EFFECT OF BOUNDARY CONDITIONS ON ELECTROTHERMAL CONVECTION IN A POROUS MEDIUM WITH VARIABLE VISCOSITY

The effect of thermal and velocity boundary conditions on electrothermal convection in a dielectric fluidsaturated layer of Brinkman porous medium with temperature dependent viscosity (TDV) has been studied. The lower rigid/free boundary is either fixed temperature or fixed heat flux with respect to temperature perturbations, while at the upper rigid/free boundary the Robin type of thermal boundary condition is invoked. The eigenvalue problem is solved numerically using the Galerkin-type of weighted residual method. The instability threshold depends significantly on boundary conditions and dimensionless physical parameters namely, the Biot number, temperature dependent viscosity and permeability parameters. The critical gravity thermal or electric thermal Rayleigh number making the onset of electrothermal convection is found to be higher for fixed temperature conditions and also for both boundaries rigid while it is lower for fixed heat flux conditions and also for both boundaries free. Some known results are recovered as special cases from the present study.


Introduction
Electrohydrodynamics (EHD) is the branch of fluid mechanics which deals with more complex interactions among fluid, heat and electric fields [1].Electrohydrodynamics has wide applications in flow and heat transfer control, enhancement of heat and mass transfer, micro-electromechanical systems (MEMS) and some other industrial processes [2].Recently, two conceptual designs were established with applications in the cooling system of laptops and devices on a flight in space [3,4].The relative backwardness of such applications with the EHD technique is attributed to the lack of complete mastering of the characteristics of flow motion.
The study of convective instability in the presence of electric field and buoyancy effects is called the Rayleigh-Bénard-electroconvection or electrothermal convection (see Taylor [5]).Turnbull and Melcher [6] studied the natural convective instability problem of electroconvection under an applied AC or DC electric field and also successfully carried an experiment to verify the theoretical prediction.Roberts [7] studied theoretically the onset of electroconvection in an insulating fluid layer subject to temperature gradients and electrical potential differences across the fluid layer.Turnbull [8] analyzed the effect of dielectrophoretic force on the onset of natural convection by considering electrical conductivity is temperature dependent.Many researchers investigated Rayleigh-Bénard electroconvection considering various effects [9][10][11][12][13][14][15][16][17].Shivakumara et al. [18] investigated the problem of electrothermal convection in a rotating dielectric fluid layer by taking different boundary conditions while the effect of couple stresses on the aforementioned problem has been presented by Shivakumara et al. [19].
Thermal convection in a layer of dielectric fluid-saturated porous medium under a uniform vertical AC electric field has also attracted significant attention in the literature due to its importance in geophysics and porous materials modeling.Moreno et al. [20] studied on fluid flow in a porous medium subjected to an external electric field, particular importance in view of its possibility of reduction of fluid viscosity in enhancing petroleum production.Rio and Whitaker [21] developed the frequency-dependent governing equations for EHD in saturated porous medium.Rudraiah and Gayathri [22] investigated the temperature modulation effect on electroconvection in a dielectric fluid-saturated porous medium in the presence of a uniform vertical AC electric field.Shivakumara et al. [23] analyzed the onset of electrothermal convection in a dielectric fluidsaturated Brinkman porous layer for different velocity boundary conditions while the additional effect of rotation on the above study is considered by Shivakumara et al. [24].In all of these investigations, the fluid viscosity is considered to be a constant.In reality, the viscosity is a strong function of temperature and it affects the stability of the system significantly.
The aim of the present paper is to determine analytically the effect of temperature dependent viscosity (TDV) on thermal convection in a dielectric fluid-saturated porous medium in the presence of a uniform vertical AC electric field for different temperature and velocity boundary conditions.The lower rigid/free surface is considered to be either conducting (constant temperature) or insulating with respect to temperature perturbations (constant heat flux) while at the upper rigid/free surface the Robin type of thermal boundary condition is utilized.The outline of the present paper is as follows.The mathematical formulation of the problem is described in section 2. The basic state equations are derived in section 3. The linear instability analysis using normal mode expansion procedure is also handled in this section.In the same section the eigenvalue problem involving the system of differential equations and the boundary conditions are also specified in this section.The numerical results obtained using the ninth-order Galerkin weighted residual method are discussed and explained in detail in section 4. Finally, some conclusions are documented in section 5.

Mathematical Formulation
The schematic geometry of the problem considered is presented in Fig. 1.We consider an incompressible dielectric fluid-saturated Brinkman porous medium under a uniform AC electric field acting perpendicular to the horizontal porous layer bounded by the surfaces 0 z = and .zd = The lower and upper surfaces are maintained at temperatures 0 TT = and 10 A Cartesian coordinate system (x, y, z) is chosen such that the origin is at the bottom of the porous layer.Gravity is acting in the negative vertical z-direction.
The viscosity  is considered to be varying linearly The governing equations in the relevant context are: Mass conservation: Momentum conservation: ( ) Energy conservation: Electrical equation: Maxwell equation: The last term in Eq. ( 2) is the electric force induced by the electrical field which is of the form: In Eq. ( 6), the first term stands for Coulomb force exerted by an electric field upon the free charge within the bulk liquid.The second term is dielectrophoretic (or dielectric) force and the third term is electrostrictive force which is a conservative vector and can be conveniently combined with static pressure.The additional quantities appeared in the above set of governing equations are defined in the nomenclature.The standard linear stability analysis procedure leads to (for details see [8] and [17]) where where x a and y a are wave numbers in x and y directions, respectively,  is the growth factor which is complex, in general, while W ,  and  are the amplitudes of perturbed velocity, temperature and electric potential, respectively.Using Eq. ( 10) in Eqs. ( 7)-( 9) and nondimensionalizing the resulting equations by applying the definitions: , W=( / ) *, t= t* we obtain the stability equations (after ignoring the asterisks) in the form where 1 fz = +  .By performing qualitative analysis on the oscillatory instability, Shivakumara et al. [23] have shown that the principle of exchange of stability is valid for the onset of Darcy-Brinkman electrothermal convection irrespective of the nature of velocity boundary conditions.This is expected as there are no physical mechanisms to set up oscillatory motions when a dielectric fluid-saturated porous medium under a uniform vertical AC electric field is heated from below.Here, the motion, temperature and electric fields are all in phase and no restoring force exists and hence oscillatory convection is not possible.Similar is the situation in the present paper and of course the variation in viscosity with respect to temperature does not introduce oscillatory motions.Therefore, the principle of exchange of stability is considered to be valid in the present case as well and take 0 The above equations are solved by imposing the following types of boundary conditions: (i) Both boundaries rigid (R-R boundaries):

Method of solution
Equations (13a-c) together with the chosen boundary conditions constitute an eigenvalue problem which has been solved numerically using the Galerkin technique.Accordingly, the unknown variables are written in series of trial (basis) functions as


; and integrating the resulting relations over the region {0 1} V z =   and using the boundary conditions, we obtain a system of algebraic equations which can be written in the form 0 00 0 0 where, , (1),   =  .The system of equations given by Eq. ( 18) has a non-trivial solution if and only if the determinant of the coefficient matrix is zero.That is, 00 The eigenvalue has to be extracted from Eq. (19).For this, we select the following trial functions satisfying the boundary conditions: (i) R-R boundaries: (ii) R-F boundaries: (iii) F-F boundaries:

Results and Discussion
The Galerkin-type of weighted residual method is used to obtain the critical values of To validate the results obtained by applying the numerical procedure, a comparison with some existing results is made (see Tables 1 and 2 Ra are compared with those of Roberts [7] in Table 3 for selected values of Chiang [26] are also presented and note that there is an excellent agreement between the present results and those of [26].Here, we find that tc R decreases with increasing AC electric field strength. In Figs.3-9, the plots of ec R and ec a against Bi are illustrated.In these figures, the solid curves correspond to temperature boundary condition of the type (0) 0  = while the dotted curves correspond to the condition of the type (0) 0 D = .Figure 3  Bi is to increase the critical electrical thermal Rayleigh number and thus its effect is to delay the onset of electrothermal convection.This may be attributed to the fact that with increasing Bi , the thermal disturbances can easily dissipate into the ambient surrounding due to a better convective heat transfer coefficient at the top surface and hence higher heating is required to make the system unstable.On the upper free surface, for small values of Bi , these perturbations are very prone to heat transfer coefficient and for large values of Bi , these can be regarded as an imposed constant temperature that causes ec R to approach this asymptotic value.The critical wave number reported in Fig. 4  It is seen that increase in the value of permeability parameter 2  σ is to delay the onset of electrothermal convection.Here, we note that the results for two types of temperature boundary conditions differ only quantitatively and the system is found to be more stable when the lower surface is fixed at constant temperature as expected.From Fig. 6, it is seen that the critical electric wave number ec a increases as 2  σ increases.Therefore, increase in the porous parameter is to reduce the size of convection cells.
The effect of temperature dependent viscosity parameter Γ on the onset of electrothermal convection in a dielectric fluid saturated porous medium is presented in Fig. 7 for fixed values 2 Bi = , Here, we focus on the strengths between buoyancy and electric forces on the stability of the system.If there is an increase in the strength of one, then there is a decrease in the other.Thus the strength of AC electric field leads to destabilizing effect on the system.This result is true for all the boundary conditions considered.A closer inspection of the figures further reveals that

Conclusion
The effect of variable viscosity on the onset of electrothermal convection in a porous medium under a uniform vertical AC electric field has been studied for different types of velocity and temperature boundary conditions.It is observed that: 1.The onset of electrothermal convection is to be delayed with increasing Bi 2. The effect of increasing AC electric field is to hasten the onset of convection.
3. The system is more stable for R-R surfaces while for F-F surfaces it is least stable.Also, =0 =0 ( and ) ( and )


and  are positive constants.
 +   is the horizontal Laplacian operator.We employ the normal mode expansion in the form of rate of heat from the interface to the environment to the rate of heat supply to the interface from the bulk of a fluid due to the thermal conduction at the upper boundary.Increase in Bi from 0 to  means change in the thermal condition at the upper boundary from "fixed heat flux condition" or "insulating case" (i.e., 0 D= ) to the "constant temperature" or "conducting case" (i.e., 0 = ).


represent the basis functions.On substituting Eq.(17) into Eqs.(13a-c), multiplying both sides of resulting Eq.(13a) by() 2,...... m = ) of the second kind.It is seen that the trial functions chosen satisfy the respective boundary conditions except the thermal boundary condition 0 D Bi  +  = at 1 z = but the residue is included from the differential Eqs.(13a-c).On substituting Eqs.(20)(21)(22) into Eq.(19) leads to an equation of the form:

R
for various boundary conditions (R-R, R-F and F-F), temperature dependent viscosity (TDV) and porous parameter are presented in Tables 1-3 and also shown graphically in Figs.2-8.
with those of Sparrow et al.[25].Besides, the critical stability parameters ( , ) ec c  .In Fig.2, the results of Char and

R
= .It shows that the results are bridging the space between the fixed heat flux and constant temperature at the upper surface with increasing Bi.Clearly, imposing fixed heat flux condition at the lower surface advances the electrothermal convection compared to constant temperature condition.Figure3also reveals that the system with R-R surfaces is stable compared to F-F surfaces.The values of ec R initially increases slowly with Bi and then increases quickly and approaches an asymptotic with further increasing Bi for R-R, R-F and F-F boundaries, respectively when the lower surface is held at constant temperature while the asymptotic values for the said boundaries are found to be 1271.99ec R = , 928.858 and 814.86 when the lower surface is held at fixed heat flux condition.It is also evident that the dielectric fluid layer under an AC electric field becomes more stable with increasing Bi .Besides, increase in the value of heat transfer coefficient increasing Γ indicating its effect is stabilizing on the system.That is, the effect of increasing Γ is to delay electrothermal convection in the presence of AC electric field.Whereas Fig.8reveals that the variation in ec a with Γ is insignificant.In Fig.9, the variation of ec R and tc R is plotted for fixed values of 2 Bi = , 2 σ 10 = and Γ 0.2 = .

 4 .T
The critical electric wave number ec a increases with increasing Bi and 2 σ Thus their effect is to contract the size of convection cells.Also, = temperature at upper surface V = root mean square velocity of the electric potential W = amplitude of perturbed vertical velocity component x, y, z = Cartesian co-ordinates = reference temperature for dielectric constant  (>0)= analog for dielectric constant of thermal expansion coefficient  = porosity  = amplitude of perturbed electric potential  = temperature dependent viscosity  = effective thermal diffusivity 0 . = amplitude of perturbed temperature

Table 1
Comparison of Rtc and ac for lower boundary rigid at fixed temperature with different values of Bi in the absence of AC electric field strength with 2

Table 2
Comparison of Rtc and ac for lower boundary rigid at fixed heat flux with different values of Bi in the absence of AC electric field strength with 2