The study of thermal convection in dielectric fluids has gained much importance in recent past due to its manifold applications in nuclear reactors, ink jet printing, coalescence and many other processes (Del Río and Whitaker [7]). Several theoretical and experimental studies on the convective instabilities in a dielectric fluid layer heated from below/above in the presence of an electric field have been carried out in the recent past. The convection produced in a dielectric fluid layer heated from above was reported by Gross and Porter [10] and Turnbull [33], which was kept under the influence of a uniform electric field. Roberts [27] investigated electroconvection by assuming the dielectric constant as a function of temperature. Castellanos and Velarde [3] studied the influence of a temperature-dependent dielectric constant in the stability analysis of a fluid layer subjected to an electric field, weak unipolar injection and temperature gradient. Maekawa et al. [14] investigated the convective instability problem in alternating current (AC) and direct current (DC) electric fields using linear stability theory. Exhaustive reviews in this domain of enquiry have been presented by Jones [12] and Saville [29].
Two different types of instabilities are observed experimentally by Gross and Porter [10] and Turnbull [34] for horizontal dielectric fluid layers heated from above; the former observed a stationary instability, whereas the latter observed the manifestation of oscillatory instability, also known as overstability.
Turnbull [33] and Bradley [2] predicted oscillatory convection by using a quadratic conductivity model and a linear conductivity model, respectively. Castellanos and Velarde [3] studied the influence of a temperature-dependent dielectric constant on the stability of a liquid layer in the presence of an electric field, weak unipolar injection and temperature gradient and predicted that oscillatory instability occurs only when the heating is from above. Martin and Richardson [15] investigated the linear instability of a unipolar charge injection model and predicted numerically that stationary instability is dominant if temperature gradient is weakly stabilising, whereas oscillatory instability is dominant if the temperature gradient is strongly stabilising. Later, Martin and Richardson [16] derived a conductivity model and predicted the manifestation of oscillatory instability by investigating numerically the linear instabilities for linear quadratic and Arhenius-type conductivity variations.
Turnbull [35] also studied the effect of dielectrophoretic forces on an insulating fluid layer heated from below. He proved the validity of the principle of the exchange of stabilities for a particular set of boundary conditions for free boundaries, which were not usually used by the subsequent researchers. Bradley [2] too discussed such Bénard-type situation by assuming Prandtl number
The investigations of convective instabilities in porous media have been an important domain of research due to its wide applications in different fields like nuclear waste repository, radioactive waste management, solid matrix compact heat exchangers, thermal insulation engineering, mantle convection, geophysical systems and many more (Nield and Bejan [18],[19]). For the copious literature related to this domain of research, one may be referred to Chaudhary and Sunil [6], Ingham and Pop [11], Nield and Bejan [18],[19], Prakash et al. [23], Prakash et al. [24] and Vafai [36]. Electrothermoconvection in a dielectric fluid layer saturating porous medium subjected to an external electric field is of particular interest in the light of its possibility to reduce the fluid viscosity, which results in increasing the petroleum production and a control of heat and mass transfer in high-voltage devices by electric field (Moreno et al. [17]. Several researchers have contributed to the electrothermoconvection studies in dielectric fluid layer saturating a porous medium. The study of convective instability of dielectric fluids in porous media is also of practical importance in many domains such as chemical engineering, material science processing, biomechanics of the design of artificial organs and purification of ground water pollution (Rudraiah and Gayathri [28]). For a broad view of the subject, one may refer to Del Río and Whitaker [7], El-Sayed et al. [9], El-Sayed et al. [8], Rudraiah and Gayathri [28] and Shivakumara et al. [30].
The problem of deriving upper limits for the complex rate of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude in various hydrodynamic stability problems is an important feature of fluid dynamics, especially when both the bounding surfaces are not free, so that the exact solutions in a closed form are not derivable. Banerjee et al. [1] formulated a method to combine the governing equations and boundary conditions for classic thermohaline convection problem, which, in turn, yields the desired bounds. Their work is further extended to different hydrodynamic configurations by Prakash [21], Prakash et al. [22], Prakash et al. [25] and Ram et al. [26]. Since the inability of finding the exact solutions in a closed form also exists for the case of electrothermoconvection problems when both the boundaries are not free, the upper limits for the complex growth rate for such configurations must also be found. The extension of Banerjee et al. [1] result in a more complex problem of electrothermoconvection in the domains of astrophysics, geophysics and terrestrial physics, wherein the liquid concerned has the property of electrical conduction and rotation are prevalent, is very much sought after in the present context. This paper, which mathematically establishes the upper limits for the complex growth rate of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude, in a dielectric fluid layer saturating a sparsely distributed porous medium wherein a uniform rotation parallel to gravity is superimposed, may be regarded as a first step in this scheme of extended investigations. Thus, novelty of the present paper mainly relies on these newly derived upper limits for the complex growth rate of a more complex problem of electrothermoconvection, which will definitely facilitate the theoretical scientists and experimentalists in their investigations.
We consider a dielectric fluid layer of infinite horizontal extension and finite vertical depth
Geometrical configuration of the problem.
The basic equations, governing the flow of dielectric fluid for the present model, are given by (Shivakumara et al. [31], Takashima, [32])
The equation of state is given by
The Maxwell’s equations relevant to the present context are
The dielectric constant is given by
Now, following the linear stability theory (Chandrasekhar [5]), using basic state solutions, linearised perturbation equations, normal mode technique ascribing, to all the quantities describing the perturbation, a dependence on
The boundaries are considered to be free and rigid. Hence, the boundary conditions are given by
Now we derive the upper limits for the complex growth rate of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude for a dielectric fluid layer saturating a sparsely distributed porous medium subjected to uniform vertical rotation for the cases when (I) layer is heated from below and (II) layer is heated from above.
We prove the following theorem:
If
Multiplying Eq. (11) by
Equating imaginary parts of both sides of Eq. (22), we get
Multiplying Eq. (14) by Φ* and integrating, by parts, we have
Shaded region shows the region of complex growth rate in the
For the case of rigid boundaries, the boundary conditions are given by Eq. (16)
Following the same procedure as is used in Theorem 1, we derive the same integrated Eq. (22) in this case also. But for the case of rigid boundaries, the third integral in the right hand side of this equation cannot be dropped from the imaginary part without justification. This is elaborated as follows:
In this case on multiplying Eq. (14) by When
When imaginary part of
If
Integrating the various terms of Eq. (21), by parts, for appropriate number of times and making use of Eqs (12), (13) and boundary conditions (16), we get,
From the physical standpoint, Theorem 2 can be stated as: the complex growth rate of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude, in a rotating dielectric fluid layer saturating sparsely distributed porous medium heated from below, lies inside the region (in the right half of the
Shaded region shows the region of complex growth rate in the
We prove the following theorem:
If
In the present case,
From the physical standpoint, Theorem 3 can be stated as: the complex growth rate of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude, in a rotating dielectric fluid layer saturating sparsely distributed porous medium heated from above, lies inside a semicircle in the right half of the
Shaded region shows the region of complex growth rate in the
Similar arguments hold for the present case as were used in subcase (ii) of case I, that is, when
If
Multiplying Eq. (44) by
which on using inequality (30) yields
Shaded region shows the region of complex growth rate in the
The linear stability theory has been used to derive the upper limits for the complex growth rates in electrothermoconvection in a dielectric fluid layer in a sparsely distributed porous medium heated from below and from above in the presence of electric field for free and rigid bounding surfaces separately. The following results are obtained:
The complex growth rate of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude, in a rotating dielectric fluid layer saturating sparsely distributed porous medium, lies inside a semicircle, in the right half of the
For the case of rigid boundaries, when
The complex growth rate of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude, in a rotating dielectric fluid layer saturating sparsely distributed porous medium, lies inside a semicircle, in the right half of the
For the case of rigid boundaries, similar arguments hold as in the case of heated from below.
Furthermore, the results derived herein involve only dimensionless quantities and are wave number independent; thus, the present results are of uniform validity and applicability.