- Email: [email protected]
A nonlinear stability analysis for thermoconvective magnetized ferrofluid with magnetic field dependent viscosity
International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
A nonlinear stability analysis for thermoconvective magnetized ferrofluid with magnetic field dependent viscosity☆ Sunil ⁎, Poonam Sharma, Amit Mahajan Department of Applied Sciences, National Institute of Technology, Hamirpur (H.P.)−177 005, India
a r t i c l e
i n f o
Available online 7 September 2008 Keywords: Magnetized ferrofluid Nonlinear stability Magnetization MFD viscosity
a b s t r a c t The generalized energy method which gives sufficient condition for the stability is developed for convection problem in a magnetized ferrofluid with magnetic field dependent (MFD) viscosity heated from below. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. Both linear and nonlinear analyses are carried out and comparison of results shows a marked difference in the stability boundaries and thus indicates that the sub-critical instabilities are possible. The effect of various parameters on the sub-critical region has also been analyzed. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The approach adopted in the present paper is by the application of energy method, pioneered and developed in its modern use way by Straughan [1,2]. Straughan [2] emphasizes that energy theory is certainly much stronger when the stability obtained is unconditional or for at least finite (non-vanishing) initial data. Straughan [3] developed a sharp nonlinear energy stability analysis for the Darcy's equations of thermal convection in a fluid saturated porous medium and the results obtained are the best possible showing sub-critical instabilities are not possible. In recent years, nonlinear stability analysis of non-magnetic fluids by using generalized energy stability theory has been considered by many authors [4–16]. Magnetic fluids or ferrofluids are colloidal suspensions of fine ferromagnetic mono domain nanoparticles in non-conducting liquids. One of the major applications of ferrofluid is its use in medical field, like the transport of drugs to an injured site, removal of tumors from the body. Other applications include its use as coolant and in devices like rotating shaft seals, loudspeakers, printers, sealed motors, acoustic devices etc. Rosensweig [17] in his monograph has given an authoritative introduction to the research on magnetic liquids and various authors [18–27] have studied convection problems of ferrofluid. The convection in ferrofluid is gaining much importance due to its astounding physical properties. One such property is viscosity of the ferrofluid. The effect of magnetic field dependent (MFD) viscosity on convection in ferrofluid by linearized theory has been studied by many authors [28–32]. Recently, Sunil and Mahajan [33] studied the non-linear stability analysis for mag☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (Sunil). 0735-1933/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2008.08.004
netized ferrofluid heated from below. They assume that the viscosity is isotropic and independent of magnetic field. The effect of MFD viscosity on nonlinear stability analysis of magnetized ferrofluid cannot be ignored. In the present analysis, an attempt is made to see the effect of MFD viscosity on magnetized ferrofluid heated from below. We analyzed both the nonlinear stability analysis via generalized energy method and linear instability results. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. Here, the energy method gives critical magnetic thermal Rayleigh numbers just slightly below the ones given by linear theory. This shows the possibility of the existence of sub-critical instability. On the other hand, in case of non-ferrofluid the nonlinear and linear stability results coincide. This in turn implies exclusion of occurrence of sub-critical instability. Comparison of the results obtained, respectively, by the energy method and the linear stability analysis has been discussed finally. This problem, to the best of our knowledge, has not been investigated yet. 2. Perturbation equations and boundary conditions Here, we consider an infinite horizontal layer of thickness ‘d’ of an electrically non-conducting incompressible thin ferrofluid having a variable viscosity, given by µ1 = µ(1 + δ · B) heated from below. Here, µ is taken as viscosity of the fluid when the applied magnetic field is absent and µ1 is magnetic field dependent (MFD) viscosity. The variation of magnetic field dependent viscosity δ has been taken to be isotropic, δ = δ1 = δ2 = δ3. The fluid is assumed to occupy the layer za − 2d ; 2d with gravity acting in the negative z-direction and magnetic ˆ acts outside the layer. field intensity, H ¼ H0ext k, To describe nonlinear energy stability results, we begin with the relevant equations and basic state as given in (Sunil and Mahajan [33]). In order to analyze the stability of the system, let the basic state
1282
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
Nomenclature Latin symbols b Subscript; basic state B Magnetic induction (T) B Magnitude of B (T) C0 Specific heat at constant pressure (J/kg K) d Thickness of the ferrofluid layer (m) g Acceleration due to gravity (m/s2) H Magnetic field intensity (Amp/m) Hext External magnetic field intensity (Amp/m) 0 H′ The perturbation in magnetic field intensity (Amp/m) H Magnitude of H (Amp/m) H0 Reference imagnetic field intensity (Amp/m) h K1 ¼ − AM The pyromagnetic coefficient (Amp/mK) AT H ;T ˆ k Unit vector in the z-direction M Magnetization (Amp/m) M′ The perturbation in the magnetization (Amp/m) M Magnitude of M (Amp/m) M0 Magnetization when magnetic field is H0 and temperature Ta (Amp/m) p The fluid pressure (psi) p′ The perturbation in fluid pressure (psi) q Velocity of the ferrofluid (m/s) q′=(u, v, w)The perturbation in velocity (m/s) t Time (s) T Temperature (K) TL Constant average temperature at the bottom surface z = −d/2 (K) TU Constant average temperature at the upper surface z = +d/2 (K) Ta Average temperature (K) 0
be perturbed. Let q′ = (u, v, w), ρ′, p′, θ, H′ = (H1′, H2′, H3′) and M′ = (M1′, M2′, M3′) represent the perturbations in velocity, density, pressure, temperature, magnetic field intensity and magnetization, respectively. Then the nonlinear perturbation equations become ρ0
μ 0 K1 β Aq′ ¼ −∇p′þ μ 1 þ δμ 0 ðH0 þ M0 Þ ∇2 q′þ ρ0 αgθ kˆ − 1þχ At h i M0 2 ˆ ˆ ð1 þ χÞ′zk−K1 θk þ μδμ 0 1 þ ′x ∇ q′þ μδμ 0 H0 M0 1þ ′y ∇2 q′þ μδμ 0 ð1 þ χÞ′z ∇2 q′− μδμ 0 K1 θ∇2 q′ H0 þμ 0
M0 M0 ′x ∇′x þ μ 0 ′y ∇′y þ μ 0 χ′z ∇′z H0 H0 ð1Þ
−μ 0 K1 θ∇′z −ρ0 q′ ∇q′;
a
∇ q ′¼ 0;
ð2Þ
Aθ þ q ′ ∇θ ¼ κ∇2 θ þ βw; At M0 M0 − χ ′zz ¼ K1 θz : ∇2 ′− 1þ H0 H0
ð3Þ ð4Þ
Here, H′¼ ∇′ ½6∇ H ¼ 0; where ϕ′ is the perturbed magnetic potential. The boundary conditions are w ¼ 0; uz ¼ 0; vz ¼ 0; θ ¼ 0; ′z ¼ 0
d at z ¼ F ; 2
ð5Þ
with q′; θ; ′satisfying a plane tiling periodic boundary condition in x and y. 3. Nonlinear stability analysis
Greek letters α Coefficient of thermal expansion (K− 1) β A uniform temperature gradient β (= |dT/dz|) (K/m) δ Coefficient of MFD viscosity (T− 1) κ Thermal diffusivity (m2/s) µ1 Dynamic MFD viscosity of fluid (kg/m s) µ0 Magnetic permeability of free space (H/m) µ Reference viscosity of fluid (kg/m s) ρ Fluid density (kg/m3) ρ0 Density at the ambient temperature (kg/m3) ρ′ The perturbation in density ρ(kg/m3) θ The perturbation in temperature T(K) χ The magnetic susceptibility ▿ Gradient operator (m− 1) ϕ′ The perturbed magnetic potential (Amp)
Non-dimensional parameters M1 Buoyancy magnetization parameter M3 Magnetic parameter (measures the departure of linearity in the magnetic equation of state) Mδ Ratio of viscous to gravitational forces Ncℓ Critical linear magnetic thermal Rayleigh number Nce Critical energy magnetic thermal Rayleigh number Rℓ Linear thermal Rayleigh number Re Energy thermal Rayleigh number x Dimensionless wave number
To investigate the nonlinear stability analysis, the perturbed equations in non-dimensional form (dropping ⁎) can be written as Aq þ q ∇q ¼ −∇p þ ð1 þ δM3 Þ∇2 q þ R1=2 ð1 þ M1 Þθ kˆ At
ð6Þ
−R1=2 M1 z kˆ þ δM3 Mδ x ∇2 q þδM3 Mδ y ∇2 q þ δMδ z ∇2 q−δMδ θ∇2 q −M1 θ∇z þ M1 M3 −
1 ∇ 1þχ x x 1 χ y ∇y þ M1 ∇ : þM1 M3 − 1þχ 1þχ z z On multiplying Eq. (6) by q and integrating over V, we find (after using the boundary conditions and the divergence theorem) 1 djjqjj2 ¼ −jj∇qjj2 ð1 þ δM3 Þ þ R1=2 ð1 þ M1 Þhwθi−R1=2 M1 hwz i 2 dt þ M1 hq ∇θz i þ δM3 Mδ hx q ∇2 qi
ð7Þ
þδM3 Mδ hy q ∇2 qi þ δMδ hz q ∇2 qi−δMδ hθq ∇2 qi χ M1 hz q ∇z i þ 1þχ 1 1 hx q ∇x i þ M1 M3 − hy q ∇y i: þM1 M3 − 1þχ 1þχ Other relevant equations are same as mentioned in (Sunil and Mahajan [33]). Here, 〈·〉 denotes the integration over V and ||.|| denotes the L2 (V) norm, where V denotes a typical periodicity cell.
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
The non-dimensional quantities and non-dimensional parameters used in the above equations are same as in the work of (Sunil and Mahajan [33]). However, one additional quantity and one additional parameter is involved due to effect of MFD viscosity and are given as δ⁎ = µ0δH0(1 + χ) and Mδ ¼ ð1þχK1ÞRβd 1=2 H ; respectively. 0 Here, Mδ is a ratio of viscous to gravitational forces. Further, the process has been carried out by using the techniques of Straughan [3] and Sunil and Mahajan [33]. To study the nonlinear stability of basic state, an L2 energy, E(t), is constructed and the evolution of E(t) is given by dE ¼ I0 −D0 þ N0 ; dt
where I1 ¼ R1=2 ð2 þ M1 Þh∇w ∇θi−R1=2 M1 h∇w ∇z i−R1=2 h∇w ∇3 θi;
ð18Þ
D1 ¼ ð1 þ δM3 Þjj∇2 qjj2 þ jj∇2 θjj2 þ jj∇3 θjj2 ;
ð19Þ
N1 ¼ hq ∇θ∇2 θi þ hq ∇q ∇2 qi þ h∇q ∇θ ∇3 θi þhq ∇3 θ∇2 θi−δM3 Mδ hx ∇2 q ∇2 qi
ð8Þ
−δM3 Mδ hy ∇2 q ∇2 qi−δMδ hz ∇2 q ∇2 qi χ hz ∇2 q ∇z i þδMδ hθ∇2 q ∇2 qi−M1 1þχ
where
E¼
1 λ1 jjθjj2 þ jjqjj2 ; 2 2
ð9Þ
I0 ¼ R1=2 f1 þ λ1 ð1 þ M1 Þghwθi−R1=2 λ1 M1 hwz i−λ2 hθz i
−M1 M3 −
ð10Þ
1 1 hx ∇2 q ∇x i−M1 M3 − hy ∇2 q ∇y i 1þχ 1þχ
þM1 hθ∇2 q ∇z i: 2
2
2
1283
ð20Þ
2
D0 ¼ jj∇θjj þ λ1 ð1 þ δM3 Þjj∇qjj þ λ2 M3 jj∇jj −λ2 ðM3 −1Þjjz jj ; ð11Þ
N0 ¼ λ1 M1 hq ∇θz i þ λ1 δM3 Mδ hx q ∇2 qi þ λ1 δM3 Mδ hy q ∇2 qi þ λ1 δMδ hz q ∇2 qi 1 hx q ∇x i −λ1 δMδ hθq ∇2 qi þ λ1 M1 M3 − 1þχ 1 hy q ∇y i þ λ1 M1 M3 − 1þχ χ hz q ∇z i; þλ1 M1 ð12Þ 1þχ
H
I0 ; D0
ð13Þ
ð14Þ
ð15Þ
Here b0 is a positive coupling parameter to be chosen and the complementary energy E1(t) is given by E1 ðt Þ ¼
1 1 1 jj∇θjj2 þ jj∇qjj2 þ jj∇2 θjj2 : 2 2 2
ð16Þ
The evolution of Vg(t) is given by dVg ≤ −a0 D0 þ N0 þ b0 I1 −b0 D1 þ b0 N1 ; dt
with À given by ! ! # 1=2 " 2 λ1 λ1 2χ−1 A` ¼ 2C 4 4 þ 2δMδ ðM3 þ 1Þ 1=2 þ 1 þ M1 1 þ 1=2 ; 2M3 þ b0 1þχ b0 a0
b0 ¼
a0 π2 n o; 2 3R π þ ð2 þ M1 Þ2 þM12
and defined
with a0 = 1 − m(N0). In order to dominate the nonlinear terms and for studying the (conditional) nonlinear stability, we now introduce the generalized energy functional as Vg ðt Þ ¼ Eðt Þ þ b0 E1 ðt Þ:
ð21Þ
where we have chosen
where H is the space of admissible solutions. Then we require m b 1 so that dE ≤ −a0 D0 þ N0 ; dt
: ` Vg1=2 ; V g ðt Þ ≤ −D2 1−A
ð22Þ
with λ1, λ2 two positive coupling parameters. We now define m ¼ max
Using inequalities and embedding theorem (Sunil and Mahajan [33]), the Cauchy–Schwartz and the Young inequalities (Hardy et al. [34]), we have
D2 ¼
a0 b0 D0 þ D1 : 2 2
ð23Þ
Estimate (21) enables us to prove the following theorem of conditional nonlinear stability criterion. Theorem. Let 0 b m b 1;
ð24Þ
−2
ð25Þ
Vg ð0ÞbA` ;
with À given by Eq. (22). Then, there exists a positive constant K⁎, such that 4 ` 1=2 Vg ðt Þ ≤ Vg ð0Þe−K ð1−A Vg ð0ÞÞt ;
tz0:
Proof. The hypothesis and inequality (21) ensure that ð17Þ
: V g ð0Þb0:
ð26Þ
1284
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
Therefore, from inequality (21) by a recursive argument, we obtain : ` Vg1=2 ð0Þ ; 8 tz0: ð27Þ V g ðt Þ ≤ −D2 1−A
−D2 ≤−K 4Vg :
ð28Þ
By Eq. (15), using Eqs. (11), (19), (23) and by virtue of the Poincarétype inequalities, we have
2 D0 b0 b0 2 m 1 þ 2 D2 : þ D1 þ 2 D1 ≤ 2 1 þ 2 1−m 2 2π π 2 π π
λ1 Y0; λ ′2 Yð1 þ xÞ and
m ; 1−m
ð29Þ
Re ¼
2
: ` Vg1=2 ð0Þ V g ≤−K 4Vg 1−A
8
ð30Þ
tz0:
Integrating this last inequality, we deduce the theorem. Since Vg (t) in Eq. (15) does not contain the term jj∇jj2 , the kinetic energy term for magnetic potential, it is worthwhile checking as to what happens to jj∇jj2 as t → ∞. Using inequality (3.16)1 of [33], we have jj∇jj2 ≤ jjθjj2 :
ð31Þ
Thus, Eq. (15) and inequality (31) ensure the decay of jj∇jj2 , i.e. jjH′jj2 . 3.1. Variational problem We now solve the variational problem associated with the maximum problem (13). The corresponding Euler–Lagrange equations pffiffiffiffiffiffi for the problem (13) after taking transformations qˆ ¼ λ1 q and ˆ ¼ pffiffiffiffiffiffi λ2 (dropping caps) are 2ð1 þ δM3 Þ∇2 q þ R1=2 f1 þ λ1 ð1 þ M1 Þg
1 1=2 λ1
ˆ θ k−
1=2
R1=2 M1 λ1 1=2 λ2
ˆ −2∇p ¼ 0; z k
ð32Þ 2∇ θ þ R
1=2
f1 þ λ1 ð1 þ M1 Þg
1
wþ 1=2 λ1
1=2 λ2 z
¼ 0;
ð33Þ
wz −λ2 θz ¼ 0;
ð34Þ
1=2
2M3 ∇2 −2ðM3 −1Þzz þ
λ1 Y0: λ ′2
ð38Þ
R1=2 M1 λ1 1=2 λ2
1=2
2ð1 þ δM3 Þ∇4 w þ R1=2 f1 þ λ1 ð1 þ M1 Þg
1
∇21 θ− 1=2 λ1
1=2
λ2
ð36Þ 2
x ¼ πa2 ;
λ′2 ¼ λπ22 .
ð39Þ
ð1 þ δM3 Þð1 þ xÞ3 ð3 þ 4xM3 Þ : xð2 þ 4xM3 Þ
ð40Þ
P5 x5 þ P4 x4 þ P3 x3 þ P2 x2 þ P1 x þ P0 ¼ 0:
ð41Þ
The coefficients P0,……,P5, being quite lengthy, has not been written here and are evaluated during numerical calculations. The values of critical wave number in nonlinear stability results are determined e numerically using Newton–Raphson method by the condition dN ¼ 0. dx With x determined as a solution of Eq. (41), Eq. (40) will give the required critical magnetic thermal Rayleigh number Nce. In the absence of MFD viscosity (δ = 0), (40) reduces to Ne ¼ ð1þxÞ3 ð3þ4xM3 Þ xð2þ4xM3 Þ , which is in good agreement with previous published work (Sunil and Mahajan [33]). For analyzing the linear instability results, we return to the perturbed Eqs. (1)–(4) neglecting the non-linear terms. We again
Table 1 The variation of the critical magnetic thermal Rayleigh numbers (Ncℓ and Nce) with magnetic parameter (M3) δ
M3
xcℓ
Ncℓ
xce
Nce
Ncℓ − Nce
0
1 5 10 15 20 25 1 5 10 15 20 25 1 5 10 15 20 25 1 5 10 15 20 25
1 0.690 0.613 0.581 0.564 0.552 1 0.690 0.613 0.581 0.564 0.552 1 0.690 0.613 0.581 0.564 0.552 1 0.690 0.613 0.581 0.564 0.552
16 9.02 7.96 7.58 7.38 7.26 16.16 9.47 8.76 8.72 8.86 9.08 16.48 10.38 10.35 10.99 11.82 12.71 16.8 11.28 11.94 13.27 14.77 16.34
0.580 0.547 0.529 0.521 0.516 0.513 0.580 0.547 0.529 0.521 0.516 0.513 0.580 0.547 0.529 0.521 0.516 0.513 0.580 0.547 0.529 0.521 0.516 0.513
8.37 7.29 7.05 6.96 6.91 6.88 8.45 7.66 7.75 8.00 8.29 8.60 8.63 8.39 9.16 10.09 11.05 12.04 8.79 9.11 10.57 12.17 13.82 15.48
7.63 1.73 0.91 0.62 0.45 0.38 7.71 1.81 1.01 0.72 0.57 0.48 7.85 1.99 1.19 0.9 0.77 0.67 8.01 2.17 1.37 1.10 0.95 0.86
ð35Þ
ð1 þ δM3 Þð1 þ xÞ2 f4ð1 þ xÞð1 þ xM3 Þ−λ′2 g i; Re ¼ h x λ11 ð1 þ xM3 Þf1 þ λ1 ð1 þ M1 Þg2 þ M12 ð1 þ xÞ λλ1′2 −M1 f1 þ λ1 ð1 þ M1 Þg
:
As a function of x, Ne given by Eq. (40) attains its minimum when
∇21 z ¼ 0:
The Eqs. (35), (33) and (34) are solved using same procedure as in Sunil and Mahajan [33]. With the plane tiling form and the boundary conditions as described in the above referenced paper, we obtained the Rayleigh number as
xf4ð1 þ M1 Þð1 þ xM3 Þ−2M1 g
Ne ¼ R e M 1 ¼
0.01
1=2
R1=2 M1 λ1
n o M1 ð1 þ δM3 Þð1 þ xÞ3 4ð1 þ xM3 Þ− 1þM 1
For M1 sufficiently large, we obtain the magnetic thermal Rayleigh number
where p is a Lagrange's multiplier introduced, since q is solenoidal. On taking curlcurl of Eq. (32) and taking third component of the resulting equation, we find
where Re ¼ πR4 ;
ð37Þ
Using Eqs. (37) and (38) in Eq. (36), we have
and assuming K 4 ¼ 2kπ0 þ3, from Eqs. (27) and (28), we have
2
ð1 þ xÞM1 1 ; λ′2 ¼ : 1 þ M1 1 þ M1
The classical results in respect of Newtonian fluids (non-magnetic fluid) can be obtained when M1 = 0 which imply λ1 = 1, λ2′ = 0. Thus, λ1 = 1 is the optimal value, i.e. that value which maximizes Re (Straughan [1], pp 59–62). Since, we are dealing with magnetized ferrofluid, so for sufficiently large values of buoyancy magnetization (M1), we have
Let k0 N 0, such that k0 z
The values of λ1 and λ2′ are determined by the conditions dRe ¼ 0; dλ ¼ 0, respectively, and are found to be ′2
λ1 ¼
Now, we prove that there exists K⁎ N0, such that
Vg ≤
dRe dλ1
0.03
0.05
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
1285
Fig. 1. The variation of critical magnetic thermal Rayleigh numbers (Ncℓ and Nce) with magnetic parameter (M3).
perform the standard, stationary, mode analysis and look for the solution of these equations with the same boundary conditions. Following the procedure as stated earlier in the energy stability case, we have Rℓ ¼
ð1 þ δM3 Þð1 þ xÞ3 ð1 þ xM3 Þ : xð1 þ xM3 þ xM1 M3 Þ
ð42Þ
For M 1 sufficiently large, the critical magnetic thermal Rayleigh number, in linear case, is Nℓ ¼
ð1 þ δM3 Þð1 þ xÞ3 ð1 þ xM3 Þ : x2 M3
ð43Þ
In the3 absence of MFD viscosity (δ = 0), Eq. (43) reduces to 1þxM3 Þ Nℓ ¼ ð1þxÞx2ðM . 3
This is exactly Eq. (21) of Finlayson [18]. There are instances in which the two theories coincide. This is true for the classical Bénard problem. In the absence of magnetic parameters (M1 = 0 and M3 = 0), we obtain Rℓ ¼
ð1 þ xÞ3 ¼ Re ; x
i.e., the linear instability boundary ≡ the nonlinear stability boundary. Here, the energy method leads to the strong result that arbitrary sub-critical instabilities are not possible which is in good agreement with the previous published work (Joseph [35–36]). Thus, for lower values of magnetic parameters, this coincidence is immediately lost.
Fig. 2. The variation of critical magnetic thermal Rayleigh numbers (Ncℓ and Nce) with coefficient of magnetic field dependent viscosity (δ) for M3 = 5.
1286
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
4. Discussion of results and conclusions
Acknowledgements
The critical wave numbers xcℓ, xce and critical magnetic thermal Rayleigh numbers Ncℓ, Nce depend on M3 and coefficient of MFD viscosity δ. The variation of xcℓ, xce and Ncℓ, Nce with various parameters is given in Table 1 and the results are further illustrated in Figs. 1 and 2. Fig. 1 represents the plot of critical magnetic thermal Rayleigh numbers Ncℓ and Nce versus magnetic parameter M3 for various values of δ. It is depicted from Fig. 1 that the magnetization hastens the onset of convection in the absence of MFD viscosity because as M3 increases, the values of Ncℓ and Nce decrease, whereas for small values of δ, both Ncℓ and Nce decrease for lower values of M3 and increase for higher values of M3, and for higher values of δ, Ncℓ decreases for lower values of M3 and increases for higher values of M3 and Nce always increases with increase in M3. We also note that the values of Ncℓ are always higher than those of Nce and thus the difference between the values of Ncℓ and Nce reveals that there is a band of Rayleigh numbers where sub-critical instabilities may arise. We note that this band decreases as M3 increases (Table 1). Fig. 2 represents the plot of critical magnetic thermal Rayleigh numbers Ncℓ, Nce with δ at M3 = 5. This figure shows that the MFD viscosity has a stabilizing effect on convection as δ increases, the values Ncℓ and Nce increase. We note that the subcritical region expands a little for small values of δ and expands significantly for large values of δ. In order to investigate our results, we must review the results and its physical explanations. As M3 increases, departure of the linearity in the magnetic equation of state increases, which causes the increase in the velocity of the fluid perpendicular to the planes, and it renders the system prone to instability. In other words, M3 destabilizes the flow. This variation in magnetization releases extra energy, which adds up to thermal energy to destabilize the system. Presence of MFD viscosity gives rise to a resistive-type force. This force has the tendency to slow down the motion of the fluid in the boundary layer, thus inducing the heat transfer from bottom to top. The decrease in heat transfer is responsible for delaying the onset of convection. Thus, MFD viscosity promotes stabilization. The principal conclusions from the above analysis are as under:
The financial assistance to Dr. Sunil in the form of Research and Development Project [No. 25(0165)/08/EMR-II] and to Mr. Amit Mahajan in the form of a Senior Research Fellowship (SRF) of the Council of Scientific and Industrial Research (CSIR), New Delhi and to Miss Poonam Sharma in the form of Research Fellowship (RF) from NIT Hamirpur, are gratefully acknowledged.
(i) The result we establish is that boundaries of nonlinear stability and linear instability analyses do not coincide. (ii) It is found that the MFD viscosity always delay the onset of convection. (iii) We also observed that when viscosity is isotropic and independent of the magnetic field intensity, magnetization has always a destabilizing effect, whereas in the presence of MFD viscosity, magnetization has a dual role. (iv) Comparison between the linear and energy stability results for magnetized ferrofluid shows that the linear critical magnetic thermal Rayleigh number is higher in values than the nonlinear critical magnetic thermal number and thus indicates the possibility of existence of sub-critical instability region. The existence of sub-critical instability region for magnetized ferrofluid can also be seen in previous published paper (Sunil and Mahajan [33]). (v) It is also found that sub-critical region of instability can be induced by magnetic mechanism alone. (vi) It is important to realize that this region decreases as magnetization increases whereas as coefficient of MFD viscosity increases, the gap between the linear and energy stability results widens. (vii) In non-ferrofluids, a best possible result is verified in that we show that the global nonlinear stability Rayleigh number is exactly the same as that for linear instability.
References [1] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer Verlag, New York, 2004. [2] B. Straughan, Explosive Instabilities in Mechanics, Springer, Berlin, 1998. [3] B. Straughan, A sharp nonlinear stability threshold in rotating porous convection, Proc. Roy. Soc. Lond. A 457 (2001) 87–93. [4] G.P. Galdi, Nonlinear stability of the magnetic Bénard problem via a generalized energy method, Arch. Ration. Mech. Anal. 87 (1985) 167–186. [5] G.P. Galdi, B. Straughan, A nonlinear analysis of the stabilizing effect of rotation in the Bénard problem, Proc. Roy. Soc. Lond. Ser A 402 (1985) 257–283. [6] S. Rionero, G. Mulone, A nonlinear stability analysis of the magnetic Bénard problem through the Lyapunov direct method, Arch. Ration. Mech. Anal. 103 (1988) 347–368. [7] G. Mulone, S. Rionero, On the nonlinear stability of the rotating Bénard problem via the Lyapunov direct method, J. Math. Anal. Appl. 144 (1989) 109–127. [8] G.P. Galdi, M. Padula, A new approach to energy theory in the stability of fluid motion, Arch. Ration. Mech. Anal. 110 (1990) 187–286. [9] Y. Qin, P.N. Kaloni, Nonlinear stability problem of a rotating porous layer, Quart. Appl. Math. 53 (1995) 129–142. [10] J. Guo, Y. Qin, P.N. Kaloni, Nonlinear stability problem of a rotating doubly diffusive fluid layer, Int. J. Eng. Sci. 32 (1994) 1207–1219. [11] J. Guo, P.N. Kaloni, Nonlinear stability problem of a rotating doubly diffusive porous layer, J. Math. Anal. Appl. 190 (1995) 373–390. [12] S. Venkatasubramanian, P.N. Kaloni, Stability and uniqueness of magnetic fluid motions, Proc. Roy. Soc. Lond. A 458 (2002) 1189–1204. [13] P.N. Kaloni, Z. Qiao, Nonlinear convection with inclined temperature gradient and horizontal mass flow, Int. J. Eng. Sci. 35 (1997) 299–309. [14] P.N. Kaloni, Z. Qiao, Non-linear stability of convection in a porous medium with inclined temperature gradient, Int. J. Heat Mass Transfer 40 (1997) 1611–1615. [15] P.N. Kaloni, Z. Qiao, Non-linear convection in a porous medium with inclined temperature gradient and variable gravity effects, Int. J. Heat Mass Transfer 44 (2001) 1585–1591. [16] L.E. Payne, B. Straughan, Unconditional nonlinear stability in temperaturedependent viscosity flow in a porous medium, Stud. Appl. Math. 105 (2000) 59–81. [17] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, Cambridge, 1985. [18] B.A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech. 40 (1970) 753–767. [19] D.P. Lalas, S. Carmi, Thermoconvective stability of ferrofluids, Phys. Fluids 14 (1971) 436–437. [20] M.I. Shliomis, Magnetic fluids, Sov. Phys. Usp. 17 (1974) 153–169. [21] L. Schwab, U. Hildebrandt, K. Stierstad, Magnetic Bénard convection, J. Magn. Magn. Mater. 39 (1983) 113–114. [22] P.J. Stiles, M.J. Kagan, Thermo-convective instability of a horizontal layer of ferrofluid in a strong vertical magnetic field, J. Magn. Magn. Mater. 85 (1990) 196–198. [23] P. Blennerhassett, F. Lin, P. Stiles, Heat transfer through strongly magnetized ferrofluids, Proc. Roy. Soc. Lond. Ser. A 433 (1991) 165–177. [24] S. Venkatasubramanian, P.N. Kaloni, Effects of rotation on the thermoconvective instability of a horizontal layer of ferrofluids, Int. J. Eng. Sci. 32 (1994) 237–256. [25] Sunil, Divya, R.C. Sharma, Effect of dust particles on thermal convection in ferromagnetic fluid saturating a porous medium, J. Magn. Magn. Mater. 288 (2005) 183–195. [26] Sunil, Divya, V. Sharma, Effect of dust particles on rotating ferromagnetic fluid heated from below saturating a porous medium, J. Colloid Interface Sci. 291 (2005) 152–161. [27] Sunil, A. Sharma, R.C. Sharma, Effect of dust particles on ferrofluid heated and soluted from below, Int. J. Therm. Sci. 45 (4) (2006) 347–358. [28] G. Vaidyanathan, R. Sekar, R. Vasanthakumari, A. Ramanathan, The effect of magnetic field dependent viscosity on ferroconvection in a rotating sparsely distributed porous medium, J. Magn. Magn. Mater. 250 (2002) 65–76. [29] A. Ramanathan, P. Surendra, Effect of magnetic field on viscosity of ferroconvection in an isotropic sparsely distributed porous medium, Ind. J. Pure Appl. Phys. 41 (2003) 522–526. [30] Sunil, Anupama, R.C. Sharma, The effect of magnetic field dependent viscosity on thermosolutal convection in ferromagnetic fluid, Appl. Math. Comput. 163 (2005) 1197–1214. [31] Sunil, Divya, R.C. Sharma, The effect of magnetic field dependent viscosity on thermosolutal convection in ferromagnetic fluid saturating a porous medium, Trans. Porous. Media 60 (2005) 251–274.
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287 [32] Sunil, A. Sharma, D. Sharma, P. Kumar, Effect of magnetic field-dependent viscosity on thermal convection in a ferromagnetic fluid, Chem. Eng. Commun. 195 (2008) 571–583. [33] Sunil, A. Mahajan, A nonlinear stability analysis for magnetized ferrofluid heated from below, Proc. Roy. Soc. Lond. A 464 (2008) 83–98. [34] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1994.
1287
[35] D.D. Joseph, On the stability of the Boussinesq equations, Arch. Ration. Mech. Anal. 20 (1965) 59–71. [36] D.D. Joseph, Nonlinear stability of the Boussinesq equations by the method of energy, Arch. Ration. Mech. Anal. 22 (1966) 163–184.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
A nonlinear stability analysis for thermoconvective magnetized ferrofluid with magnetic field dependent viscosity☆ Sunil ⁎, Poonam Sharma, Amit Mahajan Department of Applied Sciences, National Institute of Technology, Hamirpur (H.P.)−177 005, India
a r t i c l e
i n f o
Available online 7 September 2008 Keywords: Magnetized ferrofluid Nonlinear stability Magnetization MFD viscosity
a b s t r a c t The generalized energy method which gives sufficient condition for the stability is developed for convection problem in a magnetized ferrofluid with magnetic field dependent (MFD) viscosity heated from below. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. Both linear and nonlinear analyses are carried out and comparison of results shows a marked difference in the stability boundaries and thus indicates that the sub-critical instabilities are possible. The effect of various parameters on the sub-critical region has also been analyzed. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The approach adopted in the present paper is by the application of energy method, pioneered and developed in its modern use way by Straughan [1,2]. Straughan [2] emphasizes that energy theory is certainly much stronger when the stability obtained is unconditional or for at least finite (non-vanishing) initial data. Straughan [3] developed a sharp nonlinear energy stability analysis for the Darcy's equations of thermal convection in a fluid saturated porous medium and the results obtained are the best possible showing sub-critical instabilities are not possible. In recent years, nonlinear stability analysis of non-magnetic fluids by using generalized energy stability theory has been considered by many authors [4–16]. Magnetic fluids or ferrofluids are colloidal suspensions of fine ferromagnetic mono domain nanoparticles in non-conducting liquids. One of the major applications of ferrofluid is its use in medical field, like the transport of drugs to an injured site, removal of tumors from the body. Other applications include its use as coolant and in devices like rotating shaft seals, loudspeakers, printers, sealed motors, acoustic devices etc. Rosensweig [17] in his monograph has given an authoritative introduction to the research on magnetic liquids and various authors [18–27] have studied convection problems of ferrofluid. The convection in ferrofluid is gaining much importance due to its astounding physical properties. One such property is viscosity of the ferrofluid. The effect of magnetic field dependent (MFD) viscosity on convection in ferrofluid by linearized theory has been studied by many authors [28–32]. Recently, Sunil and Mahajan [33] studied the non-linear stability analysis for mag☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (Sunil). 0735-1933/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2008.08.004
netized ferrofluid heated from below. They assume that the viscosity is isotropic and independent of magnetic field. The effect of MFD viscosity on nonlinear stability analysis of magnetized ferrofluid cannot be ignored. In the present analysis, an attempt is made to see the effect of MFD viscosity on magnetized ferrofluid heated from below. We analyzed both the nonlinear stability analysis via generalized energy method and linear instability results. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. Here, the energy method gives critical magnetic thermal Rayleigh numbers just slightly below the ones given by linear theory. This shows the possibility of the existence of sub-critical instability. On the other hand, in case of non-ferrofluid the nonlinear and linear stability results coincide. This in turn implies exclusion of occurrence of sub-critical instability. Comparison of the results obtained, respectively, by the energy method and the linear stability analysis has been discussed finally. This problem, to the best of our knowledge, has not been investigated yet. 2. Perturbation equations and boundary conditions Here, we consider an infinite horizontal layer of thickness ‘d’ of an electrically non-conducting incompressible thin ferrofluid having a variable viscosity, given by µ1 = µ(1 + δ · B) heated from below. Here, µ is taken as viscosity of the fluid when the applied magnetic field is absent and µ1 is magnetic field dependent (MFD) viscosity. The variation of magnetic field dependent viscosity δ has been taken to be isotropic, δ = δ1 = δ2 = δ3. The fluid is assumed to occupy the layer za − 2d ; 2d with gravity acting in the negative z-direction and magnetic ˆ acts outside the layer. field intensity, H ¼ H0ext k, To describe nonlinear energy stability results, we begin with the relevant equations and basic state as given in (Sunil and Mahajan [33]). In order to analyze the stability of the system, let the basic state
1282
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
Nomenclature Latin symbols b Subscript; basic state B Magnetic induction (T) B Magnitude of B (T) C0 Specific heat at constant pressure (J/kg K) d Thickness of the ferrofluid layer (m) g Acceleration due to gravity (m/s2) H Magnetic field intensity (Amp/m) Hext External magnetic field intensity (Amp/m) 0 H′ The perturbation in magnetic field intensity (Amp/m) H Magnitude of H (Amp/m) H0 Reference imagnetic field intensity (Amp/m) h K1 ¼ − AM The pyromagnetic coefficient (Amp/mK) AT H ;T ˆ k Unit vector in the z-direction M Magnetization (Amp/m) M′ The perturbation in the magnetization (Amp/m) M Magnitude of M (Amp/m) M0 Magnetization when magnetic field is H0 and temperature Ta (Amp/m) p The fluid pressure (psi) p′ The perturbation in fluid pressure (psi) q Velocity of the ferrofluid (m/s) q′=(u, v, w)The perturbation in velocity (m/s) t Time (s) T Temperature (K) TL Constant average temperature at the bottom surface z = −d/2 (K) TU Constant average temperature at the upper surface z = +d/2 (K) Ta Average temperature (K) 0
be perturbed. Let q′ = (u, v, w), ρ′, p′, θ, H′ = (H1′, H2′, H3′) and M′ = (M1′, M2′, M3′) represent the perturbations in velocity, density, pressure, temperature, magnetic field intensity and magnetization, respectively. Then the nonlinear perturbation equations become ρ0
μ 0 K1 β Aq′ ¼ −∇p′þ μ 1 þ δμ 0 ðH0 þ M0 Þ ∇2 q′þ ρ0 αgθ kˆ − 1þχ At h i M0 2 ˆ ˆ ð1 þ χÞ′zk−K1 θk þ μδμ 0 1 þ ′x ∇ q′þ μδμ 0 H0 M0 1þ ′y ∇2 q′þ μδμ 0 ð1 þ χÞ′z ∇2 q′− μδμ 0 K1 θ∇2 q′ H0 þμ 0
M0 M0 ′x ∇′x þ μ 0 ′y ∇′y þ μ 0 χ′z ∇′z H0 H0 ð1Þ
−μ 0 K1 θ∇′z −ρ0 q′ ∇q′;
a
∇ q ′¼ 0;
ð2Þ
Aθ þ q ′ ∇θ ¼ κ∇2 θ þ βw; At M0 M0 − χ ′zz ¼ K1 θz : ∇2 ′− 1þ H0 H0
ð3Þ ð4Þ
Here, H′¼ ∇′ ½6∇ H ¼ 0; where ϕ′ is the perturbed magnetic potential. The boundary conditions are w ¼ 0; uz ¼ 0; vz ¼ 0; θ ¼ 0; ′z ¼ 0
d at z ¼ F ; 2
ð5Þ
with q′; θ; ′satisfying a plane tiling periodic boundary condition in x and y. 3. Nonlinear stability analysis
Greek letters α Coefficient of thermal expansion (K− 1) β A uniform temperature gradient β (= |dT/dz|) (K/m) δ Coefficient of MFD viscosity (T− 1) κ Thermal diffusivity (m2/s) µ1 Dynamic MFD viscosity of fluid (kg/m s) µ0 Magnetic permeability of free space (H/m) µ Reference viscosity of fluid (kg/m s) ρ Fluid density (kg/m3) ρ0 Density at the ambient temperature (kg/m3) ρ′ The perturbation in density ρ(kg/m3) θ The perturbation in temperature T(K) χ The magnetic susceptibility ▿ Gradient operator (m− 1) ϕ′ The perturbed magnetic potential (Amp)
Non-dimensional parameters M1 Buoyancy magnetization parameter M3 Magnetic parameter (measures the departure of linearity in the magnetic equation of state) Mδ Ratio of viscous to gravitational forces Ncℓ Critical linear magnetic thermal Rayleigh number Nce Critical energy magnetic thermal Rayleigh number Rℓ Linear thermal Rayleigh number Re Energy thermal Rayleigh number x Dimensionless wave number
To investigate the nonlinear stability analysis, the perturbed equations in non-dimensional form (dropping ⁎) can be written as Aq þ q ∇q ¼ −∇p þ ð1 þ δM3 Þ∇2 q þ R1=2 ð1 þ M1 Þθ kˆ At
ð6Þ
−R1=2 M1 z kˆ þ δM3 Mδ x ∇2 q þδM3 Mδ y ∇2 q þ δMδ z ∇2 q−δMδ θ∇2 q −M1 θ∇z þ M1 M3 −
1 ∇ 1þχ x x 1 χ y ∇y þ M1 ∇ : þM1 M3 − 1þχ 1þχ z z On multiplying Eq. (6) by q and integrating over V, we find (after using the boundary conditions and the divergence theorem) 1 djjqjj2 ¼ −jj∇qjj2 ð1 þ δM3 Þ þ R1=2 ð1 þ M1 Þhwθi−R1=2 M1 hwz i 2 dt þ M1 hq ∇θz i þ δM3 Mδ hx q ∇2 qi
ð7Þ
þδM3 Mδ hy q ∇2 qi þ δMδ hz q ∇2 qi−δMδ hθq ∇2 qi χ M1 hz q ∇z i þ 1þχ 1 1 hx q ∇x i þ M1 M3 − hy q ∇y i: þM1 M3 − 1þχ 1þχ Other relevant equations are same as mentioned in (Sunil and Mahajan [33]). Here, 〈·〉 denotes the integration over V and ||.|| denotes the L2 (V) norm, where V denotes a typical periodicity cell.
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
The non-dimensional quantities and non-dimensional parameters used in the above equations are same as in the work of (Sunil and Mahajan [33]). However, one additional quantity and one additional parameter is involved due to effect of MFD viscosity and are given as δ⁎ = µ0δH0(1 + χ) and Mδ ¼ ð1þχK1ÞRβd 1=2 H ; respectively. 0 Here, Mδ is a ratio of viscous to gravitational forces. Further, the process has been carried out by using the techniques of Straughan [3] and Sunil and Mahajan [33]. To study the nonlinear stability of basic state, an L2 energy, E(t), is constructed and the evolution of E(t) is given by dE ¼ I0 −D0 þ N0 ; dt
where I1 ¼ R1=2 ð2 þ M1 Þh∇w ∇θi−R1=2 M1 h∇w ∇z i−R1=2 h∇w ∇3 θi;
ð18Þ
D1 ¼ ð1 þ δM3 Þjj∇2 qjj2 þ jj∇2 θjj2 þ jj∇3 θjj2 ;
ð19Þ
N1 ¼ hq ∇θ∇2 θi þ hq ∇q ∇2 qi þ h∇q ∇θ ∇3 θi þhq ∇3 θ∇2 θi−δM3 Mδ hx ∇2 q ∇2 qi
ð8Þ
−δM3 Mδ hy ∇2 q ∇2 qi−δMδ hz ∇2 q ∇2 qi χ hz ∇2 q ∇z i þδMδ hθ∇2 q ∇2 qi−M1 1þχ
where
E¼
1 λ1 jjθjj2 þ jjqjj2 ; 2 2
ð9Þ
I0 ¼ R1=2 f1 þ λ1 ð1 þ M1 Þghwθi−R1=2 λ1 M1 hwz i−λ2 hθz i
−M1 M3 −
ð10Þ
1 1 hx ∇2 q ∇x i−M1 M3 − hy ∇2 q ∇y i 1þχ 1þχ
þM1 hθ∇2 q ∇z i: 2
2
2
1283
ð20Þ
2
D0 ¼ jj∇θjj þ λ1 ð1 þ δM3 Þjj∇qjj þ λ2 M3 jj∇jj −λ2 ðM3 −1Þjjz jj ; ð11Þ
N0 ¼ λ1 M1 hq ∇θz i þ λ1 δM3 Mδ hx q ∇2 qi þ λ1 δM3 Mδ hy q ∇2 qi þ λ1 δMδ hz q ∇2 qi 1 hx q ∇x i −λ1 δMδ hθq ∇2 qi þ λ1 M1 M3 − 1þχ 1 hy q ∇y i þ λ1 M1 M3 − 1þχ χ hz q ∇z i; þλ1 M1 ð12Þ 1þχ
H
I0 ; D0
ð13Þ
ð14Þ
ð15Þ
Here b0 is a positive coupling parameter to be chosen and the complementary energy E1(t) is given by E1 ðt Þ ¼
1 1 1 jj∇θjj2 þ jj∇qjj2 þ jj∇2 θjj2 : 2 2 2
ð16Þ
The evolution of Vg(t) is given by dVg ≤ −a0 D0 þ N0 þ b0 I1 −b0 D1 þ b0 N1 ; dt
with À given by ! ! # 1=2 " 2 λ1 λ1 2χ−1 A` ¼ 2C 4 4 þ 2δMδ ðM3 þ 1Þ 1=2 þ 1 þ M1 1 þ 1=2 ; 2M3 þ b0 1þχ b0 a0
b0 ¼
a0 π2 n o; 2 3R π þ ð2 þ M1 Þ2 þM12
and defined
with a0 = 1 − m(N0). In order to dominate the nonlinear terms and for studying the (conditional) nonlinear stability, we now introduce the generalized energy functional as Vg ðt Þ ¼ Eðt Þ þ b0 E1 ðt Þ:
ð21Þ
where we have chosen
where H is the space of admissible solutions. Then we require m b 1 so that dE ≤ −a0 D0 þ N0 ; dt
: ` Vg1=2 ; V g ðt Þ ≤ −D2 1−A
ð22Þ
with λ1, λ2 two positive coupling parameters. We now define m ¼ max
Using inequalities and embedding theorem (Sunil and Mahajan [33]), the Cauchy–Schwartz and the Young inequalities (Hardy et al. [34]), we have
D2 ¼
a0 b0 D0 þ D1 : 2 2
ð23Þ
Estimate (21) enables us to prove the following theorem of conditional nonlinear stability criterion. Theorem. Let 0 b m b 1;
ð24Þ
−2
ð25Þ
Vg ð0ÞbA` ;
with À given by Eq. (22). Then, there exists a positive constant K⁎, such that 4 ` 1=2 Vg ðt Þ ≤ Vg ð0Þe−K ð1−A Vg ð0ÞÞt ;
tz0:
Proof. The hypothesis and inequality (21) ensure that ð17Þ
: V g ð0Þb0:
ð26Þ
1284
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
Therefore, from inequality (21) by a recursive argument, we obtain : ` Vg1=2 ð0Þ ; 8 tz0: ð27Þ V g ðt Þ ≤ −D2 1−A
−D2 ≤−K 4Vg :
ð28Þ
By Eq. (15), using Eqs. (11), (19), (23) and by virtue of the Poincarétype inequalities, we have
2 D0 b0 b0 2 m 1 þ 2 D2 : þ D1 þ 2 D1 ≤ 2 1 þ 2 1−m 2 2π π 2 π π
λ1 Y0; λ ′2 Yð1 þ xÞ and
m ; 1−m
ð29Þ
Re ¼
2
: ` Vg1=2 ð0Þ V g ≤−K 4Vg 1−A
8
ð30Þ
tz0:
Integrating this last inequality, we deduce the theorem. Since Vg (t) in Eq. (15) does not contain the term jj∇jj2 , the kinetic energy term for magnetic potential, it is worthwhile checking as to what happens to jj∇jj2 as t → ∞. Using inequality (3.16)1 of [33], we have jj∇jj2 ≤ jjθjj2 :
ð31Þ
Thus, Eq. (15) and inequality (31) ensure the decay of jj∇jj2 , i.e. jjH′jj2 . 3.1. Variational problem We now solve the variational problem associated with the maximum problem (13). The corresponding Euler–Lagrange equations pffiffiffiffiffiffi for the problem (13) after taking transformations qˆ ¼ λ1 q and ˆ ¼ pffiffiffiffiffiffi λ2 (dropping caps) are 2ð1 þ δM3 Þ∇2 q þ R1=2 f1 þ λ1 ð1 þ M1 Þg
1 1=2 λ1
ˆ θ k−
1=2
R1=2 M1 λ1 1=2 λ2
ˆ −2∇p ¼ 0; z k
ð32Þ 2∇ θ þ R
1=2
f1 þ λ1 ð1 þ M1 Þg
1
wþ 1=2 λ1
1=2 λ2 z
¼ 0;
ð33Þ
wz −λ2 θz ¼ 0;
ð34Þ
1=2
2M3 ∇2 −2ðM3 −1Þzz þ
λ1 Y0: λ ′2
ð38Þ
R1=2 M1 λ1 1=2 λ2
1=2
2ð1 þ δM3 Þ∇4 w þ R1=2 f1 þ λ1 ð1 þ M1 Þg
1
∇21 θ− 1=2 λ1
1=2
λ2
ð36Þ 2
x ¼ πa2 ;
λ′2 ¼ λπ22 .
ð39Þ
ð1 þ δM3 Þð1 þ xÞ3 ð3 þ 4xM3 Þ : xð2 þ 4xM3 Þ
ð40Þ
P5 x5 þ P4 x4 þ P3 x3 þ P2 x2 þ P1 x þ P0 ¼ 0:
ð41Þ
The coefficients P0,……,P5, being quite lengthy, has not been written here and are evaluated during numerical calculations. The values of critical wave number in nonlinear stability results are determined e numerically using Newton–Raphson method by the condition dN ¼ 0. dx With x determined as a solution of Eq. (41), Eq. (40) will give the required critical magnetic thermal Rayleigh number Nce. In the absence of MFD viscosity (δ = 0), (40) reduces to Ne ¼ ð1þxÞ3 ð3þ4xM3 Þ xð2þ4xM3 Þ , which is in good agreement with previous published work (Sunil and Mahajan [33]). For analyzing the linear instability results, we return to the perturbed Eqs. (1)–(4) neglecting the non-linear terms. We again
Table 1 The variation of the critical magnetic thermal Rayleigh numbers (Ncℓ and Nce) with magnetic parameter (M3) δ
M3
xcℓ
Ncℓ
xce
Nce
Ncℓ − Nce
0
1 5 10 15 20 25 1 5 10 15 20 25 1 5 10 15 20 25 1 5 10 15 20 25
1 0.690 0.613 0.581 0.564 0.552 1 0.690 0.613 0.581 0.564 0.552 1 0.690 0.613 0.581 0.564 0.552 1 0.690 0.613 0.581 0.564 0.552
16 9.02 7.96 7.58 7.38 7.26 16.16 9.47 8.76 8.72 8.86 9.08 16.48 10.38 10.35 10.99 11.82 12.71 16.8 11.28 11.94 13.27 14.77 16.34
0.580 0.547 0.529 0.521 0.516 0.513 0.580 0.547 0.529 0.521 0.516 0.513 0.580 0.547 0.529 0.521 0.516 0.513 0.580 0.547 0.529 0.521 0.516 0.513
8.37 7.29 7.05 6.96 6.91 6.88 8.45 7.66 7.75 8.00 8.29 8.60 8.63 8.39 9.16 10.09 11.05 12.04 8.79 9.11 10.57 12.17 13.82 15.48
7.63 1.73 0.91 0.62 0.45 0.38 7.71 1.81 1.01 0.72 0.57 0.48 7.85 1.99 1.19 0.9 0.77 0.67 8.01 2.17 1.37 1.10 0.95 0.86
ð35Þ
ð1 þ δM3 Þð1 þ xÞ2 f4ð1 þ xÞð1 þ xM3 Þ−λ′2 g i; Re ¼ h x λ11 ð1 þ xM3 Þf1 þ λ1 ð1 þ M1 Þg2 þ M12 ð1 þ xÞ λλ1′2 −M1 f1 þ λ1 ð1 þ M1 Þg
:
As a function of x, Ne given by Eq. (40) attains its minimum when
∇21 z ¼ 0:
The Eqs. (35), (33) and (34) are solved using same procedure as in Sunil and Mahajan [33]. With the plane tiling form and the boundary conditions as described in the above referenced paper, we obtained the Rayleigh number as
xf4ð1 þ M1 Þð1 þ xM3 Þ−2M1 g
Ne ¼ R e M 1 ¼
0.01
1=2
R1=2 M1 λ1
n o M1 ð1 þ δM3 Þð1 þ xÞ3 4ð1 þ xM3 Þ− 1þM 1
For M1 sufficiently large, we obtain the magnetic thermal Rayleigh number
where p is a Lagrange's multiplier introduced, since q is solenoidal. On taking curlcurl of Eq. (32) and taking third component of the resulting equation, we find
where Re ¼ πR4 ;
ð37Þ
Using Eqs. (37) and (38) in Eq. (36), we have
and assuming K 4 ¼ 2kπ0 þ3, from Eqs. (27) and (28), we have
2
ð1 þ xÞM1 1 ; λ′2 ¼ : 1 þ M1 1 þ M1
The classical results in respect of Newtonian fluids (non-magnetic fluid) can be obtained when M1 = 0 which imply λ1 = 1, λ2′ = 0. Thus, λ1 = 1 is the optimal value, i.e. that value which maximizes Re (Straughan [1], pp 59–62). Since, we are dealing with magnetized ferrofluid, so for sufficiently large values of buoyancy magnetization (M1), we have
Let k0 N 0, such that k0 z
The values of λ1 and λ2′ are determined by the conditions dRe ¼ 0; dλ ¼ 0, respectively, and are found to be ′2
λ1 ¼
Now, we prove that there exists K⁎ N0, such that
Vg ≤
dRe dλ1
0.03
0.05
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
1285
Fig. 1. The variation of critical magnetic thermal Rayleigh numbers (Ncℓ and Nce) with magnetic parameter (M3).
perform the standard, stationary, mode analysis and look for the solution of these equations with the same boundary conditions. Following the procedure as stated earlier in the energy stability case, we have Rℓ ¼
ð1 þ δM3 Þð1 þ xÞ3 ð1 þ xM3 Þ : xð1 þ xM3 þ xM1 M3 Þ
ð42Þ
For M 1 sufficiently large, the critical magnetic thermal Rayleigh number, in linear case, is Nℓ ¼
ð1 þ δM3 Þð1 þ xÞ3 ð1 þ xM3 Þ : x2 M3
ð43Þ
In the3 absence of MFD viscosity (δ = 0), Eq. (43) reduces to 1þxM3 Þ Nℓ ¼ ð1þxÞx2ðM . 3
This is exactly Eq. (21) of Finlayson [18]. There are instances in which the two theories coincide. This is true for the classical Bénard problem. In the absence of magnetic parameters (M1 = 0 and M3 = 0), we obtain Rℓ ¼
ð1 þ xÞ3 ¼ Re ; x
i.e., the linear instability boundary ≡ the nonlinear stability boundary. Here, the energy method leads to the strong result that arbitrary sub-critical instabilities are not possible which is in good agreement with the previous published work (Joseph [35–36]). Thus, for lower values of magnetic parameters, this coincidence is immediately lost.
Fig. 2. The variation of critical magnetic thermal Rayleigh numbers (Ncℓ and Nce) with coefficient of magnetic field dependent viscosity (δ) for M3 = 5.
1286
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287
4. Discussion of results and conclusions
Acknowledgements
The critical wave numbers xcℓ, xce and critical magnetic thermal Rayleigh numbers Ncℓ, Nce depend on M3 and coefficient of MFD viscosity δ. The variation of xcℓ, xce and Ncℓ, Nce with various parameters is given in Table 1 and the results are further illustrated in Figs. 1 and 2. Fig. 1 represents the plot of critical magnetic thermal Rayleigh numbers Ncℓ and Nce versus magnetic parameter M3 for various values of δ. It is depicted from Fig. 1 that the magnetization hastens the onset of convection in the absence of MFD viscosity because as M3 increases, the values of Ncℓ and Nce decrease, whereas for small values of δ, both Ncℓ and Nce decrease for lower values of M3 and increase for higher values of M3, and for higher values of δ, Ncℓ decreases for lower values of M3 and increases for higher values of M3 and Nce always increases with increase in M3. We also note that the values of Ncℓ are always higher than those of Nce and thus the difference between the values of Ncℓ and Nce reveals that there is a band of Rayleigh numbers where sub-critical instabilities may arise. We note that this band decreases as M3 increases (Table 1). Fig. 2 represents the plot of critical magnetic thermal Rayleigh numbers Ncℓ, Nce with δ at M3 = 5. This figure shows that the MFD viscosity has a stabilizing effect on convection as δ increases, the values Ncℓ and Nce increase. We note that the subcritical region expands a little for small values of δ and expands significantly for large values of δ. In order to investigate our results, we must review the results and its physical explanations. As M3 increases, departure of the linearity in the magnetic equation of state increases, which causes the increase in the velocity of the fluid perpendicular to the planes, and it renders the system prone to instability. In other words, M3 destabilizes the flow. This variation in magnetization releases extra energy, which adds up to thermal energy to destabilize the system. Presence of MFD viscosity gives rise to a resistive-type force. This force has the tendency to slow down the motion of the fluid in the boundary layer, thus inducing the heat transfer from bottom to top. The decrease in heat transfer is responsible for delaying the onset of convection. Thus, MFD viscosity promotes stabilization. The principal conclusions from the above analysis are as under:
The financial assistance to Dr. Sunil in the form of Research and Development Project [No. 25(0165)/08/EMR-II] and to Mr. Amit Mahajan in the form of a Senior Research Fellowship (SRF) of the Council of Scientific and Industrial Research (CSIR), New Delhi and to Miss Poonam Sharma in the form of Research Fellowship (RF) from NIT Hamirpur, are gratefully acknowledged.
(i) The result we establish is that boundaries of nonlinear stability and linear instability analyses do not coincide. (ii) It is found that the MFD viscosity always delay the onset of convection. (iii) We also observed that when viscosity is isotropic and independent of the magnetic field intensity, magnetization has always a destabilizing effect, whereas in the presence of MFD viscosity, magnetization has a dual role. (iv) Comparison between the linear and energy stability results for magnetized ferrofluid shows that the linear critical magnetic thermal Rayleigh number is higher in values than the nonlinear critical magnetic thermal number and thus indicates the possibility of existence of sub-critical instability region. The existence of sub-critical instability region for magnetized ferrofluid can also be seen in previous published paper (Sunil and Mahajan [33]). (v) It is also found that sub-critical region of instability can be induced by magnetic mechanism alone. (vi) It is important to realize that this region decreases as magnetization increases whereas as coefficient of MFD viscosity increases, the gap between the linear and energy stability results widens. (vii) In non-ferrofluids, a best possible result is verified in that we show that the global nonlinear stability Rayleigh number is exactly the same as that for linear instability.
References [1] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer Verlag, New York, 2004. [2] B. Straughan, Explosive Instabilities in Mechanics, Springer, Berlin, 1998. [3] B. Straughan, A sharp nonlinear stability threshold in rotating porous convection, Proc. Roy. Soc. Lond. A 457 (2001) 87–93. [4] G.P. Galdi, Nonlinear stability of the magnetic Bénard problem via a generalized energy method, Arch. Ration. Mech. Anal. 87 (1985) 167–186. [5] G.P. Galdi, B. Straughan, A nonlinear analysis of the stabilizing effect of rotation in the Bénard problem, Proc. Roy. Soc. Lond. Ser A 402 (1985) 257–283. [6] S. Rionero, G. Mulone, A nonlinear stability analysis of the magnetic Bénard problem through the Lyapunov direct method, Arch. Ration. Mech. Anal. 103 (1988) 347–368. [7] G. Mulone, S. Rionero, On the nonlinear stability of the rotating Bénard problem via the Lyapunov direct method, J. Math. Anal. Appl. 144 (1989) 109–127. [8] G.P. Galdi, M. Padula, A new approach to energy theory in the stability of fluid motion, Arch. Ration. Mech. Anal. 110 (1990) 187–286. [9] Y. Qin, P.N. Kaloni, Nonlinear stability problem of a rotating porous layer, Quart. Appl. Math. 53 (1995) 129–142. [10] J. Guo, Y. Qin, P.N. Kaloni, Nonlinear stability problem of a rotating doubly diffusive fluid layer, Int. J. Eng. Sci. 32 (1994) 1207–1219. [11] J. Guo, P.N. Kaloni, Nonlinear stability problem of a rotating doubly diffusive porous layer, J. Math. Anal. Appl. 190 (1995) 373–390. [12] S. Venkatasubramanian, P.N. Kaloni, Stability and uniqueness of magnetic fluid motions, Proc. Roy. Soc. Lond. A 458 (2002) 1189–1204. [13] P.N. Kaloni, Z. Qiao, Nonlinear convection with inclined temperature gradient and horizontal mass flow, Int. J. Eng. Sci. 35 (1997) 299–309. [14] P.N. Kaloni, Z. Qiao, Non-linear stability of convection in a porous medium with inclined temperature gradient, Int. J. Heat Mass Transfer 40 (1997) 1611–1615. [15] P.N. Kaloni, Z. Qiao, Non-linear convection in a porous medium with inclined temperature gradient and variable gravity effects, Int. J. Heat Mass Transfer 44 (2001) 1585–1591. [16] L.E. Payne, B. Straughan, Unconditional nonlinear stability in temperaturedependent viscosity flow in a porous medium, Stud. Appl. Math. 105 (2000) 59–81. [17] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, Cambridge, 1985. [18] B.A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech. 40 (1970) 753–767. [19] D.P. Lalas, S. Carmi, Thermoconvective stability of ferrofluids, Phys. Fluids 14 (1971) 436–437. [20] M.I. Shliomis, Magnetic fluids, Sov. Phys. Usp. 17 (1974) 153–169. [21] L. Schwab, U. Hildebrandt, K. Stierstad, Magnetic Bénard convection, J. Magn. Magn. Mater. 39 (1983) 113–114. [22] P.J. Stiles, M.J. Kagan, Thermo-convective instability of a horizontal layer of ferrofluid in a strong vertical magnetic field, J. Magn. Magn. Mater. 85 (1990) 196–198. [23] P. Blennerhassett, F. Lin, P. Stiles, Heat transfer through strongly magnetized ferrofluids, Proc. Roy. Soc. Lond. Ser. A 433 (1991) 165–177. [24] S. Venkatasubramanian, P.N. Kaloni, Effects of rotation on the thermoconvective instability of a horizontal layer of ferrofluids, Int. J. Eng. Sci. 32 (1994) 237–256. [25] Sunil, Divya, R.C. Sharma, Effect of dust particles on thermal convection in ferromagnetic fluid saturating a porous medium, J. Magn. Magn. Mater. 288 (2005) 183–195. [26] Sunil, Divya, V. Sharma, Effect of dust particles on rotating ferromagnetic fluid heated from below saturating a porous medium, J. Colloid Interface Sci. 291 (2005) 152–161. [27] Sunil, A. Sharma, R.C. Sharma, Effect of dust particles on ferrofluid heated and soluted from below, Int. J. Therm. Sci. 45 (4) (2006) 347–358. [28] G. Vaidyanathan, R. Sekar, R. Vasanthakumari, A. Ramanathan, The effect of magnetic field dependent viscosity on ferroconvection in a rotating sparsely distributed porous medium, J. Magn. Magn. Mater. 250 (2002) 65–76. [29] A. Ramanathan, P. Surendra, Effect of magnetic field on viscosity of ferroconvection in an isotropic sparsely distributed porous medium, Ind. J. Pure Appl. Phys. 41 (2003) 522–526. [30] Sunil, Anupama, R.C. Sharma, The effect of magnetic field dependent viscosity on thermosolutal convection in ferromagnetic fluid, Appl. Math. Comput. 163 (2005) 1197–1214. [31] Sunil, Divya, R.C. Sharma, The effect of magnetic field dependent viscosity on thermosolutal convection in ferromagnetic fluid saturating a porous medium, Trans. Porous. Media 60 (2005) 251–274.
Sunil et al. / International Communications in Heat and Mass Transfer 35 (2008) 1281–1287 [32] Sunil, A. Sharma, D. Sharma, P. Kumar, Effect of magnetic field-dependent viscosity on thermal convection in a ferromagnetic fluid, Chem. Eng. Commun. 195 (2008) 571–583. [33] Sunil, A. Mahajan, A nonlinear stability analysis for magnetized ferrofluid heated from below, Proc. Roy. Soc. Lond. A 464 (2008) 83–98. [34] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1994.
1287
[35] D.D. Joseph, On the stability of the Boussinesq equations, Arch. Ration. Mech. Anal. 20 (1965) 59–71. [36] D.D. Joseph, Nonlinear stability of the Boussinesq equations by the method of energy, Arch. Ration. Mech. Anal. 22 (1966) 163–184.