# Effects of Magnetic Field and Nonlinear Temperature Profile on Marangoni Convection in Micropolar Fluid

Authors: M. N. Mahmud; R. Idris; I. Hashim
Journal: Differential Equations and Nonlinear Mechanics
 (2009)

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## Abstract

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 numberM 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.


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## Body

## 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 thicknessd  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]:
(2.1)∇·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 tensionσ  at the free upper surface is(2.2)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:(2.3)(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(2.4)(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 perturbationsW, Ωz, Θ, and Hz  are assumed to be periodic waves and hence these permit a normal mode solution in the following form:(2.5)[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
(2.6)(1+N1)(D2-a2)2W+N1(D2-a2)Ω+QPrPm(D2-a2)DHz=0,(2.7)N1(D2-a2)W-N3(D2-a2)Ω+2N1Ω=0,(2.8)(D2-a2)Θ+f(z)(W-N5Ω)=0,(2.9)(D2-a2)Hz+PmPrDW=0,
where D≡d/dz.EliminatingHz  between (2.6) and (2.9), we obtain(2.10)(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:
(2.11)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(2.12)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(2.13)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.Table 1
Reference steady-state temperature gradients.
ModelReference steady-statef(z)a1*a2*a3*temperature gradient1Linear11002Inverted parabolic2(1-z)0-103Cubic 13(z-1)20014Cubic 20.6+1.02(z-1)20.600.34


## 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(3.1)M=[〈(Dθ1)2〉+a2〈θ12〉][C1(C2-Q〈(DW1)2〉)+N12C32](1+N1)a2θ(1)DW(1)C4,
where
(3.2)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:(3.3)M=f4[f2{315(1+N1)f3+132Q}-315f12]630(1+N1)[f2f6-N5f1f5],
where
(3.4)f1=115N1(4+1128a2),f2=13(N3+110N3a2+15N1),(3.5)f3=45(21+2221a2+263a4),f4=43(1+25a2),(3.6)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 numberMc  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.Figures4–6 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.Figure 1
Plot ofMc  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.Figure 2
Plot ofMc  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.Figure 3
Plot ofMc  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.Figure 4
Plot ofMc  versus Q  for different temperature gradients with N3=2.0 and N5=1.0.Figure 5
Plot ofMc  versus Q  for different temperature gradients with N1=0.1 and N5=1.0.Figure 6
Plot ofMc  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.



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