The combined effects of a uniform vertical magnetic field and a nonuniform basic temperature profile on the onset of steady Marangoni convection in a horizontal layer of micropolar fluid are studied. The closed-form expression for the Marangoni number M for the onset of convection, valid for polynomial-type basic temperature profiles upto a third order, is obtained by the use of the single-term Galerkin technique. The critical conditions for the onset of convection have been presented graphically.
1. Introduction
Convective flow in a thin layer of fluid, free at the upper surface and heated from below, is of fundamental importance and a prototype to a more complex configuration in experiments and industrial processes. The convective flows in a liquid layer can be driven by buoyancy forces due to temperature gradients and/or thermocapillary (Marangoni) forces caused by surface tension gradients. Thermal convective problems have long been studied extensively since the pioneering experimental and theoretical works of Bénard [1], Rayleigh [2], and Pearson [3]. The instability problems have been studied in several other directions (cf. [4–18]).
Most of the previous studies were concerned with convection in Newtonian fluids. However, much less work has been done on convection in non-Newtonian fluids such as the micropolar fluids. The theory of micropolar fluids, as developed by Eringen [19], has been a field of sprightly research for the last few decades especially in many industrially important fluids like paints, polymeric suspensions, colloidal fluids, and also in physiological fluids such as normal human blood and synovial fluids. Rama Rao [20] studied the effect of a magnetic field on convection in a micropolar fluid. The onset of convection as overstable motions in a micropolar fluid was examined in [21]. Sharma and Gupta [22] studied convection in micropolar fluids in a porous medium. Ramdath [23] considered buoyancy-and thermocapillary-driven (Bénard-Marangoni) convection in a layer of micropolar fluid. The effect of throughflow on Marangoni convection in micropolar fluids was analyzed in [24]. Siddheshwar and Sri Krishna [25] presented both linear and nonlinear analyses of convection in a micropolar fluid occupying a porous medium. Sunil et al. [26] studied the effect of rotation on convection in a micropolar ferrofluid.
There has also been much less work focused on the effect of nonuniform temperature gradient on convection. Friedrich and Rudraiah [27] studied the combined effects of nonuniform temperature gradients and rotation on Marangoni convection. The combined effects of nonuniform temperature gradients and a magnetic field on Marangoni convection were investigated by Rudraiah et al. [28]. The work of Friedrich and Rudraiah [27] was further extended to include the effect of buoyancy by Rudraiah and Ramachandramurthy [29]. Dupont et al. [30] studied the effect of a cubic quasisteady temperature profile on Marangoni convection. The effects of nonuniform temperature gradients on the onset of oscillatory Marangoni and Bénard-Marangoni convection in a magnetic field were analyzed in [31, 32], respectively. Chiang [33] investigated the effect of Dupont et al. [30] temperature profile on the onset of stationary and oscillatory Bénard-Marangoni convection.
Thermal convection in micropolar fluids has also been studied. Rudraiah and Siddheshwar [34] analyzed the effects of nonuniform temperature gradients of parabolic- and stepwise-types on the onset of Marangoni convection in a micropolar fluid. This study was later extended by Siddheshwar and Pranesh [35] to include the effect of a magnetic field and buoyancy forces. Very recently, Idris et al. [36] studied the effect of Dupont et al. [30] cubic temperature profile on the onset of Bénard-Marangoni convection in a micropolar fluid.
In this paper, we shall investigate the combined effects of Dupont et al. [30] cubic temperature profile and a magnetic field on the onset of Marangoni convection in a micropolar fluid. The single-term Galerkin technique [37] is employed to obtain a closed-form expression of M (Marangoni number) for the onset of convection. Comparisons with the other polynomial-type temperature profiles normally used by previous investigators shall be undertaken.
2. Mathematical Formulation
We wish to examine the stability of a horizontal layer of quiescent micropolar fluid of thickness d in the presence of a magnetic field. We assume that the layer is bounded below by a rigid boundary, which is kept at a constant temperature, and above by a perfectly insulated, flat free surface. Moreover, the spin-vanishing boundary condition is assumed at the boundaries.
The governing equations for the problem are the continuity equation, conservation of momentum, conservation of angular momentum, conservation of energy, and magnetic induction, compare [19, 34, 35]:
∇·q⃗=0,ρ0[∂q⃗∂t+(q⃗·∇)q⃗]=-∇P+(2ζ+η)∇2q⃗+ζ∇×ω⃗+μm(H⃗·∇)H⃗,ρ0I[∂ω⃗∂t+(q⃗·∇)ω⃗]=(λ′+η′)∇(∇·ω⃗)+η′∇2ω⃗+ζ(∇×q⃗-2ω⃗),∂T∂t+[q⃗-βρ0Cυ∇×ω⃗]·∇T=χ∇2T,∂H⃗∂t+(q⃗·∇)H⃗=(H⃗·∇)q⃗+γm∇2H⃗,
where q⃗ is the velocity, ω⃗ is the spin, T is the temperature, H⃗ is the magnetic field, P=p+μmH2/2 is the hydromagnetic pressure, ζ is the coupling viscosity coefficient, η is the shear kinematic viscosity coefficient, I is the moment of inertia, λ′ and η′ are the bulk and shear spin viscosity coefficients, β is the micropolar heat conduction coefficient, Cv is the specific heat, χ is the thermal conductivity, and γm=1/μmσm is the magnetic viscosity (where σm electrical conductivity and μm magnetic permeability). All the viscosity coefficients, heat conduction coefficient and thermal conductivity are thermodynamically restricted on the assumption of Clausius-Duhem inequality (see Eringen [19]) and are all positive quantities.
The surface tension σ at the free upper surface is
σ=σ0-σ1(T-T0),
where σ0 is the unperturbed value of σ and σ1=-(dσ/dT)T0. The perturbation (2.1) are nondimensionalised using the following definition:
(x*,y*,z*)=(x,y,z)d,q⃗*=q⃗′χ/d,ω⃗*=ω⃗′χ/d2,T*=T′ΔT,H⃗*=H⃗′H0.
Following the classical lines of linear stability theory, the linearised and dimensionless governing equations are
(1+N1)∇4W+N1∇2Ωz+QPrPm∇2(∂Hz∂z)=0,N3∇2Ωz-2N1Ωz-N1∇2W=0,∇2Θ+f(z)(W-N5Ωz)=0,∇2Hz+PmPr∂W∂z=0,
where W, Ωz, Θ, and Hz are, respectively, the amplitudes of the infinitesimal perturbations of velocity, spin, temperature, and magnetic field, N1=ζ/(ζ+η) is the coupling parameter (0≤N1≤1 ), N3=η′/(ζ+η) is the couple stress parameter (0≤N3≤m, m: finite, real), N5=β/(ρ0Cvd2) is the micropolar heat conduction parameter (0≤N5≤n, n: finite, real), Q=μmH02d2/[(ζ+η)γm] is the Chandrasekhar number, Pr=(ζ+η)/χ is the Prandtl number, Pm=(ζ+η)/γm is the magnetic Prandtl number, and f(z) is a nondimensional basic temperature gradient satisfying the condition ∫01f(z)dz=1.
The infinitesimal perturbations W, Ωz, Θ, and Hz are assumed to be periodic waves and hence these permit a normal mode solution in the following form:
[W,Ωz,Θ,Hz]=[W(z),Ωz(z),Θ(z),Hz(z)]exp[i(lx+my)],
where l and m are horizontal components of the wave number a⃗.
Substituting (2.5) into (2.4), we get
(1+N1)(D2-a2)2W+N1(D2-a2)Ω+QPrPm(D2-a2)DHz=0,N1(D2-a2)W-N3(D2-a2)Ω+2N1Ω=0,(D2-a2)Θ+f(z)(W-N5Ω)=0,(D2-a2)Hz+PmPrDW=0,
where D≡d/dz.
Eliminating Hz between (2.6) and (2.9), we obtain
(1+N1)(D2-a2)2W+N1(D2-a2)Ω-QD2W=0.
Equations (2.7), (2.8), and (2.10) are solved subject to the linearized and dimensionless boundary conditions:
W=D2W+a2MΘ=DΘ=Ω=0atz=1,W=DW=Θ=Ω=0atz=0,
where M=σ1ΔTd/μχ is the Marangoni number (where ΔT is the temperature difference between the two boundaries).
Following [30], we consider the steady state temperature profile given by
T̅b=T̅OS-a1(z̅-d)-a2(z̅-d)2-a3(z̅-d)3,
which precisely represents an experimental data, where (-) denotes dimensional quantities, T̅OS is the temperature at the upper free surface, and ai, i=1,2,3 are constants. In nondimensional form, the f(z) in this case is given by
f(z)=a1*+2a2*(z-1)+3a3*(z-1)2.
The case a1*=1, a2*=0, and a3*=0 recovers the classical linear basic state temperature distribution. The different temperature gradients studied in this paper are listed in Table 1.
Reference steady-state temperature gradients.
Model
Reference steady-state
f(z)
a1*
a2*
a3*
temperature gradient
1
Linear
1
1
0
0
2
Inverted parabolic
2(1-z)
0
-1
0
3
Cubic 1
3(z-1)2
0
0
1
4
Cubic 2
0.6+1.02(z-1)2
0.6
0
0.34
3. Solution of the Linearized Problem
Equations (2.7), (2.8), and (2.10) subject to the boundary conditions (2.11) constitute an eigenvalue problem. To solve the resulting eigenvalue problem, a single-term Galerkin expansion technique [37] is used to encompass a vast parameter space. Also, the technique employed yields sufficiently accurate and useful results for the purpose in hand with minimum of mathematics [37].
First we multiply (2.7), (2.8) and (2.10) by Ω, Θ, and W, respectively. Then we integrate the resulting equations by parts with respect to z from 0 to 1. By using the boundary conditions (2.11) and taking Ω=AΩ1(z), Θ=BΘ1(z), and W=CW1(z), and in which A, B, and C are constants and Ω1(z)=z(1-z), Θ1(z)=z(2-z), and W1(z)=z2(1-z2) are trial functions, yields the eigenvalue M in the form
M=[〈(Dθ1)2〉+a2〈θ12〉][C1(C2-Q〈(DW1)2〉)+N12C32](1+N1)a2θ(1)DW(1)C4,
where
C1=N3〈(DΩ1)2〉+(N3a2+2N1)〈Ω12〉,C2=-(1+N1)[〈(D2W1)2〉+2a2〈(DW1)2〉+a4〈W12〉],C3=〈(DΩ1)(DW1)〉+a2〈W1Ω1〉,C4=〈f(z)W1θ1〉C1-N1N5〈f(z)θ1Ω1〉C3.
Now with f(z) as given in (2.13), we rewrite the expression (3.1) in the closed-form expression for M:
M=f4[f2{315(1+N1)f3+132Q}-315f12]630(1+N1)[f2f6-N5f1f5],
where
f1=115N1(4+1128a2),f2=13(N3+110N3a2+15N1),f3=45(21+2221a2+263a4),f4=43(1+25a2),f5=110(1114a3*-a2*+76a1*)a2,f6=121(a3*-3120a2*+2310a1*)a2.
We remark that (3.3) is valid for all polynomial-type basic temperature profiles up to a third order. The critical Marangoni number, Mc, for the onset of convection is the global minimum of M over a≥0.
4. Discussion
The critical Marangoni number Mc which attains its minimum at ac2 is computed from (3.3) for different volumes of Q, N1, N3, and N5 and the results are depicted in Figures 1, 2, and 3. We recover the results of Rudraiah and Siddheshwar [34] for the linear and inverted parabolic temperature gradients when Q=0. We observe that as N1 or N5 increases, Mc also increases. Obviously, the onset of convection will be delayed by increasing the concentration of the microelements or heat induced into the fluid by the microelements. But, an increase in N3 leads to a decrease in microrotation, and hence the system becomes more unstable. Also it is observed that Model 4 (Cubic 2), with a1*=0.6, a2*=0, a3*=0.34 as used by Dupont et al. [30], is less stabilizing than Model 2 (Inverted parabolic), that is, Mc4<Mc2. Based on our results, Model 3 (Cubic 1) with a1*=0, a2*=0, a3*=1 is shown to be the most stabilizing of all the considered types of temperature gradients, that is, Mc1<Mc4<Mc2<Mc3.
Figures 4–6 illustrate the variations of the critical Marangoni number Mc with the Chandrasekhar number Q for some assigned values of N1, N3, and N5, respectively. The results indicate that Mc is generally an increasing function of Q. From Figure 4, we notice that the increase in the concentration of the microelements is to stabilize the system by superposing on the effect of the magnetic field. Figure 5 shows that the effect of N3 on the system is very small compared to the effects of the other microelements. As before, Model 3 (Cubic 1) with a1*=0, a2*=0, a3*=1 is shown to be the most stabilizing of all the considered types of temperature gradients, that is, Mc1<Mc4<Mc2<Mc3.
Plot of Mc versus N1 with N3=2 and N5=1, A: Linear. Q=0; B: Linear, Q=100; C: Cubic 2, Q=0; D: Cubic 2, Q=100; E: Inv. Parabolic, Q=0; F: Inv. Parabolic, Q=100; G: Cubic 1, Q=0; H: Cubic 1, Q=100.
Plot of Mc versus N3 with N1=0.1 and N5=1.0, A: Linear, Q=0; B: Linear, Q=50; C: Cubic 2, Q=0; D: Cubic 2, Q=50; E: Inv. Parabolic, Q=0; F: Inv. Parabolic, Q=50, G: Cubic 1, Q=0, H: Cubic 1, Q=50.
Plot of Mc versus N5 with N1=0.1 and N3=2.0, A: Linear, Q=0; B: Linear, Q=50; C: Cubic 2, Q=0; D: Cubic 2, Q=50; E: Inv. Parabolic, Q=0; F: Inv. Parabolic, Q=50; G: Cubic 1, Q=0; H: Cubic 1, Q=50.
Plot of Mc versus Q for different temperature gradients with N3=2.0 and N5=1.0.
Plot of Mc versus Q for different temperature gradients with N1=0.1 and N5=1.0.
Plot of Mc versus Q for different temperature gradients with N1=0.1 and N3=2.0.
5. Conclusion
The problem of Marangoni convection in a micropolar fluid in the presence of a cubic basic state temperature profile and a vertical magnetic field has been studied theoretically. The results indicate that it is possible to delay the onset of convection by the application of a cubic basic state temperature profile. In addition, the presence of a magnetic field is to suppress Magnetomarangoni convection and hence leads to a more stable system. As expected, the presence of the micron-sized suspended particles adds to the stabilizing effect of the magnetic field.
Acknowledgment
The authors acknowledge the financial support received under the Grant UKM-GUP-BTT-07-25-173 and from Universiti Kuala Lumpur (UniKL MICET).
BénardH.Les tourbillons cellulaires dans une nappe liquide19001112611271RayleighL.On convection currents in a horizontal layer of fluid when the higher temperature is on the other side191632529546PearsonJ. R. A.On convection cells induced by surface tension19584489500ChandrasekharS.1961Oxford, UKClarendon Pressxix+654The International Series of Monographs on PhysicsMR0128226ZBL0142.44103NieldD. A.Surface tension and buoyancy effects in the cellular convection of an electrically conducting liquid in a
magnetic field19661711311392-s2.0-000033177710.1007/BF01594092TakashimaM.Nature of the neutral state in convective instability induced by surface tension and buoyancy197028810DavisS. H.HomsyG. M.Energy stability theory for free-surface problems: buoyancy-thermocapillary layers1980983527553MR58306110.1017/S0022112080000274ZBL0434.76039CharM.-I.ChiangK.-T.Boundary effects on the Bénard-Marangoni instability under an electric field19945243313542-s2.0-002844992210.1007/BF00936836HashimI.WilsonS. K.The effect of a uniform vertical magnetic field on the onset of oscillatory Marangoni convection in a horizontal layer of conducting fluid19991321–41291462-s2.0-0032785943HashimI.On competition between modes at the onset of Bénard-Marangoni convection in a layer of fluid2002433387395MR1888304ZBL1003.76026HashimI.ishak_h@ukm.myMd ArifinN.The effect of a magnetic field on the linear growth rates of Bénard-Marangoni convection2005172582-s2.0-2342422527HashimI.ishak_h@ukm.mySiriZ.Stabilization of steady and oscillatory marangoni instability in rotating fluid layer by feedback control strategy20085466476632-s2.0-4934910930610.1080/10407780802289384Awang KechilS.HashimI.Control of Marangoni instability in a layer of variable-viscosity fluid20083510136813742-s2.0-000194624610.1016/j.icheatmasstransfer.2008.06.006SiriZ.HashimI.ishak_h@ukm.myControl of oscillatory Marangoni convection in a rotating fluid layer2008359113011332-s2.0-034551801610.1016/j.icheatmasstransfer.2008.06.008Awang KechilS.HashimI.ishak_h@ukm.myOscillatory Marangoni convection in variable-viscosity fluid layer: the effect of thermal feedback control2009486110211072-s2.0-4934910930610.1016/j.ijthermalsci.2008.11.008YangW.-M.Thermal instability of a fluid layer induced by radiation19901733653762-s2.0-0025418839GelfgatA. Y.TanasawaI.Numerical analysis of oscillatory instability of buoyancy convection with the Galerkin spectral method19942566276482-s2.0-0028444611Evren-SelametE.ArpaciV. S.ChaiA. T.Thermocapillary-driven flow past the Marangoni instability19942655215352-s2.0-0028547821EringenA. C.JohnsonJ. F.PorterR. S.Micropolar theory of liquid crystals19783New York, NY, USAPlenumRama RaoK. V.Thermal instability in a micropolar fluid layer subject to a magnetic field19801857417502-s2.0-0019178350Pérez-GarcíaC.RubíJ. M.On the possibility of overstable motions of micropolar fluids heated from below1982207873878MR65264110.1016/0020-7225(82)90009-XZBL0484.76015SharmaR. C.GuptaU.Thermal convection in micropolar fluids in porous medium19953313188718922-s2.0-0013363696RamdathG.Bénard-Marangoni instability in a layer of micropolar fluid19972242993102-s2.0-0001150271MurtyY. N.Ramana RaoV. V.Effect of throughflow on Marangoni convection in micropolar fluids199913832112172-s2.0-0017092112SiddheshwarP. G.pgsiddheshwar@hotmail.comSri KrishnaC. V.cvsrikrishna@hotmail.comLinear and non-linear analyses of convection in a micropolar fluid occupying a porous medium20033810156115792-s2.0-000019499010.1016/S0020-7462(02)00120-8Sunilsunil@nitham.ac.inChandP.BhartiP. K.MahajanA.Thermal convection in micropolar ferrofluid in the presence of rotation20083203-43163242-s2.0-3374736119610.1016/j.jmmm.2007.06.006FriedrichR.RudraiahN.Marangoni convection in a rotating fluid layer with non-uniform temperature gradient19842734434492-s2.0-0021386661RudraiahN.RamachandramurthyV.ChandnaO. P.Effects of magnetic field and non-uniform temperature gradient on Marangoni convection1985288162116242-s2.0-0022112814RudraiahN.RamachandramurthyV.Effects of non-uniform temperature gradient and Coriolis force on Bénard-Marangoni's instability1986611–437502-s2.0-002276293210.1007/BF01176361DupontO.HennenbergM.LegrosJ. C.Marangoni-Bénard instabilities under non-steady conditions. Experimental and theoretical results19923512323732442-s2.0-0026990319CharM.-I.ChenC.-C.Effects of nonuniform temperature gradients on the onset of oscillatory Marangoni convection in a magnetic field20031611-217302-s2.0-003215652510.1007/s00707-002-0985-yCharM.-I.ChenC.-C.Effect of a non-uniform temperature gradient on the onset of oscillatory Bénard-Marangoni convection of an electrically conducting liquid in a magnetic field2003411517111727MR198532810.1016/S0020-7225(03)00068-5ChiangK.-T.Effect of a non-uniform basic temperature gradient on the onset of Bénard-Marangoni convection: stationary and oscillatory analyses2005321-21922032-s2.0-002699031910.1016/j.icheatmasstransfer.2004.03.023RudraiahN.nrudraiah@hotmail.comSiddheshwarP. G.Effect of non-uniform basic temperature gradient on the onset of Marangoni convection in a fluid with suspended particles2000485175232-s2.0-0034274573SiddheshwarP. G.pgsiddheshwar@hotmail.comPraneshS.spranesh@hotmail.comMagnetoconvection in fluids with suspended particles under 1 g and μ g2002621051142-s2.0-002917869610.1016/S1270-9638(01)01144-0IdrisR.OthmanH.HashimI.ishak_h@ukm.myOn effect of non-uniform basic temperature gradient on Bénard-Marangoni convection in micropolar fluid20093632552582-s2.0-001656437610.1016/j.icheatmasstransfer.2008.11.009FinlaysonB. A.1972New York, NY, USAAcademic Press