Transp Porous Med (2011) 90:509–528 DOI 10.1007/s11242-011-9797-7
The Onset of Darcy–Brinkman Electroconvection in a Dielectric Fluid Saturated Porous Layer I. S. Shivakumara · N. Rudraiah · Jinho Lee · K. Hemalatha
Received: 23 September 2010 / Accepted: 18 June 2011 / Published online: 6 July 2011 © Springer Science+Business Media B.V. 2011
Abstract The combined effect of a vertical AC electric field and the boundaries on the onset of Darcy–Brinkman convection in a dielectric fluid saturated porous layer heated either from below or above is investigated using linear stability theory. The isothermal bounding surfaces of the porous layer are considered to be either rigid or free. It is established that the principle of exchange of stability is valid irrespective of the nature of velocity boundary conditions. The eigenvalue problem is solved exactly for free–free (F/F) boundaries and numerically using the Galerkin technique for rigid–rigid (R/R) and lower-rigid and upper-free (F/R) boundaries. It is observed that all the boundaries exhibit qualitatively similar results. The presence of electric field is emphasized on the stability of the system and it is shown that increasing the AC electric Rayleigh number Rea is to facilitate the transfer of heat more effectively and to hasten the onset of Darcy–Brinkman convection. Whereas, increase in the ratio of viscosities and the inverse Darcy number Da −1 is to delay the onset of Darcy– Brinkman electroconvection. Besides, increasing Rea and Da −1 as well as decreasing are to reduce the size of convection cells. Keywords
Electroconvection · Dielectric fluid · Porous medium · Viscosity ratio
I. S. Shivakumara · J. Lee (B) School of Mechanical Engineering, Yonsei University, Seoul, 120-749, South Korea e-mail:
[email protected] I. S. Shivakumara · N. Rudraiah UGC-Centre for Advanced studies in Fluid Mechanics, Department of Mathematics, Bangalore University, 560 001 Bangalore, India e-mail:
[email protected] K. Hemalatha Department of Mathematics, Mount Carmel College, 560 052 Bangalore, India
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List of Symbols √ a = 2 + m 2 d D = d/dz Da = k/d 2 E g kˆ k , m M p Pr = νφ/κ q = (u, v, w) Rea = η2 ε0 E 02 ( T )2 d 2 /μκ ReaD (= Rea Da) Rt = αg T d 3 /νκ RtD (= Rt Da) t T T0 T1 V W (x, y, z)
I. S. Shivakumara et al.
Overall horizontal wave number Thickness of the porous layer Differential operator Darcy number Electric field Acceleration due to gravity Unit vector in z-direction Permeability of the porous medium Wave numbers in the x and y directions Ratio of heat capacities Pressure Prandtl number Velocity vector AC electric Rayleigh number AC electric Darcy–Rayleigh number Thermal Rayleigh number Thermal Darcy–Rayleigh number Time Temperature Temperature of the lower boundary Temperature of the upper boundary Electric potential Amplitude of vertical component of perturbed velocity Cartesian co-ordinates
Greek Symbols α ∇ 2 = ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 + ∂ 2 /∂z 2 ∇h2 = ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 T = T0 − T1 ε η κ = μ/μ ˜ μ μ˜ ν = μ/ρ0 φ
ρ ρe ρ0 σ ω = ωr + iωi
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Thermal expansion coefficient Laplacian operator Horizontal Laplacian operator Temperature difference between the lower upper boundaries Dielectric constant Thermal expansion coefficient of dielectric constant Effective thermal diffusivity of the fluid Ratio of viscosities Dynamic viscosity of the fluid Effective viscosity Kinematic viscosity Porosity of the porous medium Amplitude of perturbed electric potential Fluid density Free charge density Reference density at T0 Amplitude of perturbed temperature Electrical conductivity Growth rate
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Subscripts/Superscripts b Basic state c Critical f Fluid s Solid
1 Introduction Thermal instability in a porous layer heated uniformly from below in the presence of gravitational field is due to buoyancy and is referred to as Darcy–Bénard convection. Since the pioneering work of Horton and Rogers (1945) and Lapwood (1948) an extensive study has been made on thermal convection in fluid saturated porous media because of its natural occurrence as well as applications in many science, engineering and technological problems such as biomedical engineering applications, drying processes, thermal insulation, radioactive waste management, transpiration cooling, geophysical systems, and contaminant transport in groundwater, ceramic processing, solid-matrix compact heat exchangers, and many others. The copious literature covering different developments in this field is well documented in the literature; see, for example, Bear (1988), Kaviany (1995), Ingham (1998), Rees (2000), Vafai (2000), Vafai (2005), Tyvand (2002), Bejan et al. (2004), Ingham et al. (2004), Shivakumara and Venkatachalappa (2004) and Nield and Bejan (2006). Many convective instability problems occurring in practical problems often involve electrically conducting fluids. In such cases the effects of external constraint of fields like magnetic field and/or electric field become important. In particular, the effects of magnetic field become dominant on convective instability when the fluid is either finitely or highly electrically conducting and such instability in a horizontal fluid layer has been studied in detail (see Chandrasekhar 1961). Its counterpart in a porous medium has also been investigated in the past (Patil and Rudraiah 1973; Rudraiah and Vortmeyer 1978; Rudraiah 1984; Alchaar et al. 1995). The effect of temperature modulation on the onset of thermal convection in an electrically conducting fluid-saturated porous medium subjected to a vertical magnetic field is discussed by Bhadauria (2008). Recently, Bhatta et al. (2010) have studied steady magneto-convection in a horizontal mushy layer which is considered as a porous medium. To the contrary, if the fluid is dielectric then the electric forces play a major role rather than magnetic forces in driving the motion. Several studies have been carried out in the past to assess the effect of AC as well as DC electric fields on convective instability in a dielectric fluid layer. In dielectric fluids, an applied temperature gradient produces non-uniformities in the dielectric constant. The variation in dielectric constant and the electric field intense the polarization force causing fluid motion. In this case, convection can occur in a dielectric fluid layer even if the temperature gradient is stabilizing (i.e., cooling from below and heating from above) and such an instability produced by an electric field is called electroconvection (EC) which is analogous to Rayleigh–Bénard instability. In addition, if the applied temperature gradient is also destabilizing (that is heating from below and cooling from above) then such an instability problem is called electrothermoconvection (ETC). Onset of convection in a dielectric fluid layer in the presence of an electric field has been the subject of theoretical and experimental studies in the recent past. Roberts (1969) has made an individual study on electrohydrodynamic convection by considering the variation in dielectric constant as a linear function of temperature. Turnbull (1968a,b, 1969), Turnbull
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and Melcher (1969), Takashima and Aldridge (1976), Maekawa et al. (1992), and Char and Chiang (1994) have studied natural convection problem under an AC or DC electric field. An exhaustive review on this topic has been given by Jones (1978) and Saville (1997). Douiebe et al. (2001) have studied the combined effects of vertical AC electric field and uniform rotation on coupled buoyancy and thermo capillary instability in an electrically conducting fluid layer, whereas Othman (2004) has considered the stability of a rotating layer of viscoelastic dielectric liquid (Walters’ liquid B ) heated from below under an AC electric field. The problem of onset of convective instability in a horizontal layer of viscoelastic dielectric liquid with a relaxation time heated from below under the action of a vertical AC electric field is analyzed by Othman and Zaki (2003). Shivakumara et al. (2007) have investigated the effect of DC electric field on electrothermal convective instability in a heat generating dielectric fluid layer. Shivakumara et al. (2009) have analyzed the effect of internal heat generation on ETC in a horizontal dielectric fluid layer under the influence of vertical AC electric field. Electrohydrodynamic convection in a layer of porous medium under the influence of electric field has also been investigated in the past but it is still in much- to-be desired state. Moreno et al. (1996) have investigated transport-related effects of imposing AC currents over two-phase flows, oil and water, in porous rocks. The volume averaged form of the frequency-dependent governing equations for electrohydrodynamics in a saturated porous medium have been developed by del Río and Whitaker (2001). El-Sayed (2008) has analyzed the onset of electrohydrodynamic instability in a horizontal layer of Oldroydian viscoelastic liquid saturating a porous medium under the action of vertical AC electric field and a vertical temperature gradient. Rudraiah and Gayathri (2009) have investigated the effect of thermal modulation and vertical electric field on electroconvection in a horizontal dielectric fluid saturated densely packed porous layer and they have also discussed the importance of ETC in porous media. Recently, a weakly nonlinear stability analysis of wave propagation in two superposed dielectric fluids streaming through porous media in the presence of vertical electric field producing surface charges has been investigated by El-Sayed et al. (2011). The drying process in porous media is a rather complicated process as coupled heat and mass transport phenomena are involved simultaneously. Therefore, new techniques are being used to make the drying processes more efficient and one of the effective ways to improve the overall drying kinetics is to apply an electric field (Yabe et al. 1996; Lai and Lai 2002). It is thus imperative to study the onset of ETC in a layer of porous medium. The intent of the present study is, therefore, to investigate the effect of a vertical AC electric field on the criterion for the onset of Darcy–Brinkman convection in a dielectric fluid saturated horizontal layer of porous medium heated uniformly either from below or from above (known as Darcy–Brinkman electroconvection) for different types of velocity boundary conditions. In other words, the present study helps in understanding the influence of electric field on heat transfer. For a high porosity porous medium Givler and Altobelli (1994) have determined experimentally that 5.1 ≤ μ/μ ˜ ≤ 10.9, where μ˜ is the effective viscosity and μ is the fluid viscosity. Therefore, the ratio of effective viscosity to the fluid viscosity is taken to be as a separate parameter and the dielectric constant of the fluid is taken to be as a linear function of temperature. It is shown that the principle of exchange of stability is valid and the condition for the onset of Darcy–Brinkman electroconvection is analyzed using linear stability analysis for three different types of velocity boundary conditions. An exact solution to the resulting eigenvalue problem is obtained in the case of free–free (F/F) boundaries, while for the rigid–rigid (R/R), and lower-rigid and upper-free (F/R) boundaries the critical stability parameters are obtained numerically using the Galerkin method. The effect of vertical AC electric field on the criterion for the onset of Darcy–Brinkman convection is emphasized.
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The results for the Darcy porous medium have also been obtained as a particular case from the present study.
2 Mathematical Formulation We consider a dielectric fluid saturated sparsely packed horizontal porous layer of thickness d with a uniform vertical AC electric field applied across the porous layer. The lower surface is grounded and the upper surface is kept at constant potential V1 . The lower and upper boundaries of the porous layer are maintained at uniform, but different, temperatures T0 and T1 (< T0 ) respectively, and thus a constant temperature difference T (= T0 − T1 ) is maintained between the boundaries. A Cartesian coordinate system (x, y, z) is chosen with the origin at the bottom of the porous layer and z-axis normal to the porous layer in the gravitational field. The relevant basic equations for an incompressible dielectric fluid saturating a porous layer under the Boussinesq approximation are: The conservation of mass ∇ · q = 0. The conservation of linear momentum 1 ∂ q μ 1 ρ0 q · ∇) q = −∇ p + ρ g − q + μ∇ + 2 ( ˜ 2 q + fe . φ ∂t φ k
(1)
(2)
The conservation of energy M
∂T + ( q · ∇)T = κ∇ 2 T. ∂t
(3)
The equation of state ρ = ρ0 {1 − α(T − T0 )}.
(4)
Here, q the velocity vector, T the temperature, p the pressure, ρ the fluid density, κ the effective thermal diffusivity, k the permeability of the porous medium, μ the fluid viscosity, μ˜ the effective viscosity, g the acceleration due to gravity, φ the porosity of the porous medium, M = (ρ0 c)m /(ρ0 c p )f = [(1 − φ)(ρ0 c)s + φ(ρ0 c p )f ]/(ρ0 c p )f the ratio of heat capacities of the fluid saturated porous medium to that of the fluid, c the specific heat, c p the specific heat at constant pressure, α the thermal expansion coefficient, ρ0 the density at reference temperature T = T0 and fe the force of electrical origin which acts on the liquid has no unique formulation. A generally preferred expression is (see, for example, Stratton 1941; Landau and Lifshitz 1988) 1 1 ∂ε fe = ρe E − E · E ∇ε + ∇ ρ E · E (5) 2 2 ∂ρ where, E is the electric field, ρe the free charge density, and ε is the dielectric constant. In Eq. 5, the last electrostriction term can be grouped with the pressure p in Eq. 2 and it has no effect on an incompressible fluid. The first term on the right hand side is the Coulomb force due to a free charge and it is the strongest EHD force term and usually dominates when DC electric fields are present. The second term depends on the gradient of ε and this term dominates when an AC electric field is imposed on the dielectric liquid. The electrical force fe will have no effect on the bulk of the dielectric fluid if both the dielectric constant ε and
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the electrical conductivity σ are homogeneous. Since ε and σ are functions of temperature, a temperature gradient applied to a dielectric fluid produces a gradient in ε and σ . The application of a DC electric field then results in the accumulation of free charge in the liquid. The free charge increases exponentially in time with a time constant ε/σ , which is known as the electrical relaxation time. If an AC electric field is applied at a frequency much higher than the reciprocal of the electrical relaxation time, the free charge does not have time to accumulate. Moreover, the electrical relaxation times of most dielectric liquids appear to be sufficiently long to prevent the buildup of free charge at standard power line frequencies. At the same time, dielectric loss at these frequencies is so low that it makes no significant contribution to the temperature field (Jones 1978). Under the circumstances, only the force induced by non-uniformity of the dielectric constant is considered. Assuming the free charge density is negligibly small, the relevant Maxwell equations are (Turnbull 1969; Roberts 1969) =0 ∇ · (ε E)
(6)
∇ × E = 0.
(7a)
In view of Eq. 7a, E can be expressed as E = −∇V
(7b)
where, V is the electric potential. The dielectric constant is assumed to be a linear function of temperature of the form ε = ε0 [1 − η(T − T0 )]
(8)
where, η(> 0) is the thermal expansion coefficient of dielectric constant and is assumed to be small. For example, for 10 cs Silicone oil η = 2.86 × 10−3 K −1 and ε = 2.6 × 10−11 F m−1 (Maekawa et al. 1992). 2.1 Basic State The basic state is quiescent and is given by q = qb = 0, T = Tb (z), p = pb (z), E = Eb (z), ε = εb (z)
(9)
where, the subscript b denotes the basic state. Substituting Eq. 9 into Eqs. 1–7, we get E 2 pb ρb ∂εb − b ∇εb = 0 −∇ + g + ∇ Eb (10) ρ0 ρ0 ∂ρ 2ρ0 d2 Tb =0 dz 2 ρb = ρ0 {1 − α(Tb − T0 )} εb = ε0 [1 − η(Tb − T0 )] ∇ · (εb Eb ) = 0.
(11) (12) (13) (14)
Solving Eq. 11, using the boundary conditions Tb = T0 at z = 0; Tb = T1 at z = d
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(15)
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we get Tb − T0 = − T z/d.
(16)
In view of Eq. 13 and since E bx = E by = 0, it follows that εb E bz = ε0 E 0 = constant (say).
(17)
Then we have E bz =
E0 1 + T z/d
(18)
and Vb (z) = −
E0 d log(1 + η T z/d) η T
(19)
where E0 = −
V1 η T /d log(1 + η T )
(20)
is the value of the electric field at z = 0. 2.2 Perturbed State To study the stability of the basic state given by Eq. 9, we superimpose infinitesimally small perturbations on the basic state of the form q = q , p = pb + p , V = Vb + V , T = Tb + T , ρ = ρb + ρ , ε = εb + ε
(21)
where, primes denote perturbed quantities and subscript b denotes the basic state. Substituting Eq. 21 in Eqs. 1–7, linearizing the equations by neglecting the products of primed quantities, eliminating the pressure from the momentum equation by operating curl twice and retaining the vertical component, we get the required equations in the form 1 ∂ E 0 ε0 η − T μ¯ ∂V μ − ∇2 + ∇ 2 w = αg∇h2 T + ∇h2 ηE 0 T − φ ∂t ρ0 ρ0 k ρ0 d ∂z (22) T ∂ (23) w M − κ∇ 2 T = ∂t d ∂T . (24) ∇ 2 V = ηE 0 ∂z Non-dimensionalizing Eqs. 22–24 using the scales (x, y, z) by d, t by d 2 /κ, w by T by T and V by ηE 0 T d, we obtain (after neglecting the primes for simplicity) 1 ∂ ∂V 2 −1 2 2 2 − ∇ + Da ∇ w = Rt ∇h T + Rea ∇h T − Pr ∂t ∂z ∂ M − ∇2 T = w ∂t ∂T ∇2V = ∂z
κ/d,
(25) (26) (27)
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where Rt = αg T d 3 /νκ is the thermal Rayleigh number, Rea = η2 ε0 E 02 ( T )2 d 2 /μκ is the AC electric Rayleigh number, Da = k/d 2 is the Darcy number, Pr = νϕ/κ is the Prandtl number, and = μ/μ ˜ is the ratio of viscosities. The bounding surfaces of the porous layer are considered to be either rigid or free. Following Chandrasekhar (1961) and Roberts (1969), the boundary conditions are considered as follows. For convenience, we denote the no-slip boundary by R (rigid), which implies w=
∂w =T =V =0 ∂z
(28)
and the stress-free boundary is denoted by F (free), which implies w=
∂ 2w ∂V = 0. =T = 2 ∂z ∂z
(29)
It may be noted that only one type of boundary condition on V is considered on the rigid and free boundaries in analyzing the problem but any one of these conditions can be imposed on these boundaries (Turnbull 1969; Maekawa et al. 1992). We use a fractional notation with the top of the fraction being the condition at the upper boundary and the bottom of the fraction being the condition at the lower boundary. Thus, F/R means stress-free upper and no-slip lower boundaries. We consider three cases: F/F, R/R, and F/R boundaries.
3 Linear Stability Analysis To carry out the linear stability analysis, we use the normal mode procedure in which we look for the normal mode solution of the form (w, T, V ) = (W, , )(z) exp{i(x + my) + ωt}
(30)
where, and m are the horizontal wave numbers in the x and y directions, respectively, and ω is the frequency. In general, ω = ωr + iωi , is a complex quantity. Substituting Eq. 30 into Eqs. 25–27, we obtain ω − (D2 − a 2 ) + Da −1 (D2 − a 2 )W = −Rt a 2 − Rea a 2 ( − D ) (31) Pr [Mω − (D2 − a 2 )] = W (32) √
(D2 − a 2 ) = D
(33)
where D = d/dz and a = 2 + m 2 is the overall horizontal wave number. On using Eq. 30 in Eqs. 28 and 29, we get W = DW = = = 0
(34)
W = D2 W = = D = 0
(35)
on the rigid boundary, and
on the free boundary. The above set of equations together with the chosen boundary conditions is a double eigenvalue problem for Rt or Rea and ω. 3.1 Qualitative Analysis on the Oscillatory Instability It is a well established fact that oscillatory convection is not a preferred mode of instability in the classical Darcy–Bénard problem. Although oscillatory convection occurs in a porous
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layer in the presence of external constraints of magnetic field and/or rotation as well as in double diffusive systems (see Shivakumara and Venkatachalappa 2004), it is not known whether the vertical AC electric field supports oscillatory convection or not. This has been analyzed in this section. To establish this, first we eliminate from Eq. 31 by operating (D2 − a 2 ) on that equation and then using Eq. 33 we obtain ω 2 − D2 − a 2 + Da −1 D2 − a 2 W = −Rt a 2 D2 + (Rt + Rea ) a 4 . (36) Pr Also, we have
Mω − D2 − a 2 = W. (37) Now, multiplying Eq. 36 by W ∗ (the complex conjugate of W ) and integrating from z = 0 to 1, we obtain ω 2 W∗ − D2 − a 2 + Da −1 D2 − a 2 W Pr (38) = −Rt a 2 W ∗ D2 + (Rt + Rea )a 4 W ∗ 1 where · · · = 0 (· · · ) dz. On using the boundary conditions common to all the three cases of velocity boundary conditions and repeatedly applying integration by parts and simplifying, the LHS of Eq. 38 can be written as ω
2
1 + Da −1 D2 W + 2a 2 |DW |2 + a 4 |W |2 + DW ∗ D4 W − D2 W ∗ D3 W 0 Pr
2 2 (39) + D3 W + 3a 2 D2 W + 3a 4 |DW |2 + a 6 |W |2 . From Eq. 37, we have
Mω∗ − D2 − a 2 ∗ = W ∗ .
Similarly, the right hand side of Eq. 38 upon using Eq. 40 becomes
2 Rt a 2 Mω∗ |D|2 + D2 + a 2 |D|2
+ (Rt + Rea ) a 4 Mω∗ ||2 + |D|2 + a 2 ||2 .
(40)
(41)
Use of Eqs. 39 and 41 in Eq. 38 gives
2 1 ω
+ Da −1 D2 W + 2a 2 |DW |2 + a 4 |W |2 DW ∗ D4 W − D2 W ∗ D3 W 0 + Pr
2 2 + D3 W + 3a 2 D2 W + 3a 4 |DW |2 + a 6 |W |2
2 = Rt a 2 D2 + a 2 |D|2 + (Rt + Rea ) a 4 |D|2 + a 2 ||2
+Mω∗ Rt a 2 |D|2 + (Rt + Rea ) a 4 ||2 . (42) The real and imaginary parts of the above equation are equated separately. The real part, in effect, determines whether the system returns to the equilibrium position or not depending on the sign of the real part of ω. The imaginary part gives the information about the possibility of occurring oscillatory instability. Equating the imaginary parts of Eq. 42, we obtain
ωi 2 2 D W + 2a 2 |DW |2 + a 4 |W |2 Pr
(43a) = −Mωi Rt a 2 |D|2 + (Rt + Rea ) a 4 ||2 .
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Thus, oscillatory convection occurs only if,
1 2 2 D W + 2a 2 |DW |2 + a 4 |W |2 + Rt Ma 2 |D|2 Pr + (Rt + Rea ) Ma 4 ||2 = 0.
(43b)
But the above condition is never satisfied as Rea ≥ 0 and the left hand side of Eq. 43b is positive definite. This establishes that oscillatory instability cannot occur in a dielectric fluid saturated porous layer heated uniformly from below and not from above under the influence of a vertical AC electric field. Therefore, we restrict ourselves to stationary convection in our subsequent analyses and take ω = 0 in Eqs. 31 and 32 to arrive at the following stability equations: [(D2 − a 2 ) − Da −1 ](D2 − a 2 )W = Rt a 2 + Rea a 2 ( − D )
(44)
(D − a ) = −W
(45)
(D2 − a 2 )φ = D.
(46)
2
2
4 Solution The solution to the eigenvalue problem is obtained depending on the choice of boundary conditions. An exact solution is obtained in the case of F/F boundaries, while for R/R and F/R boundaries the critical stability parameters are obtained numerically using the Galerkin method. 4.1 Exact Solution for F/F Boundaries Though this case is of less physical interest, mathematically it is important since an analytical solution to the eigenvalue problem can be obtained and thereby the essential physical features of the problem can be understood. The problem here is to solve the system of Eqs. 44–46 subject to the boundary conditions W = D2 W = = Dφ = 0 at z = 0, 1.
(47)
Let us assume the trial function in the Galerkin expansion of the following form which satisfies the above boundary conditions: W = A1 sin π z, = A2 sin π z, = A3 cos π z
(48)
where A1 to A3 are constants. Substituting Eq. 48 in Eqs. 44–46, we find the condition for the existence of a non-trivial eigenvalue is 2 a + π 2 + Da −1 a 2 + π 2 −(Rt + Rea )a 2 −Rea πa 2 2 (49) 1 − a + π 2 0 = 0. 0 π a2 + π 2 Expanding the above determinant, we get 3 2 a 2 + π 2 + Da −1 a 2 + π 2 a2 . Rt = − R ea a2 a2 + π 2
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(50)
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It is interesting to check Eq. 50 for existing results in the literature under some limiting cases. In the absence of electric field (i.e., Rea = 0), the above equation reduces to the known result 3 2 a 2 + π 2 + Da −1 a 2 + π 2 (51) Rt = a2 of Brinkman convection theory. Equation 50 coincides with that of Roberts (1969) when Da → ∞ and = 1 (non-porous case). To find the minimum value of Rt with respect to the wave number a, Eq. 50 is differentiated with respect to a 2 and equated to zero. A polynomial in ac2 whose coefficients are functions of the physical parameters influencing the instability is obtained in the form 2(ac2 )5 + (Da −1 + 7π 2 )(ac2 )4 + (2Da −1 π 2 + 8π 4 )(ac2 )3
+(2π 6 − Rea π 2 )(ac2 )2 − (2Da −1 π 6 + 2π 8 )(ac2 ) − (Da −1 π 8 + π 10 ) = 0. (52)
It is seen that the electric Rayleigh number, the ratio of viscosities and the Darcy number alters the critical wave number. When Rea = 0, Eq. 52 can be written as {2ac4 + (Da −1 + π 2 )ac2 − π 2 (Da −1 + π 2 )}(π 2 + ac2 )3 = 0.
(53)
Since (π 2 + ac2 )3 = 0, from Eq. 53 it follows that 2ac4 + (Da −1 + π 2 )ac2 − π 2 (Da −1 + π 2 ) = 0.
(54)
From the above equation we note that ac2 = π 2 /2
(55a)
as Da → ∞ and when = 1 and the corresponding critical Rayleigh number from Eq. 51 is Rtc =
27π 4 4
(55b)
which are the known exact values for the classical ordinary viscous fluid case given by Chandrasekhar (1961). Equation 52 is solved numerically for various values of Rea , , and Da −1 and hence the critical wave number is obtained each time. Using this in Eq. 50, the critical Rayleigh number, above which the stationary convection sets is determined. For the Darcy case, Eq. 50 can be written as 2 2 a + π2 a2 RtD = (56a) − R eaD a2 a2 + π 2 where RtD (= Rt Da) is the thermal Darcy–Rayleigh number and ReaD (= Rea Da) is the AC electric Darcy–Rayleigh number. We note that RtD attains its critical value at a 2 = ac2 , where ac2 satisfies the equation (ac2 )4 + 2π 2 (ac2 )3 − ReaD π 2 (ac2 )2 − 2π 6 (ac2 ) − π 8 = 0.
(56b)
From Eq. 50, we readily note that dRt a2 < 0. = − 2 dRea a + π2
(57)
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Thus, the effect of increasing Rea is to decrease Rt and thus it has a destabilizing effect on the system. Similar result holds for the Darcy porous medium case. Similarly, we observe that 3 2 a + π2 dRt = >0 (58) d a2 and 2 2 a + π2 dRt = > 0. dDa −1 a2
(59)
Thus, we note that the effect of increase in the value of and Da −1 is to stabilize the system. 4.2 Numerical Solution for R/R and F/R Boundaries Unlike the F/F boundaries case, the solution for the eigenvalue problem in a closed form is not possible for these two sets of boundary conditions and therefore the eigenvalue equations are solved numerically to obtain the critical stability parameters. Here, we solve Eqs. 44–46 subject to the boundary conditions W = DW = = = 0
at z = 0, 1
(60)
for R/R boundaries, and at z = 0
(61)
W = D2 W = = D = 0 at z = 1
W = DW = = = 0
(62)
for F/R boundaries by employing the Galerkin method as explained in the book by Finlayson (1972). Accordingly, the variables are written in a series of trial functions as W = =
=
n i=1 n i=1 n
Ai Wi Bi i
(63)
Ci i
i=1
where Ai , Bi , and Ci are constants and the trial functions Wi , i , and i will be represented by the power series satisfying the respective boundary conditions. Substituting Eq. 63 into Eqs. 44–46, multiplying the resulting momentum equation (44) by W j (z), energy equation (45) by j (z), electric potential equation (46) by j (z); performing the integration by parts with respect to z between z = 0 and z = 1 and using the boundary conditions given by Eq. 60 or Eqs. 61 and 62, we obtain the following system of linear homogeneous algebraic equations: E ji Ai + F ji Bi + G ji Ci = 0 I ji Ai + J ji Bi = 0 K ji Bi + L ji Ci = 0
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(64)
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where
E ji = D2 W j D2 Wi + 2a 2 DW j DWi + a 4 W j Wi + Da −1 DW j DWi + a 2 W j Wi F ji = −(Rt + Rea )a 2 W j i G ji = Rea a 2 DW j i I ji = j Wi J ji = − D j Di + a 2 j i K ji = − D j i L ji = − D j D i + a 2 j i . 1 Here the inner product is defined as · · · = 0 (· · · )dz. The above set of homogeneous algebraic equations can have a non-trivial solution if and only if E ji F ji G ji I ji J ji 0 = 0. (65) 0 K ji L ji For both boundaries rigid, the trial functions chosen are ∗ Wi = z i+1 − 2z i+2 + z i+3 Ti−1 ∗ i = z i − z i+1 Ti−1 = i
(66)
while the trial functions chosen in the case of lower-rigid and upper-free boundaries are ∗ Wi = 2z i+3 + 3z i+1 − 5z i+2 Ti−1 ∗ i = z i − z i+1 Ti−1 (67) ∗
i = 3z i+1 − 2z i+2 Ti−1 where Ti∗ (i = 1, 2, .....n) is the modified Chebyshev polynomial of ith order such that Wi , i , and i satisfy the corresponding boundary conditions. Equation 66 or 67, as the case may be, is substituted in Eq. 65 and the inner products involved therein are evaluated analytically rather than numerically in order to avoid errors in the numerical integration. The characteristic equation is solved numerically for different values of Da −1 , Rea and using the Newton–Raphson method to obtain the thermal Rayleigh number Rt as a function of wave number a and the bisection method is built-in to locate the critical stability parameters (Rtc , ac ). The results presented here are for i = j = 6 the order at which the convergence is achieved to the desired degree of accuracy, in general.
5 Results and Discussion The effect of vertical AC electric field and boundaries on the criterion for the onset of Darcy– Brinkman convection in a dielectric fluid saturated porous layer heated uniformly from below or above is investigated. Attention is focused on three types of velocity boundary conditions keeping in mind the laboratory and geophysical problems namely, (i) both boundaries free (F/F), (ii) both boundaries rigid (R/R), and (iii) upper-free and lower-rigid (F/R) boundaries.
123
522 Table 1 Comparison of Reac and ac for different values of Rt for R/R boundaries when = 1 and Da −1 = 0
I. S. Shivakumara et al. Roberts (1969) Rt
Reac
−1000 −500
ac
Reac
ac
3370.077
3.2945
3370.0758
3.2945
2749.868
3.2598
2749.8674
3.2598
0
2128.696
3.2260
2128.6951
3.2260
500
1506.573
3.1929
1506.5732
3.1929
1000
883.517
3.1606
883.5172
3.1606
0.0
3.1162
0.0000
3.1162
1707.762
Table 2 Comparison of Rtc and ac for different values of Rea for F/R boundaries when = 1 and Da −1 = 0
Present study
Maekawa et al. (1992) Rea
Rtc
Present study ac
Rtc
ac
0
1100.650
2.682
1100.6502
2.6823
1
1100.092
2.683
1100.0928
2.6826
10
1095.074
2.686
1095.0743
2.6856
20
1089.493
2.687
1089.4937
2.6888
50
1072.722
2.699
1072.7231
2.6986
100
1044.676
2.715
1044.6771
2.7149
200
988.2311
2.747
988.2324
2.7473
500
816.146
2.844
816.1488
2.8435
1000
520.689
2.999
520.6941
2.9989
The eigenvalue problem is solved exactly in the case of F/F boundaries and for the remaining R/R and F/R boundaries the eigenvalues are extracted numerically. To validate the numerical procedure used to find the critical stability parameters, first the test computations are carried out for R/R and F/R boundaries under the limiting case of Da → ∞ and = 1 (i.e., classical viscous case) and compared with the earlier published results. The critical AC electric Rayleigh number Reac and the corresponding wave number ac obtained for different values of Rt for R/R boundaries are compared with those of Roberts (1969) in Table 1. While the critical thermal Rayleigh number Rtc and the corresponding ac obtained for different values of AC electric Rayleigh number Rea for F/R boundaries are compared with those of Maekawa et al. (1992) in Table 2. From these tables we note that the agreement is excellent and thus verifies the accuracy of the numerical method employed. The marginal stability curves in the (Rt , a)-plane for Rea = 100 = Da −1 and = 1 are presented in Fig. 1 for different boundary conditions, while in Fig. 2 the marginal stability curves are shown in the (Rea , a)-plane for two values of Rt = −500 (heating from above) and 500 (heating from below) and (= 1, 2) with Da −1 = 100 for R/R boundaries. From figures, it is observed that the marginal stability curves exhibit single but different minimum with respect to the wave number in all the cases considered. The marginal curves of F/F and F/R boundaries are skewed towards the lower wave number region when compared to (R/R) boundaries. Further the marginal curves of R/R boundaries lie above F/R and F/F boundaries indicating R/R boundaries offer more stabilizing effect against the fluid motion. This is because the disturbances are suppressed to a maximum extent in the case of R/R boundaries
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523
7500
7000
6500
Rt
6000
5500
5000
4500 1
2
3
4
5
6
a Fig. 1 Marginal stability curves for R/R (solid line), F/F (dotted line), and F/R (dashed line) boundaries with Rea = 100 = Da −1 and = 1
12000
Rt = -500 10500
500
R ea 9000
-500
7500
500
2
1 6000 1
2
3
4
5
6
7
a Fig. 2 Marginal stability curves for two values of and Rt when Da −1 = 100 for R/R boundaries
as compared to F/R boundaries and the least suppression is offered by F/F boundaries. The marginal stability curves shown in Fig. 2 reveal that the system is more stabilizing for negative values of Rt (i.e., heating from above) when compared to positive values of Rt (i.e., heating from below) and increasing is to increase the AC electric Rayleigh number. Moreover, the critical wave number is higher in the case of dielectric fluid saturated porous layer heated from above when compared to the porous layer heated from below.
123
524
I. S. Shivakumara et al. 7000
6000
Da -1 =100
5000
100 100
4000
R tc 3000
2000 1000
0
10
0
10
0
0
10 0
400
800
1200
1600
R ea Fig. 3 Rtc vs. Rea for different values of Da −1 when = 1 for R/R (solid line), F/F (dotted line) and F/R (dashed line) boundaries
The critical thermal Rayleigh number Rtc , and the corresponding wave number ac , calculated for F/F, R/R, and F/R boundaries are shown in Figs. 3, 4, 5, and 6 for various values of physical parameters. Figure 3 shows the variation of Rtc as a function of AC electric Rayleigh number Rea for different values of inverse Darcy number Da −1 (= 0, 10, 102 ) when the value of the ratio of viscosities is fixed at 1 (i.e., when = 1). From Fig. 3, it is observed that increase in the value of Da −1 is to increase the value of critical thermal Rayleigh number and hence its effect is to delay the onset of convection. This may be attributed to the fact that increasing Da −1 amounts to decrease in the permeability of the porous medium which in turn retards the fluid flow. Therefore, more heating is required to have instability in a dielectric fluid saturated porous layer with increasing Da −1 . On the contrary, when an AC electric Rayleigh number is increased the value of the critical thermal Rayleigh number decreases. That is, higher the electric field strength the less stable the system due to an increase in the destabilizing electrostatic energy to the system. In other words, the presence of electric field facilitates the transfer of heat more effectively and hence hastens the onset of Darcy– Brinkman convection at a lower value of the thermal Rayleigh number. We note that the results for different velocity boundary conditions differ only quantitatively but agree qualitatively. The system is found to be more stable when both boundaries are rigid because of suppression of disturbances, while the free boundaries are the least stable as expected on physical grounds, i.e., (Rtc )rigid−rigid > (Rtc )rigid−free > (Rtc )free−free . Figure 4 illustrates the variation of critical wave number ac as a function of Rea for different values of Da −1 and = 1. It is observed that ac increases slowly with an increase in the value of Rea and Da −1 . Thus their effect is to contract the size of convective cells. Moreover, it can be seen that (ac )rigid−rigid > (ac )rigid−free > (ac )free−free .
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The Onset of Darcy–Brinkman Electroconvection
525
3.4 Da
3.2
-1
=100 10 0
100
3.0
100
ac
10
2.8 0 10
2.6 0
2.4
2.2
0
200
400
600
800
R ea Fig. 4 ac vs. Rea for different values of Da −1 when = 1 for R/R (solid line), F/F (dotted line), and F/R (dashed line) boundaries 16000
Rea = 0
14000
1000
R tc
12000
2000
10000
1000
0
2000 0 1000
8000
2000 6000
4000 1
2
3
4
5
6
Fig. 5 Rtc vs. for different values of Rea when Da −1 = 100 R/R (solid line), F/F (dotted line), and F/R (dashed line) boundaries
This figure also exhibits the property that the deviation in the critical wave number between different boundary conditions increases with an increase in the value of Da −1 , which is more so in the case of F/F and F/R boundaries when compared to R/R boundaries. As pointed out in Sect. 1, Brinkman’s model rests on an effective viscosity μ˜ different from fluid viscosity μ denoted through in dimensionless form and it has a determining influence on the onset of Darcy–Brinkman electroconvection. The influence of viscosity ratio on the critical stability parameters is summarized in Figs. 5 and 6 for different values of Rea .
123
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I. S. Shivakumara et al. 3.3
Rea =2000
1000
3.2 0
2000
3.0
1000
ac
0
2000
2.8
1000 0
2.6
1
2
3
4
5
6
Fig. 6 ac vs. for different values of Rea when Da −1 = 100 for R/R (solid line), F/F (dotted line), and F/R (dashed line) boundaries
From Fig. 5, it is observed that increasing is to increase Rtc , irrespective of the nature of the boundaries (i.e., F/F, R/R, and F/R boundaries). Thus the effect of is to suppress the onset of Darcy–Brinkman electroconvection. This is because, increasing amounts to increase in the viscous effect which in turn retards the fluid flow. Therefore, higher heating is required for the onset of convection with increasing . Figure 6 shows the variation of critical wave number as a function of . It is observed that the critical wave number decreases with increasing and it is more so in the cases of F/F and F/R boundaries. That is, increase in the value of is to increase the size of convection cells. In the case of Darcy porous medium, the criterion for the onset of electroconvection in dielectric fluids is found to be independent of the nature of boundaries (rigid or free); a well established result observed in the case of ordinary viscous fluid saturating a Darcy porous medium.
6 Conclusions The effect of vertical AC electric field on the onset of Darcy–Brinkman convection in a dielectric fluid saturated porous layer heated uniformly from below is investigated for free– free (F/F), rigid–rigid (R/R) and lower-rigid and upper-free (F/R) boundaries. The resulting eigenvalue problem is solved analytically and numerically depending on the choice of velocity boundary conditions. Exact solution is obtained for F/F boundaries, while for R/R and F/R boundaries the eigenvalue problem is solved numerically using the Galerkin method. It is established that the principle of exchange of stability is valid irrespective of the type of velocity boundary conditions. It is observed that the stability of the fluid is reinforced with an increase in the value of inverse Darcy number Da −1 and ratio of viscosities . On the contrary, increase in the AC electric Rayleigh number Rea is to hasten the onset of Darcy– Brinkman convection due to increasing destabilizing electrostatic energy. In other words, the presence of electric field supports effective heat transfer in the dielectric fluid saturated porous medium to advance the onset of convection. The system is found to be more stable when
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527
both boundaries are rigid while the free boundaries are the least stable. Moreover, the onset of Darcy–Brinkman electroconvection is delayed if the porous layer is heated from above as compared to heated from below. The critical wave number increases with an increase in Rea and Da −1 . Thus their effect is to contract the size of convection cells. Whereas, increase in is to enlarge the dimension of convective cells. Many authors have advocated that an analysis of F/F boundaries offers results in qualitative agreement with R/R and F/R boundaries in the absence of electric field. Our paper offers not only a support for this assertion, but gives also quantitative comparisons between these three cases when the electric field is present. The nature of boundaries (rigid or free) has no influence on the onset electroconvection in dielectric fluids as well for the Darcy porous medium case. Acknowledgments One of the authors (ISS) wishes to thank the Brain Korea 21 (BK21) Program of the School of Mechanical Engineering, Yonsei University, Seoul, Korea for inviting him as a visiting Professor and also the Bangalore University for sanctioning sabbatical leave. One of the authors (KH) wishes to thank the Management and Principal of Mount Carmel College, Bangalore for their encouragement. We thank the reviewers for their useful suggestions.
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