity equal to fluid viscosity. Qin and Chadam (1995) have carried out the non-linear stability analysis of ferroconvection in a porous layer by including the inertial effects to accommodate high velocity. Sekar, Vaidynathan, and Ramathan (1996) have investigated ferroconvection in an anisotropic porous medium for stress-free boundaries by considering anisotropy only in the permeability of the porous medium. The laboratory scale experimental results of the behaviour of ferrofluids in porous media consisting of sands and sediments are presented in detail by Borglin, Mordis, and Oldenburg (2000). Ramathan and Suresh (2004) have analyzed the effects of magnetic field dependent viscosity and mechanical anisotropy of porous medium on ferroconvection. Recently, Shivakumara, Nanjundappa, and Ravisha (2008, 2009) have investigated in detail the onset of thermomagnetic convection in a ferrofluid saturated porous medium for various types of velocity and temperature boundary conditions. The onset of buoyancy-driven convection in a porous layer heated from below using the Brinkman–Lapwood extended Darcy equation with fluid viscosity different effective viscosity is investigated by Nanjundappa, Shivakumara, and Ravisha (2010). Many technological applications of practical importance involve porous materials of high porosity and for such a porous medium Givler and Altobelli (1994) have demonstrated that the effective viscosity is about ten times the fluid viscosity. Under the circumstances, a theoretical study which is general enough to yield accurate results for ferroconvection in porous media is of fundamental and practical interest. In many practical situations, it is reasonable to expect the porous materials to be anisotropic in their mechanical and thermal properties. Anisotropy is generally a consequence of preferential orientation or asymmetric geometry of porous matrix or fibers and is in fact encountered in numerous systems in industry and nature. Anisotropy can also be a characteristic of artificial porous materials like pelletting used in chemical engineering process and fiber material used in insulating purposes. Copious literature is available on thermal convection in a layer of an anisotropic porous medium because the anisotropy significantly alters the convective instability of the problem. For the literature on this problem see, for example, the survey in Nield and Bejan (2006). The main objective of the paper is, therefore, to obtain the criterion for the onset of ferroconvection in a ferrofluid saturated anisotropic Brinkman porous layer heated from below in the presence of a uniform vertical magnetic field. Keeping in mind the realistic situation, the boundaries are considered to be rigid and are either paramagnetic or ferromagnetic. The opportunity is also being used to review the results on free boundaries in the present context. The resulting eigenvalue problem is solved exactly for free–free boundaries and for the case of rigid–rigid boundaries the problem is solved numerically using a Galerkin technique. To achieve the above objectives, the paper is organized as under. Section 2 is devoted to mathematical formulation. The method of solution is discussed in Section 3. In Section 4, the numerical results presented are discussed and some important conclusions follow in Section 5.
2. Mathematical formulation We consider an initially quiescent ferrofluid saturated horizontal anisotropic porous layer of depth d in the presence of a ^ uniform applied magnetic field Ho acting in the vertical direction. A Cartesian coordinate system (x, y, z) is used with ð^i; ^j; kÞ being the standard basis and the gravity acts in the negative vertical z-direction. The lower and upper boundaries of the porous layer are maintained at constant but different temperatures T0 and Tl(
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The governing equations are as follows: The continuity equation
r ~ q¼0
ð1Þ
where, ~ q ¼ ðu; v ; wÞ is the velocity vector. The momentum equation
q0 @~ q ~~ ~ f r2~ q þ r ðH ¼ rp þ q0 ½1 at ðT T 0 Þ~ g lf k1 ~ qþl BÞ e @t
ð2Þ
~ is the magnetic field, ~ where, p is the pressure, H B is the magnetic induction, T is the temperature, q0 is the reference density, ~ ~ f is the effective viscosity, at is the thermal expansion coefg is the acceleration due to gravity, lf is the dynamic viscosity, l ficient, e is the porosity of the porous medium, r2 = @ 2/@x2 + @ 2/oy2 + @ 2/@z2 is the Laplacian operator and k is the permeabil 1 1 ^ ^ k, where kh and kv are the ity tensor. The horizontal isotropy is assumed and hence we have k1 ¼ kh ð^i^i þ ^j^jÞ þ kv k permeability in the horizontal and vertical directions respectively. The energy equation
2
~ ~ @M e4q0 C V;H l0 H @T
3
! V;H
~ 5 DT þ ð1 eÞðq0 CÞ @T þ l T @ M S 0 Dt @t @T
! V;H
~ DH ¼ r ðj rTÞ Dt
ð3Þ
~ is the magnetization, l0 is the where, C is the specific heat, CV,H is the specific heat at constant volume and magnetic field, M ^k, ^ magnetic permeability of vacuum and j is the effective thermal diffusivity tensor which is given by j ¼ jh ð^i^i þ ^j^jÞ þ jv k while jh and jv are the effective thermal diffusivity in the horizontal and vertical directions respectively. The Maxwell equations in the magnetostatic limit are:
~ ¼ 0 or H ~ ¼ ru r ~ B ¼ 0r H
ð4a;bÞ
~ and H ~ are related by where, u is the magnetic potential. Further, ~ B, M
~ ~ þ HÞ ~ B ¼ l0 ð M
ð5Þ
We assume that the magnetization is aligned with the magnetic field, but allow a dependence on the magnitude of magnetic field as well as the temperature in the form
~ ~ ¼ H MðH; T Þ M H
ð6Þ
The magnetic equation of state is linearized about H0 and T0 to become
M ¼ M 0 þ vðH H0 Þ KðT T 0 Þ
ð7Þ
where, v ¼ ð@M=@HÞH0 ;T 0 is the magnetic susceptibility of the ferrofluid, K ¼ ð@M=@TÞH0 ;T 0 is called the pyromagnetic co-effi~ and M ¼ jMj. ~ cient, M0 = M(H0, T0), H ¼ jHj The basic state is quiescent and is given by
~ qb ¼ 0 1 l M0 jb l j2 b2 2 pb ðzÞ ¼ p0 q0 gz q0 at gbz2 0 z 0 z 2 1þv 2ð1 þ vÞ2 T b ðzÞ ¼ T 0 bz Kbz Hb ðzÞ ¼ H0 1þv Kbz M b ðzÞ ¼ M0 þ 1þv
ð8Þ
where the subscript b denotes the basic state and b = DT/d is the temperature gradient. To study the linear stability of the above steady state solution, the variables are perturbed in the form
~ q ¼~ q0 ;
p ¼ pb ðzÞ þ p0 ;
T ¼ T b ðzÞ þ T 0 ;
~¼H ~b ðzÞ þ H ~0 ; H
~ ¼M ~ b ðzÞ þ M ~0 M
ð9Þ
where the primed quantities are perturbed ones and are assumed to be small. Substituting Eq. (9) into Eqs. (6) and (7) and using Eq. (5), we get (after ignoring the primes):
M0 Hx ; Hx þ M x ¼ 1 þ H0 Hz þ Mz ¼ ð1 þ vÞHz KT
Hy þ M y ¼
M0 Hy 1þ H0
ð10Þ ð11Þ
500
C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508
where (Hx, Hy, Hz) and (Mx, My, Mz) are the components of perturbed magnetic field and magnetization, respectively. In obtaining the above equations, it is assumed that Kbd (1 + v)H0. After substituting Eq. (9) into Eq. (2), linearizing, eliminating the pressure term by operating curl twice, and using Eqs. ~0 ¼ r/0 , the z-component of the resulting equation can be obtained as (after ignoring the (10) and (11) together withH primes):
q0
l l @2w @ @ 2 l0 K 2 b 2 ~ f r2 r2 w þ f r2h w þ f l ¼ l0 Kb rh u þ r T þ q0 at g r2h T 2 @t @z 1þv h kv kh @z
ð12Þ
where r2h ¼ @ 2 =@x2 þ @ 2 =@y2 is the horizontal Laplacian operator. As before, using Eq. (9) in Eq. (3) and linearizing the equation, we get (after ignoring the primes)
ðq0 CÞ1
# " @T @ @u l T0K2 @2T ¼ ðq0 CÞ2 0 bw þ jh r2h T þ jv 2 l0 T 0 K @t @t @z @z 1þv
ð13Þ
where (q0C)1 = eq0CV,H + el0H0K + (1 e)(q0C)s and (q0C)2 = eq0CV,H + el0H0K. ~0 ¼ ru0 , Eqs. (4a,b) may be written as (after Finally, after substituting Eq. (9) and using Eqs. (10) and (11) together with H ignoring the primes)
M0 @2u @T ¼0 1þ r2h u þ ð1 þ vÞ 2 K @z H0 @z
ð14Þ
Assume the normal mode expansion of the dependent variables in the form
fw; T; ug ¼ fWðzÞ; HðzÞ; UðzÞg exp½ið‘x þ myÞ þ rt
ð15Þ
where ‘ and m are wave numbers in the x and y directions, respectively and r is the growth rate. Substituting Eq. (15) into Eqs. (12)–(14) and non-dimensionalizing the variables by setting
z z ¼ ; d
W ¼
d W; mA
t ¼
m d
t; 2
2
r ¼
d
m
r; H ¼
jv bv d
H;
U ¼
ð1 þ vÞjv 2
Kbv d
U
ð16Þ
where v = lf/q0 is the kinematic viscosity and A = (q0C)1/(q0C)2 is the ratio of heat capacities, we obtain (after ignoring the asterisks)
1 KðD2 a2 Þ2 Da1 D2 a2 rðD2 a2 Þ W ¼ a2 R½M1 DU ð1 þ M 1 ÞH n
ð17Þ
ðD2 ga2 PrrÞH PrM 2 rDU ¼ ð1 M 2 AÞW
ð18Þ
2
2
ðD a M3 ÞU DH ¼ 0
ð19Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here D = d/dz is the differential operator, a ¼ ‘2 þ m2 is the overall horizontal wave number, R = atgbd4/mjvA is the thermal Rayleigh number, M1 = l0K2b/(1 + v)atq0g is the magnetic number, M2 = l0T0K2/(1 + v)(q0C)1 is the magnetic parameter, ~ f =lf is the M3 = (1 + M0/H0)/(1 + v) is the measure of nonlinearity of magnetization, Pr = m/j is the Prandtl number, K ¼ l ratio of viscosities, n = kh/kv is the anisotropic permeability parameter, g = jh/jv is the anisotropic effective thermal diffusivity parameter and Da = kv/d2 is the Darcy number. The typical value of M2 for magnetic fluids with different carrier liquids turns out to be of the order of 106 and hence its effect is neglected as compared to unity. The above equations are solved subject to the following three different types of boundary conditions: (i) Lower and upper boundaries are free with large magnetic susceptibility
W ¼ 0 ¼ D2 W;
H ¼ 0;
DU ¼ 0 at z ¼ 0; 1
ð20Þ
(ii) Lower and upper boundaries are rigid-paramagnetic
W ¼ 0 ¼ DW;
H ¼ 0 at z ¼ 0; 1
ð1 þ vÞDU aU ¼ 0 at z ¼ 0 ð1 þ vÞDU þ aU ¼ 0 at z ¼ 1
ð21aÞ ð21bÞ ð21cÞ
(iii) Lower and upper boundaries are rigid-ferromagnetic
W ¼ 0 ¼ DW;
H ¼ 0;
U ¼ 0 at z ¼ 0; 1
ð22Þ
3. Method of solution Since the principle of exchange of stability is valid irrespective of the boundary conditions considered (see Appendix A), we take r = 0 in Eqs. (17)–(19) and obtain
C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508
1 KðD2 a2 Þ2 Da1 D2 a2 W ¼ a2 R½M 1 DU ð1 þ M 1 ÞH n
501
ð23Þ
ðD2 ga2 ÞH ¼ W
ð24Þ
ðD2 a2 M3 ÞU DH ¼ 0
ð25Þ
Eqs. (23)–(25) together with different types of boundary conditions (20) or (21) or (22) constitute an eigenvalue problem with R as an eigenvalue. The resulting eigenvalue problem is solved exactly for stress free boundaries. While for realistic rigid–rigid paramagnetic/ferromagnetic boundaries the critical eigenvalues are obtained numerically using the Galerkin method. 3.1. Free–free boundaries We assume the solution satisfying the boundary conditions (20) in the form
W ¼ A0 sin pz;
H ¼ B0 sin pz and U ¼ ðC 0 =pÞ cos pz
ð26Þ
where A0, B0 and C0 are constants. Substituting Eq. (26) into Eqs. (23)–(25), and eliminatingA0, B0 and C0 from the equations, we obtain an expression for the Rayleigh number in the form
R¼
fKd4 þ Da1 ðp2 =n þ a2 Þgðp2 þ ga2 Þðp2 þ M 3 a2 Þ a2 fp2 þ M3 ð1 þ M 1 Þa2 g
ð27Þ
where d2 = p2 + a2. For an isotropic case (i.e., n = 1 = g), the above equation coincides with the one obtained by Shivakumara et al. (2008). To find the critical value of R (i.e., Rc) with respect to the wave number a, Eq. (27) is differentiated with respect to a2 and equated to zero. A polynomial in a2c whose coefficients are functions of the parameters influencing the instability is obtained as follows:
5 4 3 2
b1 a2c þ b2 a2c þ b3 a2c þ b4 a2c b5 a2c ¼ 0
ð28Þ
where,
b1 ¼ 2M 23 Kgnð1 þ M 1 Þ b2 ¼ M 3 n½3p2 Kg þ ð1 þ M1 Þfp2 KðM3 þ g þ 2M 3 gÞ þ M3 gDa1 g b3 ¼ 2p2 n½M3 gDa1 þ Kp2 ðg þ M 3 þ 2M3 gÞ h n o n oi b4 ¼ p4 p2 Kn 1 þ 2g M23 M 1 M3 ð2 þ M3 þ gÞ Da1 ð1 þ M 1 ÞM23 gn þ M 1 M 3 ðg þ nÞ b5 ¼ 2M 3 ð1 þ M 1 Þp6 ðp2 Kn þ Da1 Þ Eq. (29) is solved numerically for various values of M1, M3, K, n, g and Da, to obtain the critical value a2c . Using this a2c in Eq. (27), the critical Rayleigh number Rc is obtained, above which the ferroconvection in an anisotropic porous medium sets in. When M3 = 0 (i.e., non-magnetic case), Eq. (27) reduces to
R¼
fKd4 þ Da1 ðp2 =n þ a2 Þgðp2 þ ga2 Þ a2
ð29Þ
This is the result for the ordinary viscous fluid saturating an anisotropic porous medium. For very large M1, we obtain the results for the magnetic mechanism alone operating in the absence of buoyancy effects. The corresponding magnetic Rayleigh number Rm can be expressed as follows:
Rm ¼ RM1 ¼
fKd4 þ Da1 ðp2 =n þ a2 Þgðp2 þ ga2 Þðp2 þ M 3 a2 Þ M 3 a4
ð30Þ
We note that Rmattains its critical value Rmc at a = ac, where ac satisfies the equation
4 3
c1 a2c þ c2 a2c c3 a2c c4 ¼ 0 Here,
c1 ¼ 2M 3 ngK c2 ¼ Kp2 ðgn þ nM 3 þ 2M3 gnÞ þ M 3 gnDa1 c3 ¼ Kp6 nð2 þ g þ M 3 Þ þ p4 ðg þ n þ M3 ÞDa1 c4 ¼ 2p6 ðKp2 n þ Da1 Þ:
ð31Þ
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C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508
3.2. Rigid–rigid paramagnetic boundaries For this case, it is difficult to obtain an analytical solution in the closed form to the eigenvalue problem as in the case of free–free boundaries and we have to resort to numerical methods. To extract the critical stability parameters, the Galerkin method is employed and accordingly the dependent variables are written in series of basis functions in the form
W¼
n X
Ai W i ðzÞ;
HðzÞ ¼
n X
i¼1
Bi Hi ðzÞ and UðzÞ ¼
i¼1
n X
C i Ui ðzÞ
ð32Þ
i¼1
where the trial functions Wi(z), Hi(z) and Ui (z) will be generally chosen in such a way that they satisfy the respective boundary conditions, while Ai, Bi and Ci are constants. Substituting Eq. (32) into Eqs. (23)–(25), multiplying Eq. (23) by Wj(z), Eq. (24) by Hj(z) and Eq. (25) by Uj(z); performing the integration by parts with respect to z between z = 0 and z = 1 and using the boundary conditions (21), we obtain the following system of linear homogeneous algebraic equations:
C ji Ai þ Dji Bi þ Eji C i ¼ 0
ð33Þ
F ji Ai þ Gji Bi ¼ 0
ð34Þ
Hji Bi þ Iji C i ¼ 0
ð35Þ
The coefficients Cji Iji involve the inner products of the basis functions and are given by
! Da1 hDW j DW i i þ a2 ða2 K þ Da1 ÞhW j W i i C ji ¼ KhD W j D W i i þ 2a K þ n 2
2
2
Dji ¼ a2 Rð1 þ M 1 ÞhW j Hi i;
Eji ¼ a2 RM1 hW j DUi i;
F ji ¼ hHj W i i
2
Gji ¼ hDHj DHi i þ ga hHj Hi i; Hji ¼ hDUj Hi i a Iji ¼ ½Uj ð1ÞUi ð1Þ þ Uj ð0ÞUi ð0Þ þ hDUj DUi i þ a2 M 3 hUj Ui i 1þv R1 where the inner product is defined as h i ¼ 0 ð Þdz. The above set of homogeneous algebraic equations will have a non-trivial solution if and only if
C ji F ji 0
Dji Gji Hji
Eji 0 ¼0 Iji
ð36Þ
The eigenvalue has to be extracted from the above characteristic equation. For this, we select the trial functions as
W i ¼ ziþ3 2ziþ2 þ ziþ1 ;
Hi ¼ ziþ1 zi ;
Ui ¼ zi 1=2
ð37Þ
It may be noted that the magnetic potential Ui does not satisfy the boundary conditions (21b,c) but the boundary residuals technique is used for the function Ui (see Finlayson, 1972) and the first term in Iji represents this residual term. 3.3. Rigid–rigid ferromagnetic boundaries For this case, we consider the trial functions as
W i ¼ ziþ3 2ziþ2 þ ziþ1 ;
Hi ¼ ziþ1 zi ;
Ui ¼ ziþ2 3ziþ1 þ 2zi
ð38Þ
Here, we note that Wi, Hi and Ui satisfy all the corresponding boundary conditions (22). Hence, the boundary residuals for the function Ui become zero and in this case the coefficient Iji reduces to
Iji ¼ hDUj DUi i þ a2 M3 hUj Ui i
ð39Þ
4. Results and discussion The linear stability analysis has been carried out to investigate the onset of ferroconvection in an anisotropic horizontal porous layer heated from below in the presence of a uniform vertical magnetic field using a non-Darcian model. The lower and upper isothermal boundaries are considered to be rigid-paramagnetic/ferromagnetic or free with large magnetic susceptibility. The critical stability parameters computed numerically by the Galerkin method as explained above, are found to converge by considering six terms in the series expansion of the basis functions. The numerical solution procedure has been validated first by comparing the results with those of Stiles et al. (1992) for the non-porous case (Da1 = 0, K = 1, n = 1 and g = 1). For this a new magnetic parameter Q, independent of the temperature gradient, was introduced in the form 4 Q = Rm/R2, where Q ¼ l0 K 2 Ajv lf =½ð1 þ vÞq20 g 2 a2t d . The critical thermal Rayleigh number (Rc), critical magnetic Rayleigh number (Rmc) and the corresponding wave number (ac) computed numerically for different values of Q for the non-porous
503
C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508 Table 1 Comparison of critical stability parameters with those of Stiles et al. (1992) for different values of Q and v(non-porous case). Q
v
Stiles et al. (1992)
Present values
ac
Rc
ac
Rc
0
2 4 6
3.116 3.116 3.116
1707.8 1707.8 1707.8
0 0 0
3.116 3.116 3.116
1707.9 1707.9 1707.9
0 0 0
104
2 4 6
3.181 3.186 3.188
1567.7 1572.6 1575.1
245.8 247.3 248.1
3.181 3.186 3.188
1567.8 1572.7 1575.2
245.8 247.3 248.1
103
2 4 6
3.404 3.430 3.444
1056.0 1066.7 1072.1
1115.1 1137.9 1149.4
3.404 3.430 3.444
1056.0 1066.7 1072.1
1115.1 1137.9 1149.4
102
2 4 6 2 4 6
3.637 3.685 3.710 3.777 3.837 3.867
456.9 463.0 466.1 52.2 52.9 53.3
2087.6 2143.9 2172.4 2723.1 2801.1 2840.6
3.637 3.685 3.710 3.777 3.837 3.867
456.9 463.0 466.1 52.2 52.9 53.3
2087.6 2143.9 2172.4 2723.4 2798.4 2840.8
102
2 4 6
3.793 3.854 3.884
5.29 5.36 5.40
2795.8 2876.2 2916.9
3.794 3.854 3.884
5.29 5.37 5.40
2798.4 2883.6 2916.0
1
2 4 6
3.794 3.856 3.886
0 0 0
2804.0 2884.7 2925.5
3.794 3.856 3.886
0 0 0
2804.0 2884.7 2925.8
1
Rmc
Rmc
case are compared in Table 1 with those of Stiles et al. (1992). The comparison shows an excellent agreement on the numerical results. The results obtained for different types of boundary conditions covering a wide range of various parameters are illustrated in Figs. 1–4. The variation of Rc and the corresponding ac as a function of Da1 is shown in Figs. 1a and 1b, respectively for different values of K when M3 = 1, M1 = 5, n = 2 and g = 5. From Fig. 1a, it is seen that Rc increases monotonically with Da1 indicating the effect of decrease in the vertical permeability of the porous medium is to delay the onset of ferroconvection. Also, increase in the value of viscosity ratio K is to delay the onset of ferroconvection and this may be due to the increase in viscous diffusion. Further inspection of the figure reveals that the deviation in the Rc values for different boundaries becomes predominant as the value of K increases and the nature of boundaries plays an important role in deciding the stability of the system. The rigid-paramagnetic boundaries with higher magnetic susceptibility (v = 6) are more stable followed by rigidparamagnetic boundaries with low magnetic susceptibility (v = 0), then the rigid-ferromagnetic boundaries and the least stable is free boundaries. In other words, the rigid paramagnetic boundaries have stronger effect in stabilizing the thermally unstable ferrofluid saturated anisotropic porous layer than that of ferromagnetic boundaries because of their dampening 12000 R gi Ri g d-par a am amagnetic wit w th 1+ χ = 7 R gi Ri g d-par a am amagnetic wi w th t 1+ χ = 1 R gi Ri g d-fe ferrom omagnetic
10000
Λ=5
F ee-Fr Fr Free
8000 5
Rc
3
5
6000 3 5
3
4000 3
1
1 1
1
2000
0 10
20
30
40
-1
50 Da D 60
70
80
90
100
Fig. 1a. Variation of critical Rayleigh number Rc as a function of Da1 for different values of K when M3 = 1, M1 = 5, n = 2 and g = 5.
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C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508
3.2 Λ=5
3 3.0
Rigid-paramagnetic with 1+χ = 7
1
Rigid-paramagnetic with 1+χ = 1 R gid-fe Ri f rromagnetic
2.8
F ee-Fr Fr F ee
5
ac
3
1
2.6 1 2.4
5 3
2.2 1 2.0 10
20
30
-1
50 Da 60
40
1
Fig. 1b. Variation of critical wave number ac as a function of Da
70
80
90
100
for different values of K when M3 = 1, M1 = 5, n = 2 and g = 5.
15000 Rigid-paramagnetic with 1+ χ = 7 Rigid-paramagnetic with 1+ χ = 1 Rigid-ferromagnetic Free-Free -5
12000
M1 = 10
Rc
0.1
9000
1 1 1
6000 -5
10 0.1 3000 1
2
3
4
ξ
5
6
1 7
8
9
Fig. 2a. Variation of critical Rayleigh number Rc as a function of n for different values of M1 when g = 5, Da1 = 50, K = 2 and M3 = 1.
effect due to an increase in the magnetic induction. Also, it is evident that increasing magnetic susceptibility is to delay the onset of ferroconvection. From Fig. 1b it is evident that the critical wave numbers decrease with Da1 in the case of rigidparamagnetic/ferromagnetic boundaries but an opposite trend could be seen in the case of free boundaries. Irrespective of the magnetic boundary conditions, increasing K and v is to increase ac in the case of rigid boundaries and thus their effect is to narrow the convection cells. To the contrary, increase in K is to decrease ac and thus its effect is to widen the convection cells when the boundaries are free. The ac values are higher for large magnetic susceptibility rigid-paramagnetic boundaries compared to low susceptibility boundaries and the least for rigid-ferromagnetic boundaries. The important parameters which are affecting the stability characteristics of the system are the mechanical and thermal anisotropy parameters. The variations of critical Rayleigh number Rc and the corresponding critical wave number ac as a function of mechanical anisotropy parameter n(= kh/kv) for different boundaries are presented in Figs. 2a and 2b, respectively for different values of M1whenM3 = 1, g = 5,Da1 = 50 andK = 2. From Figs. 2a and 2b, it is evident that the critical Rayleigh number Rc and the corresponding wave number acincrease with decreasingn. This may be understood as follows: let us keep the vertical permeability kv fixed and decrease the horizontal permeability kh. Physically, this means that the conduction solution in the anisotropic porous medium becomes more stable and the critical wavelength decreases as the horizontal permeability decreases. Smaller horizontal permeability inhibits horizontal motion, and the conduction solution is thus
C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508
505
2.7
M1 = 1
2.4
1
ac
1
0.1
2.1
-5
10
Rigid-paramagnetic with 1+ χ = 7 Rigid-paramagnetic with 1+ χ = 1 Rigid-ferromagnetic Free-Free
1 1.8
0.1 -5
1.5
10 1
2
3
ξ
4
5
6
7
8
9
Fig. 2b. Variation of critical wave number ac as a function of n for different values of M1 when g = 5, Da1 = 50, K = 2 and M3 = 1.
20000 Rigid-paramagnetic with 1+ χ = 7 Rigid-paramagnetic with 1+ χ = 1 Rigid-ferromagnetic Free-Free
-5
M1 = 10
15000 0.1
Rc
1 1
10000
1 -5
10 0.1
1
5000
0 1
2
3
4
η
5
6
7
8
9
Fig. 3a. Variation of critical Rayleigh number Rc as a function of g for different values of M1 when n = 2, Da1 = 50, K = 2 and M3 = 1.
stabilized. The larger resistance to horizontal flow also leads to a shortening of the horizontal wavelength (or increase in the horizontal wave number) at the onset. Further, increasing M1is to decrease the values of Rc and to hasten the onset of ferroconvection due to an increase in the magnetic force, while opposite is the trend with respect to the critical wave number. We note that increasingM1is to reduce the size of convection cells. Figs. 3a and 3b represent the variation of Rcand ac as a function of thermal anisotropy parameter g (= jh/jv) on the stability characteristics of the system for different boundaries and also for different values of M1. The results exhibited in the figures are forM3 = 1, n = 2, Da1 = 50 and K = 2. In contrast to the effect of mechanical anisotropy parameter, it is seen that decrease in the value of g is to decrease the value of Rc, whereas the critical wave number increases with decreasing g. That is, decreasing g is to hasten the onset of ferroconvection and also to diminish the size of convection cells. This may be attributed to the fact that, as g decreases, a heated fluid parcel loses less heat in the horizontal directions, and hence retains its buoyancy better. Hence, the basic state becomes less stable which in turn augments the onset of ferroconvection, and also the wavelength is reduced. The complementary effects of buoyancy and the magnetic forces are made clear in Fig. 4 by displaying the locus of the critical Rayleigh number Rc and the critical magnetic Rayleigh Rmc for M3 = 1,Da1 = 100, n = 2, K = 2 and g = 2. We note that
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C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508
3.3 Rigid-paramagnetic with 1+ χ = 7 Rigid-paramagnetic with 1+ χ = 1 Rigid-ferromagnetic Free-Free
3.0
2.7
ac
M1 = 1
2.4
1 0.1
2.1
1
-5
10
1 1.8
0.1 -5
10 1.5 1
2
3
4
η
5
6
7
8
9
Fig. 3b. Variation of critical wave number ac as a function of g for different values of M1 when n = 2, Da1 = 50, K = 2 and M3 = 1.
10000 Rigid-paramagnetic with 1+ χ = 7 Rigid-paramagnetic with 1+ χ = 1
8000
Rigid-ferromagnetic
6000
Rc 4000
2000
0 0
5000
Rmc
10000
15000
20000
Fig. 4. Locus of Rc and Rmc for M3 = 1, g = 2, Da1 = 100, K = 2 and n = 2.
Rc is inversely proportional to Rmc. In the absence of buoyancy forces (Rc = 0), the instability sets in at higher values of Rmc indicating the system is more stable when the magnetic forces alone are present.
5. Conclusions The linear stability theory is used to investigate the onset of Brinkman ferroconvection in an anisotropic porous layer heated from below in the presence of a uniform magnetic field. The resulting eigenvalue problem is solved numerically by employing the Galerkin method when the boundaries are rigid and solved exactly when the boundaries are stress-free. From the foregoing study, the following conclusions may be drawn: (i) The system is more stabilizing against the ferroconvection if the boundaries are rigid-paramagnetic with high magnetic susceptibility and least stable if the boundaries are free. It is observed that
C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508
507
ðRc and ac Þrigid-parmagnetic > ðRc and ac Þrigid-ferromagnetic > ðRc and ac Þfree—free (ii) The effect of increasing the value of mechanical anisotropy parameter n is to hasten, while increase in the value of thermal anisotropy parameter g is to delay the onset of ferroconvection in porous media. Thus, it is possible to control (suppress or augment) ferroconvection by adjusting the mechanical and thermal anisotropy parameters. (iii) The effect of increasing the value of magnetic number M1 is to hasten the onset of ferroconvection, while increase in the value of Da1, K and v is to delay the onset of ferroconvection. (iv) The effect of increasing g, v, Da1 and M1 as well as decrease in n is to increase the critical wave number and hence their effect is to narrow the convection cells. (v) The magnetic and buoyancy forces are complementary with each other and the system is more stabilizing when the magnetic forces alone are present.
Acknowledgements One of the authors (ISS) wishes to thank the BK 21 Program of the School of Mechanical Engineering, Yonsei University, Seoul, Korea for inviting him as a visiting Professor and also the Bangalore University for sanctioning sabbatical leave. The authors (C.E.N.) and (M.R.) wish to thank the Principals of their respective colleges for their encouragement. We thank the reviewer for useful suggestions. Appendix A Here, we analyze whether ferroconvection in an anisotropic porous layer sets in as oscillatory instability or not. Following Shivakumara et al. (2008), first we eliminate U from Eq. (17) by operating (D2 a2M3) on that equation and use Eq. (19) to obtain
1 2 D a2 r W ¼ a2 R½M 1 D2 ð1 þ M 1 ÞðD2 a2 M 3 ÞH ðD2 a2 M3 Þ KðD2 a2 Þ2 Da1 n
ðA1Þ
Multiplying Eq. (A1) by W⁄ (complex conjugate of W) and integrating from z = 0 to z = 1, we get
1 2 D a2 r W ¼ a2 RhW D2 Hi a4 Rð1 þ M1 ÞM 3 hW Hi W ðD2 a2 M 3 Þ KðD2 a2 Þ2 Da1 n
ðA2Þ
R1
where hFi ¼ 0 Fdz. The left hand side of Eq. (A2), on repeatedly applying integration by parts and simplifying using the boundary conditions on the velocity can be written as 2
3
K½D W D W DW D
4
! Da1 hKjD Wj þ Ka ð2 þ M 3 Þ þ þ r jD2 Wj2 n ! ! Da1 þ r ð1 þ M 3 Þ jDWj2 þ a4 M3 ða2 K þ Da1 þ rÞjWj2 i n
W10
þ a4 Kð1 þ 2M 3 Þ þ a2
2
3
2
ðA3Þ
where jFj2 = FF⁄ and F⁄ is the conjugate of F. From Eq. (18), after neglecting the terms involving M2, we have
W þ ððD2 ga2 Prr ÞH ¼ 0
ðA4Þ
⁄
Substituting for W from Eq. (A4) on the right hand side of Eq. (A2), integrating by parts and using the boundary conditions on H we get
hjD2 Hj2 þ ðga2 þ Prr ÞjDHj2 i a2 M 3 ð1 þ M 1 ÞhjDHj2 þ ðga2 þ Pr r ÞjHj2 i
ðA5Þ
Using Eqs. (A3) and (A5) in Eq. (A2), we obtain
* 2
3
K½D W D W DW D 2
þa
4
W10
! Da1 KjD Wj þ Ka ð2 þ M3 Þ þ þ r jD2 Wj2 n ! ! 3
2
2
Da1 Ka ð1 þ 2M 3 Þ þ þ r ð1 þ M 3 Þ jDWj2 þ a4 M 3 ðKa2 þ Da1 þ r ÞjWj2 n
+
2
¼ hjD2 Hj2 i ða2 M 3 ð1 þ M 1 Þ þ ða2 þ Prr ÞÞhjDHj2 i a2 M3 ð1 þ M 1 Þða2 þ Prr Þ < jHj2 Equating the real and imaginary parts of the above equation separately, we get respectively
ðA6Þ
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C.E. Nanjundappa et al. / International Journal of Engineering Science 49 (2011) 497–508
*
K½D2 W D3 W DW D4 W > 10 KjD3 Wj2 þ a2 ð2 þ M3 Þ þ þa
2
! Da1 þ r jD2 Wj2 n +
! Da1 Ka ð1 þ 2M 3 Þ þ þ r ð1 þ M 3 Þ jDWj2 þ a4 M 3 ðKa2 þ Da1 ÞjWj2 n 2
¼ ½hjD2 Hj2 i þ ða2 M3 ð1 þ M1 Þ þ ðga2 þ PrÞÞhjDHj2 i þ a2 M3 ð1 þ M1 Þðga2 þ PrÞhjHj2 i
ðA7Þ
and
KhjD2 Wj2 þ a2 ð1 þ M 3 ÞjDWj2 þ a4 M 3 jWj2 i ¼ PrhjDHj2 i Pra2 M3 ð1 þ M 1 ÞhjHj2 i
ðA8Þ
Only the imaginary part gives the information about the possibility of occurring oscillatory instability. From Eq. (A9) it is evident that oscillatory instability occurs provided
KhjD2 Wj2 þ a2 ð1 þ M3 ÞjDWj2 þ a4 M3 jWj2 i þ PrhjDHj2 þ a2 M 3 ð1 þ M1 ÞjHj2 i ¼ 0
ðA9Þ
Since the left hand side of Eq. (A9) is a positive definite, the above condition will never be satisfied. Therefore oscillatory instability is not possible for ferroconvection in an anisotropic porous medium under the influence of vertical magnetic field. This result is true irrespective of the type of velocity (rigid or free) and magnetic (paramagnetic or ferromagnetic) boundary conditions. References Auernhammer, G. K., & Brand, H. R. (2000). Thermal convection in a rotating layer of a magnetic fluid. The European Physical Journal B, 16, 157–168. Bashtovoy, V. G., & Berkovsky, B. N. (1996). Magnetic fluids and applications hand book. New York: Begell house. Borglin, S. E., Mordis, G. J., & Oldenburg, C. M. (2000). Experimental studies of the flow of ferrofluid in porous media. Transport in Porous Media, 41, 61–80. Finlayson, B. A. (1970). Convective instability of ferromagnetic fluids. Journal of Fluid Mechanics, 40(4), 753–767. Finlayson, B. A. (1972). Method of weighted residuals and variational principles. Academic Press. Givler, R. A., & Altobelli, S. A. (1994). A determination of the effective viscosity for the Brinkman–Forcheimer flow model. Journal of Fluid Mechanics, 258, 355–370. Gotoh, K., & Yamada, M. (1982). Thermal convection in a horizontal layer of magnetic fluids. Journal of the Physical Society of Japan, 51, 3042–3048. Kaloni, P. N., & Lou, J. X. (2004). Convective instability of magnetic fluids under alternating magnetic fields. Physical Review E, 71, 066311-1–066311-12. Lalas, D. P., & Carmi, S. (1971). Thermoconvective stability of ferrofluids. Physics of Fluids, 14(2), 436–437. Nanjundappa, C. E., & Shivakumara, I. S. (2008). Effect of velocity and temperature boundary conditions on convective instability in a ferrofluid layer. ASME Journal of Heat Transfer, 130, 104502-1–104502-5. Nanjundappa, C. E., Shivakumara, I. S., & Ravisha, M. (2010). The onset of buoyancy-driven convection in a ferromagnetic fluid saturated porous medium. Meccanica, 45, 213–226. Nield, D. A., & Bejan, A. (2006). Convection in porous media (3rd ed.). Springer. Odenbach, S. (2004). Recent progress in magnetic fluid research. Journal of Physics Condensed Matter, 16, R1135-50. Qin, Y., & Chadam, J. (1995). A non-linear stability problem for ferromagnetic fluids in a porous medium. Applied Mathematics Letters, 8(2), 25–29. Ramathan, A., & Suresh, G. (2004). Effects of magnetic field dependent viscosity and anisotropy of porous medium on ferroconvection. International Journal of Engineering Science, 42, 411–425. Rerkovsky, B. M., Medvedev, V. F., & Krakov, M. S. (1993). Magnetic fluids, engineering applications. New York: Oxford University Press. Rosensweig, R. E. (1985). Ferrohydrodynamics. Cambridge: Cambridge University Press. Rosensweig, R. E., Zahn, M., & Volger, T. (1978). Stabilization of fluid penetration through a porous medium using magnetizable fluids. In B. Berkovsky (Ed.), Thermomechanics of magnetic fluids (pp. 195–211). Washington DC: Hemisphere. Schwab, L., Hilderbrandt, U., & Stierstadt, K. (1987). Magnetic Benard – Convection. Journal of Magnetism and Magnetic Materials, 65, 315–319. Sekar, R., Vaidynathan, G., & Ramathan, A. (1996). Ferrocnvection in an Anisotrophic porous medium. International Journal of Engineering Science, 34(4), 399–405. Shivakumara, I. S., Nanjundappa, C. E., & Ravisha, M. (2008). Thermomagnetic convection in a magnetic nanofluid saturated porous medium. International Journal of Applied Mathematics and Engineering Science, 2(2), 157–170. Shivakumara, I. S., Nanjundappa, C. E., & Ravisha, M. (2009). Effect of boundary conditions on the onset of thermomagnetic convection in a ferrofluid saturated porous medium. ASME Journal of Heat Transfer, 131, 101003-1–101003-9. Shivakumara, I. S., Rudraiah, N., & Nanjundappa, C. E. (2002). Effect of non-uniform basic temperature gradient on Rayleigh–Benard–Marangoni convection in ferrofluids. Journal of Magnetism and Magnetic Materials, 248, 379–395. Shliomis, M. I. (1974). Magnetic fluids. Soviet Physics, Uspekhi (Engl. Trans), 17(2), 153–169. Stiles, P. J., & Kagan, M. (1990). Thermoconvective instability of a ferrofluid in a strong magnetic field. Journal of Colloid Interface Science, 134, 435–448. Stiles, P. J., Lin, F., & Blennerhassett, P. J. (1992). Heat transfer through weakly magnetized ferrofluids. Journal of Colloid Interface Science, 151, 95–101. Vaidyanathan, G., Sekar, R., & Balasubramanian, R. (1991). Ferroconvective instability of fluids saturating a porous medium. International Journal of Engineering Science, 29, 1259–1267. Volker, T., Blums, E., & Odenbach, S. (2007). Heat and mass transfer phenomena in magnetic fluids. GAMM-Mitteilungen, 30(1), 185–194. Zhan, M., & Rosensweig, R. E. (1980). Stability of magnetic fluid penetration through a porous medium with uniform magnetic field oblique to the interface. IEEE Transactions of Magnetics, 16, 275–282.