Arch Appl Mech (2008) 78: 211–224 DOI 10.1007/s00419-007-0153-x

O R I G I NA L

M. F. El-Sayed

Onset of electroconvective instability of Oldroydian viscoelastic liquid layer in Brinkman porous medium

Received: 30 December 2006 / Accepted: 6 June 2007 / Published online: 19 July 2007 © Springer-Verlag 2007

Abstract The problem of the onset of electrohydrodynamic instability in a horizontal layer of Oldroydian viscoelastic dielectric liquid through Brinkman porous medium under the simultaneous action of a certical ac electric field and a vertical temperature gradient is analyzed. Applying linear stability theory, we derive an equation of eight order. Under somewhat suitable boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Both the cases of stationary and oscillatory instabilities are discussed if the liquid layer is heated from below or above. The effects of the porosity of porous medium, the medium permeability, the Prandtl number, the ratio of retardation time to relaxation time, the elastic number, in the presence or absence of Rayleigh number are shown graphically for both cases. Some of the known results are derived as special cases. The electrical force has been shown to be the sole agency causing instability of the considered system since it is much more important than the buoyancy force even if the medium is porous. Keywords Hydrodynamic stability · Viscoelastic fluids · Electrohydrodynamics · Convection in porous media

1 Introduction Convective instability in a porous media has been studied with great interest for more than half a century. Convection, given by buoyancy, has found increased applications in underground coal gasification, solar energy conversion, oil reservoir simulation, ground water contaminant transport, geothermal energy extraction, and many other areas [1–3]. In the beginning, interest was mainly focussed on problems in which the driving force was due to the applied temperature and/or concentration gradients at the boundaries of the system [4,5]. Recently, the focus has also been on the study of convective instabilities in reactive as well as electrified fluid [6]. The instability in electrohydrodynamic systems can lead to stationary convection as well as oscillatory convection. The stationary convection occurs when heating is done either from below or from above, depending on the sign of the thermal diffusion coefficient, whereas the oscillatory one occurs just when heating is from below [7,8]. For an excellent review about the subject of convective instability see the monograph of Nepomnyashchy et al. [9]. In recent years there has been considerable interest, from various branches of fluid mechanics and condensed matter physics, in systems that show an oscillatory instability as the first bifurcation [10]. The reasons for considering the instability in a porous medium are twofold [11]. First, a porous medium is often used as a catalyst enhancing the rate of instability. On the other hand, use of a porous medium significantly simplifies the descriptions of an average hydrodynamic flow limit and allows us to consider realistic boundary conditions for both the velocity and the applied electric field. In the Brinkman model [12–14], for flow through a porous M. F. El-Sayed (B) Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis (Roxy), Cairo, Egypt E-mail: [email protected]

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medium with a high permeability, the momentum equation for a porous medium flow must reduce to the viscous flow limit and advocate that classical frictional terms be added in Darcy’s law [15]. The electrohydrodynamic instabilities of various configurations can be considered due to the body force of electrical origin whose vector form per unit volume is [16]
 1 1 ∂ε fe = ρe E − E2 ∇ε + ∇ ρ E2 (1) 2 2 ∂ρ where ρe is the free charge density, E is the electric field, ε is the dielectric constant, and ρ is the mass density of the liquid. Since the last term in Eq. (1) is the gradient of a scalar, it can be included in the pressure term and, therefore, has no effect on the instability problem on hand. The first term is a force due to a free charge ρe , which is zero when the dielectric constant ε and the electrical conductivity σ are both homogeneous. The second term depends on the gradient in ε. Therefore, the electrical force fe can have no effect on the bulk of the liquid when ε and σ are both homogeneous. Since ε and σ are functions of temperature, a temperature gradient applied to a dielectric fluid produces a gradient in ε and σ [17]. 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. 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 [18,19]. Furthermore, since the remaining term (the second term) in Eq. (1) depends on E2 rather than E and the variation of E is very rapid, the root mean square value of E can be assumed as the effective value. In other words, we can treat the ac electric field as the dc electric field whose strength is equal to the root mean square value of the ac electric field. Thus, the case of an ac electric field is more tractable than that of a dc electric field, and has been examined by Turnbull and Melcher [20], and Turnbull [21,22]. The case of cylindrical geometry has been treated by Takashima [23]. The Jeffrey (Oldroyd) constitutive equation that is used for performing a linear stability analysis of Rayleigh–Bénard convection in clear viscoelastic fluids, i.e., in viscoelastic fluids with no porous medium to impede their flow is

  ∂ ∂ 1+λ (2) τij = 1 + λ0 ei j ∂t ∂t where τij is the shear stress, ei j = η(vi, j + v j,i ), vi is the velocity component, λ is the stress relaxation time, λ0 (< λ) is the strain retardation time, and η is the dynamic viscosity. The conservation of linear momentum with the Boussinesq approximation gives us ρ where

∂vi = τi j, j − ρgδi3 ∂t

(3)

τi j = − pδi j + τij .

(4)

In arriving at the governing equation of porous medium momentum transport based on the Jeffrey constitutive equation, we will have to make use of the Dupuit’s equation, viz., v f i = mvi

(5)

where m is the porosity, vi is the actual velocity component in the absence of a porous medium, and v f i is the filter velocity. Substituting Eq. (4) into Eq. (3), averaging the resulting equation in a fashion similar to that done in the derivation of the classical Darcy law for low-porosity media, and then using Eq. (5) in the resulting equation, we get [24]

 
 ρ ∂ ∂vi ∂  η ∂ (6) 1+λ = 1+λ − p,i − ρgδi3 − vi 1 + λ0 m ∂t ∂t ∂t k1 ∂t where k1 is the medium permeability. It is important to note here that the porosity is absorbed into the permeability tensor, and hence does not appear explicitly in the last term on the right-hand side of Eq. (6). We

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have dropped the subscript f with the velocity, for simplicity, in the above equation. In vector form, Eq. (6) may be written as

 
 ρ ∂ ∂v ∂  η ∂ 1+λ = 1+λ −∇ p + ρg − v. 1 + λ0 m ∂t ∂t ∂t k1 ∂t

(7)

The purpose of the research treated here is to examine theoretically the problem of the onset of electrohydrodynamic instability in a horizontal layer of Oldroydian viscoelastic dielectric liquid through Brinkman porous medium under the simultaneous action of a vertical ac electric field and a vertical temperature gradient. In this case, the electrical force can be added to the first term on the right-hand side of Eq. (7), see [25–29]; while the usual viscous term in Navier–Stokes equation of motion, divided by the porosity m, can be added to the second term [30], as indicated in the next section. In Eq. (7), terms of the second order in various parameters have been neglected, since we will apply linear stability theory to the liquid layer at rest. This means that in this problem there is no distinction between Oldroyd liquid A and Oldroyd liquid B [31,32].

2 Mathematical formulation of the problem We consider an infinite horizontal layer of Oldroydian viscoelastic dielectric fluid of thickness d through porous medium. The lower surface at z = 0, and the upper surface at z = d, are maintained at constant temperatures θ0 and θ1 , respectively. In addition to a temperature gradient, a vertical ac electric field is also imposed across the layer; the lower surface is grounded, and the upper surface is kept at an alternating potential, where root mean square value is φ0 . The equations of motion and continuity are 


  ρ η 2 ∂ η ∂ ∂v ∂ 1 2 1+λ = 1+λ −∇ + ρg − E ∇ε + 1 + λ0 ∇ v− v m ∂t ∂t ∂t 2 ∂t m k1 ∂ρ + ∇ · (ρv) = 0 m ∂t

(8) (9)

where v = (u, v, w) is the velocity of the liquid, g = (0, 0, −g) is the gravitational acceleration, E is the root mean square value of the electric field, and  is the modified pressure defined by [33] 1 ∂ε  = p − ρ E2 . 2 ∂ρ

(10)

The equation of energy, neglecting the dissipation terms, is [34–36] 

 ∂θ 1 ρc + (v · ∇)θ = k∇ 2 θ ∂t m

(11)

where θ is the temperature, c is the specific heat, and k is the thermal conductivity. Assume that the free charge density ρ0 is neglected, then the required electrical equations can be written in the form ∇ · (εE) = 0 ∇ × E = 0,

or

E = −∇φ

(12) (13)

where φ is the root mean square value of the electric potential. Finally the mass density ρ, and the dielectric constant ε are assumed to be functions of temperature [37, 38], i.e., ρ = ρ0 [1 − α(θ − θ0 )] ε = ε0 [1 − e(θ − θ0 )]

(14) (15)

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where α and e are positive constants. It is obvious that the following steady solutions, correspond to the system of Eqs. (8)–(15): v θ ρ ε

=0 = θ0 − βz = ρ0 (1 + αβz) = ε0 (1 + eβz)

E x = E y = 0, E z = φ=−

(16) (17) (18) (19) E0 (1 + eβz)

E0 ln(1 + eβz) eβ

(20) (21)

where β = (θ0 − θ1 )/d is the adverse temperature gradient, and E0 = −

eβφ1 ln(1 + eβd)

(22)

is the root mean square value of the electric field at z = d. The modified pressure  can be determined from the equation the condition 1 2 ∇ = ρg − E ∇ε. (23) 2 Note that Eq. (16) satisfies the condition that the liquid layer is initially at rest; Eq. (17) satisfies the condition that there exists a vertical temperature gradient such that θ = θ0 at z = 0, and θ = θ1 at z = d; Eqs. (18) and (19) satisfy the conditions that ρ = ρ0 and ε = ε0 , respectively, at z = 0, while they reduce to Eqs. (14) and (15), respectively, at z = d; Eq. (20) satisfy the condition that there exists a vertical electric field, i.e., E x = E y = 0, while E z is derived from Eq. (12) since ∇ · (εE) = ∂(εE z )/∂z = 0, i.e., ε(∂ E z /∂z) + (∂ε/∂z)E z = 0, which on using Eq. (19) reduces to (1 + eβz)(∂ E z /∂z) + eβ E z = 0, and this equation is automatically satisfied by the solution E z = E 0 /(1 + eβz)given by Eq. (20); Eq. (21) results from Eq. (13), since E = −∇φ, i.e., E z = −(∂φ/∂z), and hence φ = − E z dz. Finally, Eq. (22) can be obtained directly from Eq. (21) using the condition that φ = φ1 at z = d. 3 Perturbation scheme Let the initial steady state be slightly perturbed. Following the classical lines of linear stability theory [39], we assume that v = (u  , v  , w  ), θ = θ + θ  , ρ = ρ + ρ  , ε = ε + ε , E = E + E , and  =  +  , where the prime refer to the perturbed small quantities. Substituting these physical quantities into Eqs. (8), (11), and (12), respectively, and making use of Eqs. (9), (10), and (13)–(21), then the equations governing small perturbations may be written as


ε0 E 02 e2 β 1 ∂ ∂ 2  2  ∇H θ (∇ w ) − αg + 1+λ ∂t m ∂t ρ0 

 ν 4  ∂ ε0 E 0 eβ 2 ∂φ  ν 2  − ∇H = 1 + λ0 ∇ w − ∇ w (24) ρ0 ∂z ∂t m k1 β ∂θ  − K ∇ 2 θ  = w (25) ∂z m ∂θ  =0 (26) ∇ 2 φ + E0 e ∂z 2 = ∂ 2 /∂ x 2 + where ν = η/ρ0 is the kinematic viscosity, K = k/(cρ0 ) is the thermometric conductivity, ∇ H 2 2 ∂ /∂ y is the horizontal Laplacian, and the prime refers to perturbation quantities. Here, small terms have been neglected using the fact that |αβz|  1 and |eβz|  1. It should be noted here that the steady solutions (16)–(21) are disappeared in Eqs. (24)–(26) due to the linearization procedure, and only three scalar fields

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w  , θ  , and φ  exist. The derivation of Eqs. (26)–(28) is obtained earlier by many authors, in similar physical problems of interest, e.g., Takashima [23], and Takashima for example, the  and Ghosh [33], among others,  linearized form of Eq. (11) can be written in the form ρ (∂θ  /∂t) + (w  /m)(∂θ  /∂z) = (k/c)∇ 2 θ  , which   on using Eq. (17) and the condition that |αβz|  1, reduces to ρ0 (∂θ  /∂t) − (β/m)w  = (k/c)∇ 2 θ  , and hence Eq. (25) holds. Equations (24) and (26) can be obtained in the same manner after some straightforward but lengthy calculations. Equations (24)–(26) are first rendered dimensionless by choosing d, d 2 /K , K /d, βd, d 2 , and E 0 d as the units of length, time, velocity, temperature, medium permeability, and electric potential, respectively, and by introducing the following dimensionless numbers: P = ν/K is the Prandtl number,  = λK /d 2 is an elastic number which may be interpreted as a “Fourier number” in terms of λ, µ = λ0 /λ is the ratio of the retardation time to the relaxation time, R = αβgd 4 /(ν K ) is the Rayleigh number, L = L 1 L 2 = ε0 (eβ E 0 d 2 )2 /(ρ0 ν K ) is the electrical Rayleigh number, L 1 = ε0 eβ E 02 d 3 /(ρ0 ν K ), L 2 = eβd. They are, then, simplified in the usual manner by decomposing the solution in terms of normal modes, so that [40]       w , θ , φ = [W (z), (z), (z)] exp i(ax x + a y y) + ωt (27) where ax and a y are the wavenumbers, and ω is the time constant. Thus, with all variables now made dimensionless, we obtain the following equations: ω

(1 + ω) P −1 (D 2 − a 2 )W + (R + L)a 2  + L 1 a 2 D m   1 1 2 (28) (D − a 2 )2 W − (D 2 − a 2 )W = (1 + µω) m k1   1 ω − (D 2 − a 2 )  = W (29) m (30) (D 2 − a 2 ) + L 2 D = 0 where D = d/dz, and a 2 = ax2 + a 2y . In seeking solutions of these equations, we must impose certain boundary conditions at the lower and upper surfaces, i.e., z = 0 and z = 1, respectively. The most realistic boundary conditions may be written as [41] W = DW =  =  = 0

at z = 0, 1.

(31)

In this paper, however, we shall use somewhat different boundary conditions given by W = D 2 W =  = D = 0

at z = 0, 1.

(32)

This case, although it admittedly is an artificial one to consider, is of importance since its exact solution is readily obtained. Furthermore, past experience with problems of this kind (see, for example, [21,23]) suggest that the essential features of the problem can be disclosed by a discussion of this case. Equations (28)–(30) and the boundary conditions (32) constitute an eigenvalue system for the present problem. It is evident that when µ = 0, the system reduces to that for a Maxwell liquid [42]. Also, when  = 0 or µ = 1, then the system reduces to that for an ordinary viscous liquid [39].

4 Solution of the eigenvalue system Equations (28)–(30), and the boundary conditions (32) can be readily combined to yield [(D 2 − a 2 )(D 2 − a 2 − ω){(D 2 − a 2 )2 − ωG P −1 (D 2 − a 2 ) − (m/k1 )} +G Ra 2 (D 2 − a 2 ) − G La 4 ] = 0

(33)

 = D 2N  = 0

(34)

together with at z = 0, 1 (N = 1, 2, 3, . . . )

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where G=

1 + ω . 1 + µω

(35)

Here, the relation L = L 1 L 2 has been used. Examination of Eqs. (33) and (34) indicates that the relevant solution for  is given by [43] =

∞ 

An sin nπ z

(36)

n=1

substitution of this solution in Eq. (33) leads to the required eigenvalue relationship of the form   (n 2 π 2 + a 2 )2 (n 2 π 2 + a 2 + ω) (n 2 π 2 + a 2 ) ω m R L= + − − 2 (n 2 π 2 + a 2 ). a4 G P k1 G(n 2 π 2 + a 2 ) a

(37)

Let us now divide the right-hand side of Eq. (37) into the real and the imaginary parts after setting ω = i S, with S real. Then, we have L = X + i SY (38) where   (n 2 π 2 + a 2 )2 S 2 P(1 − µ) 2 2 X= 4 P(1 + µS  ) + a P(1 + S 2  2 ) (n 2 π 2 + a 2 )    m R 2 2 2 2 2 2 2 (39) × (n π + a ) − − S (1 + S  ) − 2 (n 2 π 2 + a 2 ) k1 a   P(1 + µS 2  2 ) (n 2 π 2 + a 2 )2 −P(1 − µ) + Y = 4 a P(1 + S 2  2 ) (n 2 π 2 + a 2 )    m (40) × (n 2 π 2 + a 2 )2 − + (n 2 π 2 + a 2 )(1 + S 2  2 ) . k1 Since L must be real, then S and Y must be zero. When S = 0, we obtain from Eqs. (38) and (39) that   (n 2 π 2 + a 2 )2 m R 2 2 2 2 L= π + a ) − (41) (n − 2 (n 2 π 2 + a 2 ) 4 a k1 a which is the eigenvalue relationship for a stationary instability. When Y = 0, we obtain from Eq. (40) that  −1  mµP 2 S 2 =  2 (1 + Pµ) − P(1 − µ)(n 2 π 2 + a 2 ) k1 (n 2 π 2 + a 2 )2  mP 2 2 2 −(1 + P) + [1 + (1 − µ)(n π + a )] (42) k1 (n 2 π 2 + a 2 )2 and this gives L=

  −1 (n 2 π 2 + a 2 )2 m 1 + Pµ 1 − a 4 P 2 k1 (n 2 π 2 + a 2 )2  × [Pµ 2 (1 + Pµ)(n 2 π 2 + a 2 )2 + (1 + 2Pµ + P 2 µ)(n 2 π 2 + a 2 ) Pm [µ 2 (1 + 2Pµ)(n 2 π 2 + a 2 )2 k1 + a 2 )2 P 2 m 2 µ +2µ(1 + P)(n 2 π 2 + a 2 ) + 1] + 2 2 2 [1 k1 (n π + a 2 )3  R 2 2 2 +µ(n π + a )] − 2 (n 2 π 2 + a 2 ) a −(1 + P)] −

(n 2 π 2

(43)

which is the eigenvalue relationship for an oscillatory instability. Note that, in the case of purely fluids (when k1 → ∞), Eqs. (41)–(43) reduce to the same equations obtained earlier by Takashima and Ghosh [33], and their results are therefore recovered.

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5 Numerical results and discussion It is clear from Eqs. (41) and (43) that the lowest value of L occurs when n = 1. For a fixed value of R, the lowest point of L defined by Eq. (41) as a function of a, m, and k1 , gives the critical electrical Rayleigh number L c as   (π 2 + a 2 )2 m R 2 2 2 Lc = (44) (π + a ) − − 2 (π 2 + a 2 ) 4 a k1 a and the corresponding critical wavenumber ac for a stationary instability. Minimizing Eq. (44) with respect to a 2 , yields the fourth order equation in a 2 as       4k1 (a 2 )4 + 8π 2 k1 (a 2 )3 + 2π 2 2m + k1 R − 4π 4 (a 2 ) + 4π 4 2m + k1 R − 4π 4 = 0. (45) Solving the Eq. (45) for a 2 gives the critical wavenumber ac2 . Further, substituting this critical wavenumber ac2 into Eq. (44) yields the critical electrical Rayleigh number L c for stationary instability. The values of L c and ac for an oscillatory instability are similarly determined from Eq. (43) for fixed values of P, , µ, m, k1 , and R, as   −1 (π 2 + a 2 )2 m Lc = 1 + Pµ 1 − a 4 P 2 k1 (π 2 + a 2 )2  × [Pµ 2 (1 + Pµ)(π 2 + a 2 )2 + (1 + 2Pµ + P 2 µ)(π 2 + a 2 ) Pm [µ 2 (1 + 2Pµ)(π 2 + a 2 )2 k1 + a 2 )2  P 2 m 2 µ R 2 2 2 2 +2µ(1 + P)(π + a ) + 1] + 2 2 [1 + µ(π + a )] − 2 (π 2 + a 2 ) 2 3 a k1 (π + a )

−(1 + P)] −

(π 2

(46)

and the corresponding critical frequency Sc is determined from Eq. (42) as  Sc2 =  2 (1 + Pµ) −

mµP 2 k1 (π 2 + a 2 )2

−1  P(1 − µ)(π 2 + a 2 )

 mP 2 2 −(1 + P) + [1 + (1 − µ)(π + a )] . k1 (π 2 + a 2 )2

(47)

On minimizing Eq. (46) with respect to a 2 , yields the eighth order equation in a 2 as [44] γ1 (a 2 )8 + γ2 (a 2 )7 + γ3 (a 2 )6 + γ4 (a 2 )5 + γ5 (a 2 )4 + γ6 (a 2 )3 + γ7 (a 2 )2 + γ8 (a 2 ) + γ9 = 0

(48)

where the coefficients γ1 –γ9 are given in the Appendix by Eqs. (A.1)–(A.9). We solve Eq. (48) for a 2 which gives the critical wavenumber ac2 , and substituting this critical wavenumber into Eq. (46) yields the critical electrical Rayleigh number L c for the oscillatory instability. Substituting these critical wavenumber and critical electrical Rayleigh number of oscillatory instability into the Eq. (47) gives the critical frequency of oscillatory case The type of instability which takes place in practice will be that corresponds to the lower value of L c . It should be noted here that when  = 0 or µ = 1, then Y cannot vanish, and, therefore, S must be zero. This means that, in the case of an ordinary viscous fluid, the principle of the exchange of stabilities is valid even in the presence of porous medium. Note also that, as far a stationary instability is concerned, there is no distinction between a viscous fluid and a viscoelastic fluid, and also between the presence or absence of the electric field. The region of stationary instability for different values of the porosity of porous medium m is shown in Fig. 1. We observe from this figure that an increase in the value of porosity of porous medium decreases the critical electrical Rayleigh number of the stationary mode. Figure 2 illustrates the variation of critical electrical Rayleigh number L c with the Rayleigh number R for different values medium permeability k1 for stationary case. We find from this figure that an increase in the value of medium permeability increases the critical electrical Rayleigh number, and hence increases the region of marginal stability. The values of L c when R = 0

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Fig. 1 Variation of critical electrical Rayleigh number L c (in the stationary case) as a function of R for various values of m (=0.001, 15, 35) when k1 = 1

Fig. 2 Variation of critical electrical Rayleigh number L c (in the stationary case) as a function of R for various values of k1 (=0.005, 0.01, 0.5) when m = 0.3

Fig. 3 Variation of critical electrical Rayleigh number L c (in the stationary case) as a function of m for various values of k1 (0.1, 2, 35) when R = 0

is shown in Fig. 3 as function of m for various values of k1 , for the case of stationary instability. We observe from Fig. 3 that for fixed value of the porosity of porous medium m, an increase in the value of the medium permeability k1 increases the critical electrical Rayleigh number for stationary state. It can be seen also from Fig. 3 that for fixed value of the medium permeability k1 , an increase of the value of the porosity of porous medium m decreases the region of stable state for stationary mode. In this case, it can be seen from this figure that the effects of the parameters m and k1 are similar to their effects in the general case when R  = 0 given by Figs. 1 and 2, respectively, except that their effects hold at higher values of the critical electrical Rayleigh

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Fig. 4 Variation of critical electrical Rayleigh number L c (in the oscillatory case) as a function of R for various values of m (=0.01, 15, 35) when k1 = 1,  = 1, µ = 0.1, and P = 100

Fig. 5 Variation of critical electrical Rayleigh number L c (in the oscillatory case) as a function of R for various values of k1 (=0.007, 0.01, 0.5) when m = 0.5,  = 1, µ = 0.1, and P = 100

Fig. 6 Variation of critical electrical Rayleigh number L c (in the oscillatory case) as a function of R for various values of P (=3, 30, 100) when k1 = 0.1, µ = 0.1, m = 1, and  = 1

number. Note that, when R = 0, higher values of the medium permeability (k1 → ∞) which correspond the case of purely fluids, have no effect on the stability of the considered system, where the critical electrical Rayleigh number L c will has a constant value, i.e., the obtained curve will be a horizontal straight line, which confirms the same result obtained earlier by Takashima and Ghosh [33]. Figures 4, 5, 6, 7, and 8 show the variation of critical electrical Rayleigh number L c with the Rayleigh number R for different values of the porosity of porous medium m, the medium permeability k1 , the Prandtl number P, the ratio of the retardation time to the relaxation time µ, and the elastic number , respectively, for the oscillatory case. Here the values of P and µ have been taken as P = 100 and µ = 0.1 which are based

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Fig. 7 Variation of critical electrical Rayleigh number L c (in the oscillatory case) as a function of R for various values of µ (=0.01, 0.1, 2) when k1 = 0.1,  = 1, m = 1, and P = 50

Fig. 8 Variation of critical electrical Rayleigh number L c (in the oscillatory case) as a function of R for various values of  (=0.04, 0.09, 1) when k1 = 0.1, µ = 0.1, m = 1, and P = 100

on the data given by Toms and Strawbridge [45] for a dilute solution of polymethyl methacrylate in n-butyl acetate. The similar results for other values of P and µ are not shown in this paper to save space. We observe from Fig. 4 that an increase of the value of the porosity of porous medium m decreases the region of stable state for oscillatory mode. It can be seen from Figs. 5–7 that an increase of the values of medium permeability k1 , Prandtl number P, and the ratio of the retardation time to the relaxation time µ increase the critical electrical Rayleigh number L c for oscillatory mode. The effect of the elastic number  on the oscillatory instability is shown in Fig. 8. We find that an increase in the value of  decrease the critical electrical Rayleigh number L c of the oscillatory state. Hence its effect is destabilizing. Figures 7 and 8 show that the liquid layer can become stable and unstable, respectively, even if it is heated from above (i.e., R < 0), while Figs. 4–6 show that the parameters m, k1 , and P have no effects on the stability of the considered system if the liquid layer is heated from above. We can also observe from Figs. 1–8 that the effect of increasing the value of Rayleigh number R is to decrease the critical electrical Rayleigh number L c for both stationary and oscillatory state. Thus the effect of R is to destabilize the system. From the preceding discussion and the arguments of Takashima and Ghosh [33] to neglect the buoyancy force, we conclude that the electrical force can be regarded as the sole agency causing instability, since it is much more important than the buoyancy force, even in the presence of porous medium. The values of L c when R = 0 and P = 100 are shown in Figs. 9, 10 and 11 as a function of  for various values of m, k1 , and µ, respectively, for the case of oscillatory instability. In this case, it can be seen from these figures that the effects of the parameters m, k1 , µ, and  are similar to their effects in the general case when R  = 0 given by Figs. 4, 5, 7, and 8, respectively, except that the effects hold at higher values of the critical electrical Rayleigh number. It is worthwhile pointing that when R = 0 and P > 100 the results are almost the same as those for P = 100 unless µ is exceedingly small. On the other hand, when R = 0 and P < 100, the change in P gives a considerable change to the values of L c . However, the results for the case when P > 100 are not shown in this case since the value of P is quite high for most viscoelastic liquids.

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Fig. 9 Variation of critical electrical Rayleigh number L c (in the oscillatory case) as a function of  for various values of m (=0.01, 3, 6) for the same system considered in Fig. 4, but with R = 0

Fig. 10 Variation of critical electrical Rayleigh number L c (in the oscillatory case) as a function of  for various values of k1 (=0.009, 0.03, 0.5) for the same system considered in Fig. 5, but with R = 0

Fig. 11 Variation of critical electrical Rayleigh number L c (in the oscillatory case) as a function of  for various values of µ (=0.02, 0.1, 2) for the same system considered in Fig. 7, but with R = 0

6 Concluding remarks In the present study we have employed a linear theory analysis for the problem of the onset of electrohydrodynamic instability in a horizontal layer of Oldroydian viscoelastic dielectric liquid through Brinkman porous medium under the simultaneous action of a vertical ac electric field and a vertical temperature gradient. Analytical expressions have found for the onset of stationary, oscillatory instabilities and oscillatory frequency, which depend on the porosity of porous medium, the medium permeability, the Prandtl number, the ratio of the

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retardation time to the relaxation time, and the elastic number  as well as the Rayleigh number. The critical wavenumber also depends on these quantities. In the stationary convection case, we found that (1) The porosity of porous medium has a destabilizing effect, while the medium permeability has a stabilizing effect on the considered system even if the liquid layer is heated from above. (2) In absence of the Rayleigh number, the parameters of the porosity of porous medium, and the medium permeability will behave in a similar manner as their effects on the stability of the system in the presence of the Rayleigh number, except that their effects (when R = 0) hold at higher values of the critical electrical Rayleigh number (when R  = 0). In the oscillatory convection case, we conclude the following: (1) The porosity of porous medium and the medium permeability have the same effects as in the case of stationary convection, except that they did not affect the stability of the considered system if the liquid layer is heated from above. (2) The Prandtl number has a stabilizing effect if the liquid layer is heated from below, while it has no effect on the stability of the system if the liquid layer is heated from above. (3) The ratio of the retardation time to the relaxation time has a stabilizing effect, while the elastic number has a destabilizing effect on the considered system even if the liquid layer is heated from above. (4) The electrical force can be regarded as the sole agency causing instability of the system even in the presence of porous media. (5) In absence of the Rayleigh number, the parameters of the porosity of porous medium, the medium permeability, and the ratio of the retardation time to the relaxation time will behave in a similar manner as their effects on the stability of the system in the presence of the Rayleigh number, except that their effects (when R = 0) hold at higher values of the critical electrical Rayleigh number (when R  = 0). In closing this paper, it should be noted that the Rayleigh number has a destabilizing effect for both the stationary and oscillatory cases. Acknowledgments I would like to thank both the referees for their interest in this work and their useful comments that improved the original manuscript.

Appendix The coefficients γ1 –γ9 appear in Eq. (48) are defined as γ1 = 2k13 Pµ 2 (1 + Pµ)2    γ2 = k13 (1 + Pµ) 1 + Pµ 2 + P + 12π 2 (1 + Pµ)      γ3 = 4k12 (1 + Pµ) − m P 2 µ2 + k1 π 2 1 + Pµ 2 + P + 7π 2 (1 + Pµ)      γ4 = k12 − m Pµ 1 − 2π 2  + P −2 + µ(4 + P + 12π 2  + 14Pπ 2 µ)

(A.1) (A.2) (A.3)

 +k1 π 2 (1 + Pµ){2 + 3π 2  + P(2 + (3π 2 µ(2 + P) + (1 + Pµ)(R + 28π 4 µ)))}

γ5 = 2k1 [m  P µ 2 2

3 3

− k12 π 4 (1 + 2

2

2

+k1 m P{1 + µ(1 + π  + 5π 4  2 + P(2 + 2π 2 (3 − 2µ)) + P 2 π 2 µ(1 − 5π 2 µ))}] γ6 = k1 [−m P µ{3 + µ(P − 2 + 4π 2

2

(A.4)

Pµ){−5 + 5π  + P(−5 + 5π µ(2 + P) − 2R (1 + Pµ))} 2

2

)} − k12 π 6 (1 + 4

(A.5)

Pµ){5(5π  − 4) 2

+P(−20 + (25π 2 µ(2 + P) + 2(1 + Pµ)(14π µ − 3R)))} +2k1 m Pπ 2 {4 + µ(2 + 9π 2  + 10π 4  2 + P 2 µ(9π 2 − R + 10π 4 µ) +P(6 + (−R + 20π 4 µ + 2π 2 (7 + 2µ))))}]

(A.6)

γ7 = −4k1 π 2 (1 + Pµ)[m 2 P 2 µ(2 + 3π 2 µ) + k1 m Pπ 2 {−3 − µ(8π 2 (1 + P) − P R −5π 4 (1 + 2Pµ))} + k12 π 6 {−5 + 6π 2  + P(−5 +(6π 2 µ(2 + P) + (1 + Pµ)(−R + 7π 4 µ)))}]

(A.7)

Onset of electroconvective instability of Oldroydian viscoelastic liquid layer in Brinkman porous medium

223

γ8 = m 3 P 3 µ2 (1 + 2π 2 µ) + k1 m 2 P 2 π 2 µ[−2 − {−P Rµ +4π 4 µ(3 + 4Pµ) + π 2 (7 + 6µ + 13Pµ)}] − k13 π 10 (1 + Pµ) ×[10 + 11π 2  + P{−10 + (11π 2 µ(2 + P) + (1 + Pµ)(−R +12π 2 µ))}] + k1 m Pπ 6 [8 + µ{−4 + π 2 (23 + 10π 2 ) +P 3 µ(23π 2 − 2R + 26π 4 µ) + 2P(2 + (−R + π 2 (9 + 2(7 + 9π 2 ))))}]

(A.8)

γ9 = 2k12 π 8 [m P + k1 π 4 (1 + P − π 2 )] + 2k1 Pπ 4 (k1 π 4 − m) ×[−m P(1 + π 2 ) + k1 π 4 µ{−1 − P + π 2 (3 + P) + π 4  2 }] −2P 2 π 2 µ2 (m − k1 π 4 )2 [−m P + k1 π 4 (2 + P + 2π 2 )] − 2P 3 π 4  2 µ3 (k1 π 4 − m)3 .

(A.9)

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