Transp Porous Med (2012) 92:727–743 DOI 10.1007/s11242-011-9930-7
Effect of Internal Heat Generation on the Onset of Marangoni Convection in a Fluid Layer Overlying a Layer of an Anisotropic Porous Medium I. S. Shivakumara · S. P. Suma · R. Indira · Y. H. Gangadharaiah
Received: 26 May 2011 / Accepted: 15 December 2011 / Published online: 29 December 2011 © Springer Science+Business Media B.V. 2011
Abstract Linear stability analysis has been performed to investigate the effect of internal heat generation on the criterion for the onset of Marangoni convection in a two-layer system comprising an incompressible fluid-saturated anisotropic porous layer over which lies a layer of the same fluid. The upper non-deformable free surface and the lower rigid surface are assumed to be insulated to temperature perturbations. The fluid flow in the porous layer is governed by the modified Darcy equation and the Beavers–Joseph empirical slip condition is employed at the interface between the two layers. The resulting eigenvalue problem is solved exactly. Besides, analytical expression for the critical Marangoni number is also obtained by using regular perturbation technique with wave number as a perturbation parameter. The effect of internal heating in the porous layer alone exhibits more stabilizing effect on the system compared to its presence in both fluid and porous layers and the system is least stable if the internal heating is in fluid layer alone. It is found that an increase in the value of mechanical anisotropy parameter is to hasten the onset of Marangoni convection while an opposite trend is noticed with increasing thermal anisotropy parameter. Besides, the possibilities of controlling (suppress or augment) Marangoni convection is discussed in detail. Keywords Marangoni convection · Composite layer · Anisotropic porous layer · Internal heat generation
I. S. Shivakumara UGC Centre for Advanced Studies in Fluid Mechanics, Department of Mathematics, Bangalore University, Bangalore 560 001, India e-mail:
[email protected] S. P. Suma (B) · Y. H. Gangadharaiah Department of Mathematics, New Horizon College of Engineering, Bangalore 560 103, India e-mail:
[email protected] Y. H. Gangadharaiah e-mail:
[email protected] R. Indira Department of Mathematics, Nitte Meenakshi Institute of Technology, Bangalore, India
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List of Abbreviations Variables √ a Horizontal wave number, l 2 + m 2 A Ratio of heat capacity D Differential operator d/dz d Thickness of the fluid layer dm Thickness of the porous layer 2 Da Darcy number K v /dm h Heat transfer coefficient K Permeability tensor ∼ l, m M p Pr Pr m T T0 V W qf qm Q Qm κ
Wave number in x and y-directions, respectively Marangoni number σT (T0 − Tu )d/μκ Pressure Prandtl number for fluid layer, v/κ Porous medium Prandtl number, vφ/κmv Temperature Temperature at the interface Velocity vector (u, v, w) Amplitude of perturbed vertical velocity Heat source in fluid layer Heat source in porous layer Dimensionless heat source in the fluid layer qf d 2 /2κ(T0 − Tu ) 2 /2κ (T − T ) Dimensionless heat source in the porous layer qm dm mv l 0 Thermal diffusivity
Greek Symbols β Slip parameter ∇h2 Horizontal Laplacian operator ∇h2 = ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 ∇ 2 Laplacian operator ∇ 2 = ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 + ∂ 2 /∂z 2 εT Ratio of thermal diffusivities κ/κmv η Thermal anisotropy parameter κmh /κmv φ Porosity of the porous medium κ Thermal diffusivity ˆ + κmv kˆ kˆ κ Effective thermal diffusivity tensor κmh (iˆiˆ + jˆ j) m ∼ θ μ ρ0 σ v ξ ζ
Amplitude of perturbed temperature Fluid viscosity Fluid density Temperature-dependent surface tension Kinematic viscosity μ/ρ0 Mechanical anisotropy parameter K h /K v Depth ratio d/dm
Subscripts b Basic state h Horizontal
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Effect of Internal Heat Generation
l m u mv mh
729
Lower Porous medium Upper Vertical direction Horizontal direction
1 Introduction Buoyancy and/or surface tension-driven convection in a composite fluid layer overlying a layer of isotropic porous medium saturated with the same fluid has been investigated by several authors because of its relevance and importance in the manufacture of composite materials used in aircraft structures and automobile industries, geophysics, bioconvection, nuclear reactors, solid-matrix heat exchangers, crystal growth, directional solidification of alloys, aerosol production, and electronics cooling to mention a few (Nield 1977; Vasseur et al. 1989; Taslim and Narusawa 1989; Chen 1990; McKay 1998; Nield 1998; Straughan 2001, 2002; Khalili et al. 2001; Desaive et al. 2001; Shivakumara et al. 2006; Shivakumara and Chavaraddi 2007; Alloui and Vasseur 2010). Majority of the studies pertaining to convective instability in superposed fluid and porous layers system considered the porous medium to be isotropic. Nonetheless, many porous materials are anisotropic in their mechanical and thermal properties as a consequence of a preferential orientation or asymmetric geometry of a grain. Anisotropy can also be a characteristic of artificial porous materials like pelletting used in chemical engineering process and fiber material used in insulating purposes. During the solidification of alloys a dendritic region known as mushy zone separating the melt from the solid forms and this region is regarded as a porous medium in which the permeability and possibly the thermal conductivity may be anisotropic. Realizing the importance of porous medium anisotropy on the onset of convection, Chen et al. (1991) have discussed the onset of buoyancy-driven convection in a system consisting of a fluid layer overlying a porous layer with anisotropic permeability and thermal diffusivity. An exhaustive review on this topic can be found in the book by Nield and Bejan (2006). Recently, Shivakumara et al. (2011) have investigated the criterion for the onset of surface tension-driven convection in the presence of temperature gradients in a two-layer system comprising a fluid-saturated anisotropic porous layer over which lies a layer of fluid. It is shown that decreasing the mechanical anisotropy parameter and increasing the thermal anisotropy parameter leads to stabilization of the system. When the fluid is heated internally by a uniform distribution of heat sources a thermally unstable situation arises, similar to that caused by the decay of radioactive matter in the earth’s mantle. Moreover, an internal heating can give rise to a situation where one part of a fluid/fluid-saturated porous layer is naturally convecting while the other remains stable, hence penetrative convection can occur. Straughan (1993) has surveyed the circumstances under which this type of convection can occur. In particular, convective instability due to volumetric heating has received considerable attention in the recent past because of its prevalence in geophysics and energy-related engineering problems, such as heat removal from nuclear fuel debris, underground disposal of radioactive waste materials, storage of food-stuff, exothermic chemical reactions in packed-bed reactor and so on. Many references can be found in which the effect of internal heat generation on the onset of Benard and/or Marangoni convection in a fluid or a porous layer is investigated (Lam and Bayazitoglu 1987; Char and Chiang 1994; Straughan and Walker 1996; Hill et al. 2007; Capone et al. 2008; Shivakumara et al.
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2009, 2010 and references therein). However, the corresponding problem in superposed fluid and porous layers has not received due attention in the literature despite its relevance and importance in numerous important applications as mentioned above. Somerton and Catton (1982) have presented a solution for the problem of predicting the onset of convection for a system consisting of a volumetrically heated isotropic porous bed saturated with and overlaid with a fluid, heated or cooled from below. Carr and Straughan (2003) have employed a quadratic equation of state to simulate penetrative convection in the porous medium–fluid system. Carr (2004) has studied penetrative convection via internal heating in a composite two-layer system in which a layer of fluid overlies and saturates a layer of isotropic porous medium. It is found that a heat source/sink in the fluid layer has a destabilizing effect on the porous layer, whereas one in the porous medium has a stabilizing influence on the fluid and the behavior is explained and illustrated with a range of streamlines. The objective of this study is to analyze the effect of internal heat generation on the criterion for the onset of surface tension-driven convection in an anisotropic porous layer over which lies a layer of fluid since the internal heating changes convective motions drastically. Both mechanical and thermal anisotropies of the porous medium are considered in investigating the problem. A modified Darcy equation is employed to describe the flow regime in the anisotropic porous medium and at the interface of porous and fluid media the Beavers–Joseph classical slip condition is used. The lower rigid and upper free boundaries are considered to be insulated to temperature perturbations. The resulting eigenvalue problem is solved exactly and also by regular perturbation technique with wave number a as a perturbation parameter.
2 Formulation of the Problem The physical configuration consists of a horizontal anisotropic porous layer of thickness dm underlying a fluid layer of thickness d with no lateral boundaries as shown in Fig. 1. The lower boundary of the anisotropic porous layer is taken to be rigid, while the upper free boundary of the fluid layer at which the surface tension acts is assumed to be non-deformable since for most liquids the capillary number is very small, commonly ranging from 10−6 to 10−2 . The surface tension is assumed to vary linearly with temperature in the form: σ = σ0 − σT (T − T0 ),
(1)
where σ0 is the unperturbed value and −σT is the rate of change of surface tension with temperature. The temperature of the lower and upper boundaries is taken to be uniform and
Fig. 1 Physical model
123
Effect of Internal Heat Generation
731
equal to Tl and Tu respectively with Tl > Tu . A Cartesian coordinate system (x, y, z) is chosen such that the origin is at the interface between the fluid layer and the anisotropic porous layer and the z-axis is vertically upward. The governing equations for the fluid layer are: ρ0
∇ · V = 0 ∂V + (V · ∇)V = −∇ p + μ∇ 2 V ∂t ∂T + (V · ∇)T = κ∇ 2 T + qf ∂t
(2) (3) (4)
and those for porous layer are:
A
∇m · Vm = 0
(5)
ρ0 ∂ Vm −1 = −∇m pm − μK · Vm ∼ φ ∂t
(6)
∂ Tm · ∇m Tm ) + qm . + (Vm · ∇m )Tm = ∇m · (κ ∼m ∂t
(7)
Here, V is the velocity vector, p is the pressure, T is the temperature, and qf is the heat source in the fluid layer, while Vm , pm , Tm , and qm are the corresponding quantities in the porous layer, κ is the thermal diffusivity, μ is the fluid viscosity, φ is the porosity of the porous medium, A is the ratio of heat capacities, ρ0 is the fluid density, K is the permeability tensor, ∼ and ∼ κ m is the thermal diffusivity tensor.The permeability and thermal diffusivity tensors of the porous medium are assumed to be constants and have principal axis aligned with the coor−1 = K −1 iˆiˆ + K −1 jˆ jˆ + K −1 kˆ kˆ and κ = κ iˆiˆ + κ ˆ dinate system so that K mx my jˆ jˆ + κmv kˆ k. x y v ∼ ∼m We restrict to horizontal isotropic porous media and consider K x = K y (= K h ) and κmx = κmy (= κmh ). It may be noted that the permeability and effective thermal diffusivity in the horizontal and vertical directions in an anisotropic porous layer are denoted by K h , κmh and K v , κmv , respectively. The basic steady state is assumed to be quiescent and temperature distributions are found to be: (T0 − Tu ) qf d qf 2 Tb (z) = T0 − − z+ z 0≤z≤d (8) d 2κ 2κ (Tl − T0 ) qm dm qm 2 zm + − dm ≤ z m ≤ 0, − z (9) Tmb (z m ) = T0 − dm 2κmv 2κmv m
where T0 = [2(κmv Tl d + κ Tu dm ) + ddm (qf d + qm dm )]/2(κmv d + κdm ) is the interface temperature and suffix b denotes the basic state. To investigate the stability of the basic solution, infinitesimal disturbances are introduced in the form: V = V , T = Tb (z) + T , p = pb (z) + p Vm = Vm , Tm = Tmb (z) + Tm , pm = pmb (z) + pm ,
(10) (11)
where the primed quantities are the perturbations and assumed to be small. Equations 10 and 11 are substituted in Eqs. 2–7 and linearized in the usual manner. The pressure term is eliminated from Eqs. 3 and 6 by taking curl twice on these two equations and only the vertical component is retained. The variables are then non-dimensionalized using d, d 2 /κ, κ/d and T0 − Tu as the units of length, time, velocity, and temperature in the fluid layer and Tl − T0 as the corresponding characteristic quantities in the porous layer. Note that separate length
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scales are chosen for the two layers so that each layer is of unit depth and the thermal diffusivity in the vertical direction is used as a reference value to render the quantities dimensionless in the porous layer. In this manner, the detailed flow fields in both the fluid and porous layers can be clearly discerned for all depth ratios ζ = d/dm and the non-dimensional disturbance equations are now given by: 1 ∂ (12) − ∇2 ∇2W = 0 Pr ∂t ∂ (13) − ∇ 2 T = W [1 − Q(1 − 2z)] ∂t Da ∂ ∂2 2 + ξ ∇mh + 2 Wm = 0 (14) Prm ∂t ∂z m ∂2 ∂ 2 − 2 Tm = Wm [1 + Q m (1 + 2z m ], (15) A − η∇mh ∂t ∂z m 2 is the Darcy number, Q = q d 2 /2κ (T − T ) and Q = q d 2 /2κ where Da = K v /dm f v 0 u m m m mv (Tl − T0 ) are the dimensionless heat source strengths in the fluid and porous layers, respectively, Pr = v/κ is the Prandtl number, Pr m = vφ/κmv is the porous medium Prandtl number, ξ = K h /K v is the mechanical anisotropy parameter and η = κmh /κmv is the thermal anisotropy parameter. The boundary conditions are:
W =
∂T = 0 at z = 1 ∂z
∂2W = M∇h2 T at z = 1 ∂z 2 Wm =
∂ Tm = 0 at z m = −1. ∂z m
(16)
(17)
(18)
Here, M = σT (T0 − Tu )d/μκ is the Marangoni number. At the interface (i.e., z = 0) the normal component of velocity, temperature, heat flux, and normal stress are continuous. Since, there is no viscous stress term in the Darcy equation, continuity of shear stress across the interface cannot be used. Instead, we use the experimentally suggested slip condition proposed by Beavers and Joseph (1967). Accordingly, the following conditions at the interface are used: ζ Wm εT εT T = Tm ζ ∂T ∂ Tm = ∂z ∂z m 2 ∂W ∂ ζ 4 ∂ Wm 3∇h2 + 2 =− ∂z ∂z εT Daξ ∂z m W =
∂2W βζ ∂ W βζ 3 ∂ Wm = √ , − √ 2 ∂z Daξ ∂z εT Daξ ∂z m
123
(19) (20) (21) (22) (23)
Effect of Internal Heat Generation
733
where εT = κ/κmv is the ratio of thermal diffusivities β is the slip parameter and ∇h2 = ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 is the horizontal Laplacian operator. Since, the principle of exchange instability holds for surface tension-driven convection either in fluid layer (see Pearson 1958; Vidal and Acrivos 1966) or in a porous layer (Hennenberg et al. 1997; Rudraiah and Prasad 1998), it is reasonable to assume that it holds good even for the present configuration as well. Further, the numerical calculations carried out for a wide range of parameters by Straughan (2001) also corroborates the validity of principle of exchange of stability for the superposed system. Hence, the time derivatives are dropped conveniently from Eqs. 12–15. Then performing a normal mode expansion of the dependent variables in both fluid and porous layers as: (W, T ) = [W (z), θ (z)] exp[i(lx + my)] ˜ + m˜ y)] (Wm , Tm ) = [Wm (z), θm (z)] exp[i(lx
(24) (25)
and substituting them in Eqs. 12–15 (with ∂/∂t = 0), we obtain the following ordinary differential equations: (D2 − a 2 )2 W = 0
(26)
(D − a )θ = −W [1 − Q(1 − 2z)]
(27)
2
2
(D2m (D2m
2 − ηam )θm
2 − ξ am )Wm
=0
(28)
= −Wm [1 + Q m (1 + 2z m )]
(29) √ 2 2 where D and Dm denote differentiation with respect to z and z m , respectively, a = l + m 2 2 ˜ and am = l + m˜ are correspondingly the overall horizontal wave numbers in the fluid and porous layers. If matching of the solutions in the two layers is to be possible, the wave numbers must be the same for the fluid and porous layers, so that we have a/d = am /dm and hence ζ = a/am . The ten boundary conditions after using Eqs. 24 and 25 take the form: W = Dθ = 0 at z = 1
(30)
D2 W + Ma2 θ = 0 at z = 1
(31)
Wm = Dm θm = 0 at z m = −1
(32)
and those at the interface (i.e., z = 0) are ζ Wm εT Dθ = Dm θm εT θm θ= ζ −ζ 4 Dm Wm [D2 − 3a 2 ]DW = DaεT ξ −βζ 3 βζ D W = √ Dm W m . D2 − √ Daξ εT Daξ W =
(33) (34) (35) (36) (37)
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I. S. Shivakumara et al.
3 Method of Solution The system of basic governing equations (26–29) along with the boundary and interfacial conditions (30–37) constitutes an eigenvalue problem which is solved exactly with M as an eigenvalue. Besides, an analytical expression for the critical Marangoni number is also obtained by regular perturbation technique with wave number as a perturbation parameter. 3.1 Exact Solution It is possible to solve Eqs. 26 and 28 directly as they are independent of θ and θm . Therefore, solving these equations we get the general solution in the form: W = A1 cosh(az) + A2 sinh(az) + A3 z cosh(az) + A4 z sinh(az)
Wm = Am1 cosh( ξ am z m ) + Am2 sinh ξ am z m
(38) (39)
where A1 − A4 , Am1 and Am2 are constants determined using the boundary conditions given by Eqs. 30, 32, 34, 36, and 37 to obtain W and Wm as: W = A1 [ 1 cosh(az) + 2 sinh(az) + 3 z cosh(az) + 4 z sinh(az)]
ξ am z m + coth ξ am sinh ξ am z m Wm = A1 cosh
(40) (41)
where
1 =
2 = k1 coth
ζ εT
ξ am
√ √ k3 coth ξ am sinh a +2a 1 cosh a − 1 a 2 sinh a +k1 a sinh a coth ξ am (k2 − 2)
3 = (k2 sinh a −2a cosh a)
√ k1 coth ξ am sinh a + 3 cosh a + 1 cosh a
4 = − sinh a k1 =
ζ 4 am √ 3 2a DaεT ξ
−βζ k2 = √ Daξ
βζ 3 am . k3 = − √ DaεT
Substituting for W and Wm thus obtained in Eqs. 27 and 29, respectively, and using coupled boundary condition given by the Eq. 31, an expression for Marangoni number is finally obtained in the form:
M =−
k1 a 2 coth
√
ξ am +2a 3 + 4 a 2 sinh a + (2a 4 + 3 a 2 + 1 a 2 ) cosh a (k14 sinh a + k15 cosh a) (42)
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Effect of Internal Heat Generation
where
1 k1
4
3 3 − − Q + ξ a coth ξ a − m m 2a 4a 2 2a 2 2a 4a 2 2a 3 √
coth ξ am − Qm ξ am coth ξ am − 1 k5 = √ 2 ξ am k = k4 − k5 6 √ √ √ √ √ 1 ξ am sinh ξ am coth ξ am m + cosh √ ξ am + √ coth ξ a√ = √ − sinh ξ am − ξ am cosh ξ am 2 ξ am
Q m 2 −1 ξ a coth ξ a k72 = m m 2 2ξ am
−2ξ am ) cosh ξ am + coth ξ am sinh ξ am
k4 = k1
k71
√
735
coth
ξ am
−
k7 = k71 − k72
a
1 a k1
4 2 3 k81 = ξ am + + 2 coth + (a − 1) 2 2a 4 4
a k1
3
4 a 2
4 a 3 ξ am − coth − − + sinh a −Q 2 4 2a 12 4
a
1 a 2 k1
3 2 4 ξ am + + coth + (a − 1) k82 = 2 2a 4 4
1
3
4
3 a 2 −Q + − + cosh a 2 4 2a 12 k8 = k81 + k82
k9 = k6 coth ξ am − k 7 √ √ √ k10 = a 3 1 ηam cosh a + sinh ηam + a 4 cosh ηam sinh a √ √ k11 = 1 k8 ηam sinh ηam + a 3 k9 sinh a k11 k12 = k10 k13 = k8 − a 3 k12 cosh a
a 3
4 k1 a 3
4 a 1 4 − + +Q coth − − = a 2 k12 − ξ am + 2 4 4 2 4 12 2a
ak a
a 1 3 4 1 3 4 4 2 = a k13 − ξ am + coth − +Q + − − 2 4 4 2 4 2a 12
k14 k14
3.2 Solution by Regular Perturbation Technique For the assumed boundary conditions the eigenvalue problem is solved by using regular perturbation technique with wave number as a perturbation parameter. This method provides a cross verifications of the results obtained by solving the eigenvalue problem exactly. The dependent variables in both the fluid and porous layers are now expanded in powers of in the form: (W, θ ) =
N (a 2 )i (Wi , θi )
(43)
i=0
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I. S. Shivakumara et al.
(Wm , θm ) =
N 2 i a i=0
ζ2
(Wmi , θmi ).
(44)
Substitution of Eqs. 43 and 44 into Eqs. 26–29 and the boundary conditions 30–37 yields a sequence of equations for the unknown functions Wi (z), θi (z), Wmi (z) and θmi (z) for i = 0, 1, 2, . . . At the leading order in Eqs. 26–29 become, respectively, D4 W0 = 0
(45)
D θ0 = −W0 f (z)
(46)
D2m Wm0 = 0
(47)
D2m θm0 = −Wm0 g(z)
(48)
2
and the boundary conditions 30–37 become Wm0 = 0, Dm θm0 = 0, at z m = −1
(49)
W0 = 0, Dθ0 = 0, D W0 = 0 at z = 1.
(50)
2
And at the interface ζ Wm0 εT εT θm0 θ0 = ζ Dθ0 = Dm θm0 βζ −βζ 3 DW0 = √ Dm Wm0 D2 W0 − √ Daξ εT Daξ −ζ 4 Dm Wm0 . D3 W0 = εT Daξ W0 =
(51) (52) (53) (54) (55)
The solution to the zeroth order equations is given by: εT ζ = 1.
W0 = 0, θ0 = Wm0 = 0, θm0
(56) (57)
At the first order in, Eqs. 26–29 then reduce to D4 W 1 = 0 εT D2 θ1 − = −W1 f (z) ζ
(58)
D2m Wm1 = 0
(60)
D2m θm1
− η = −Wm1 g(z).
(59)
(61)
and the boundary conditions 30–37 become Wm1 = 0,
Dm θm1 = 0, at z m = −1 εT W1 = 0, Dθ1 = 0, D2 W1 + M = 0 at z = 1. ζ
123
(62) (63)
Effect of Internal Heat Generation
737
And at the interface 1 Wm1 ζ εT εT θ1 = 3 θm1 ζ 1 Dθ1 = 2 Dm θm1 ζ βζ −βζ DW1 = √ Dm Wm1 D2 W1 − √ Daξ εT Daξ −ζ 2 Dm Wm1 . D3 W 1 = εT Daξ W1 =
(64) (65) (66) (67) (68)
Integrating Eq. 59 between z = 0 and 1, and Eq. 61 between z m = −1 and 0, using the relevant boundary conditions and adding the resulting equations, we obtain the following solvability condition: 1 0
1 f (z)W1 dz + 2 ζ
0 g(z)Wm1 dz = −1
η εT + 2, ζ ζ
(69)
where f (z) = [1 − Q(1 − 2z)] and g(z) = [1 + Q m (1 + 2z)]. The general solution of Eqs. 58 and 60 are, respectively, given by: W1 = M[P1 + P2 z + P3 z 2 + P4 z 3 ]
(70)
Wm1 = M[P5 + P6 z m ].
(71)
The constants P1 –P6 are determined using the boundary conditions and are found to be: −βζ −ζ 2 −εT (2 − c2 εT ) , c3 = , c4 = , c5 = √ 6ε Daξ ζ (c c + c2 εT + c1 c2 c3 εT ) εT Daξ T 1 2 c4 P1 = P3 c5 , P3 = , P4 = c1 c3 c5 P3 , P2 = −(c5 P1 + P3 + P4 ), (2 + 6c1 c3 c5 ) P5 = P6 = c1 c3 P3 . c1 = εT ζ, c2 =
Substituting for W1 and Wm1 from Eqs. 70 and 71, respectively, in Eq. 69 and performing the integration, we obtain an expression for the critical Marangoni number MC in the form:
η εT ζ + ζ2 MC = . (72) P1 + P2 (Q + 3)/6 + P3 (Q + 2)/6 + P4 (3Q + 5)/20 + P5 (3 − Q m )/6 The critical Marangoni number MC for various values of ζ, Q, Q m , Da, εT , β, ξ , and η is computed and the results are exhibited graphically in Tables 1 and 2 and also in Figs. 2, 3, 4, and 5.
4 Results and Discussion The effect of internal heat generation on the criterion for the onset of Marangoni convection in a fluid-saturated anisotropic porous layer over which lies a layer of the same fluid is investigated theoretically. The resulting eigenvalue problem is solved exactly. It is observed
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I. S. Shivakumara et al.
Table 1 Comparison of critical Marangoni number with those of Shivakumara et al. (2011) for different values of Q and ζ when Da = 0.001, η = 0.5 = ξ , εT = 0.725, β = 1, and Q m = 0 ζ
Shivakumara et al. (2011)
MC (Present study) with Q m = 0
MC
Q=0
Q=2
Q=4
Q=6
Q=8
0.5
68.934
68.934
64.004
55.14
48.407
43.150
1
72.414
72.414
54.791
43.626
36.242
30.995
1.5
66.136
66.136
48.520
38.195
31.493
26.792
2
62.091
62.091
45.1072
35.369
29.090
24.704
2.5
59.465
59.465
43.000
33.652
27.642
23.454
Table 2 Critical values of Marangoni number for different values of Q and Q m when Da = 0.001, η = 0.5, ξ = 0.5, εT = 0.725, and β = 1 Qm
Q
0
ζ = 0.5
ζ = 0.6
ζ = 0.8
ζ =1
ζ = 10
ζ = 50
MC
MC
MC
MC
MC
MC
0
68.934
78.038
76.635
72.414
50.967
48.497
1
69.128
68.178
66.749
62.831
42.536
40.424 25.598
5
51.548
48.452
43.195
39.593
34.268
100
7.196
5.800
4.607
4.046
2.447
2.312
0
79.803
81.042
77.979
73.920
50.967
48.497
1
1
72.520
71.745
67.266
63.882
42.536
40.424
5
53.126
49.404
43.294
39.007
34.268
25.598
100
7.226
5.853
4.616
4.051
2.447
2.312
0
87.887
85.225
79.196
74.156
50.967
48.497
3
1
79.134
75.286
68.452
63.092
42.536
40.424
5
56.59
50.620
43.6797
39.863
34.268
25.598
5.822
4.641
4.059
2.447
2.312
100
7.2873
that the critical wave number is exceedingly small and this fact is exploited as well to obtain the solution using a regular perturbation technique with wave number a as a perturbation parameter. It is noted that the results obtained from these two methods complement with each other. The critical Marangoni number computed for different values of Q and ζ when Q m = 0 (i.e., absence of internal heat source in the porous layer), εT = 0.725, Da = 0.001, β = 1, ξ = 0.5, and η = 0.5 are tabulated in Table 1. The results of Shivakumara et al. (2011) are also exhibited in the Table for the sake of comparison. It is seen that our results are in good agreement with those of Shivakumara et al. (2011) in the absence of internal heat generation (i.e., when Q = 0). Further, the system becomes destabilizing with increasing Q (i.e., the value of critical Marangoni number MC decreases as Q increases) because of increase in the energy supply to the system. The critical values of MC for different values of ζ , Q, and Q m are tabulated in Table 2 when εT = 0.725, Da = 0.001, β = 1, ξ = 0.5, and η = 0.5 since our main interest is to look at the dramatic effects of internal heating in fluid and/or porous layers on the onset of Marangoni convection. From the Table it is seen that (i) the critical Marangoni number
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Fig. 2 Variation of MC with ζ for different values of Da when η = 0.5 = ξ , εT = 0.725, and β = 1
Fig. 3 Variation of MC with ξ for different values of Q, Q m , and η when β = 1 for ζ = 1, εT = 0.725, and Da = 0.003
MC reaches a maximum value with increasing ζ depending on the strengths of Q and Q m , (ii) with increasing values of Q and/or Q m , MC decreases monotonically as the value of ζ increases, (iii) increasing Q m is to delay the onset of Marangoni convection, and (iv) the values of MC for ζ = 10 and 50 only vary with Q but remain the same for all values of Q m . That is, the internal heating in the porous layer has no effect on the onset of Marangoni convection when the depth of the fluid layer is dominant compared to the porous layer depth. The influence of permeability of the porous medium on the onset of Marangoni convection is exhibited in Fig. 2 when εT = 0.725, β = 1, ξ = 0.5, and η = 0.5 for the cases of internal heating in fluid and/or porous layer. Three cases have been considered in analyzing
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Fig. 4 Variation of MC with ξ for different values of Q, Q m , and β for ζ = 1, εT = 0.725, and Da = 0.003
Fig. 5 Perturbed velocity eigen functions W and Wm for different values of Q and Q m with β = 1 for η = 0.5, ζ = 1, εT = 0.725, and Da = 0.003
the problem, namely (i) Q m = 2, Q = 0 (internal heating only in the porous layer), (ii) Q = 2, Q m = 0 (internal heating only in the fluid layer), and (iii) Q = 2, Q m = 2 (internal heating in both fluid and porous layers). The curves of MC for different D for case (i) lie above case (iii), while those of case (ii) lie below case (iii). Thus, it indicates that the presence of volumetric distribution of heat source in the fluid layer alone is to hasten and in the porous layer alone is to delay the onset of Marangoni convection when compared to the simultaneous
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presence of volumetric internal heating in both fluid and porous layers. It is further seen that decrease in the value of Da is to increase the critical Marangoni number and hence its effect is to delay the onset of Marangoni convection. Besides, the curves of MC for case (ii) and (iii) and also for different D coalesce at higher values of ζ . The influence of porous medium anisotropy on the stability characteristics of the system is exhibited in Fig. 3. The figure shows the variation of MC as a function of mechanical anisotropy parameter ξ for different values of η when Q = 2 = Q m , Q = 2, Q m = 0 and Q = 0, Q m = 2 with ζ = 1 = β, εT = 0.725 and Da = 0.003. It is noted that MC attains higher values at lower values of ξ for the above-said combinations of internal heating. That is, decrease in the mechanical anisotropy parameter is to delay the onset of Marangoni convection. This is because, decrease in ξ corresponds to smaller horizontal permeability which in turn hinder the motion of fluid in the horizontal direction. As a consequence, the conduction process in the porous medium becomes more stable and hence higher values of MC are needed for the onset of Marangoni convection. To the contrary, in the same figure it is observed that decreasing η is to hasten the onset of Marangoni convection. This may be attributed to the fact that the decrease in η amounts to decrease in the horizontal thermal diffusivity. Thus, heat cannot be transported through the porous layer and hence the horizontal temperature variations in the fluid required to sustain convection are less efficiently dissipated for small η. Hence, the base state becomes less stable leading to lower values of critical Marangoni number. The effect of slip parameter β on the stability characteristics of the system is made clear in Fig. 4 by displaying MC as a function of ξ for different values of β when η = 1, ζ = 1, εT = 0.725, and Da = 0.003 for different combinations of internal heating, namely Q = 2 = Q m , Q = 2, Q m = 0 and Q = 0, Q m = 2. It is evident that increase in the value of slip parameter is to delay the onset of Marangoni convection. Figure 5 depicts the perturbed vertical velocity eigen functions W and Wm for different values of Q and Q m for β = 1, η = 0.5 = ξ , ζ = 1, εT = 0.725, and Da = 0.003. The presence of volumetric heating has no noticeable influence on Wm and the presence of internal heating in the porous layer alone is to accelerate W compared to its presence in the fluid layer as well as in both fluid and porous layers.
5 Conclusions The effect of internal heating on the onset of Marangoni convection in two-layer system comprising an incompressible fluid-saturated anisotropic porous layer over which lies a layer of the same fluid is analyzed theoretically. The effect of internal heating in a fluid layer alone and a porous layer alone is to respectively hasten and delay the onset of Marangoni convection in superposed fluid-porous layers system. The effect of increasing internal heating in a porous layer is found to have no influence on the stability characteristics of the system if the depth of the porous layer is small compared to the fluid layer thickness (i.e., as the value of ζ increases). It is observed that the mechanical and thermal anisotropy parameters influence the stability of the system significantly. Increasing the mechanical anisotropy parameter has a destabilizing effect on the system, while an opposite trend is noticed with an increase in the value of thermal anisotropy parameter. The effect of increasing slip parameter is to delay the onset of Marangoni convection. Thus, it is possible to either augment or suppress the onset of Marangoni convection by suitably choosing the parametric values.
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Acknowledgments The authors S. P. Suma and Y. H. Gangadharaiah express their heartfelt thanks to the management of New Horizon College of Engineering, Bangalore for their encouragement. We thank the reviewers for their constructive comments which have helped in improving the quality of the article.
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