On the upper limits for complex growth rate in rotatory electrothermoconvection in a dielectric fluid layer saturating a sparsely distributed porous medium

1. Introduction

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 P_r to be finite and a nameless number $P=\frac{\epsilon \nu }{c^{*}{d}^2}$ , (c^* is conductivity) and showed that the layer cannot be unstable for all top-heavy density distributions (all adverse temperature gradients), although it is plausible that they will be overstable if the electric field is sufficiently strong. Castellanos et al. [4] studied the oscillatory and steady convection in dielectric liquid layer subjected to unipolar injection heated underside.

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.

2. Formulation of the Problem

We consider a dielectric fluid layer of infinite horizontal extension and finite vertical depth d saturating a sparsely distributed porous medium, heated from below (Fig. 1). The fluid layer is subjected to a rotation about vertical axis with a constant angular velocity $\overrightarrow{\Omega }=\left(0,0,\Omega \right)$ . A uniform vertical AC electric field is applied across the fluid layer. The Darcy–Brinkman model has been used to mathematically analyse this problem.

Figure 1:

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 x, y and t of the form exp [i(k_xx + k_yy) + nt], we derive the non-dimensional equations given by

The boundaries are considered to be free and rigid. Hence, the boundary conditions are given by

3. Mathematical Analysis

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.

3.1. Case I: When the fluid layer is heated from below

Subcase (i): Free boundaries

We prove the following theorem:

Theorem 1

If R_t > 0, R_ea > 0, P_r > 0, T_a > 0, A > 0, D_a > 0, n = n_r + in_i, n_r ≥ 0 and n_i ≠ 0, then a necessary condition for the existence of a non-trivial solution (w, θ, ζ, Φ, n) of Eqs (11)–(14) together with the boundary conditions (15) is

$${\left.n\right|}^2<max\left(T_a{P}_r^2,\frac{R_{\mathit{ea}}{P}_r}{A}\right).$$

Proof

Multiplying Eq. (11) by w^* (the superscript ^* henceforth denotes the complex conjugation) and integrating the resulting equation from z = 0 to z = 1, we have

(17)

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}\left(D^2-a^2\right)\left(\Lambda \left(D^2-a^2\right)-D_a^{-1}-\frac{n}{P_r}\right)\mathit{wdz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\left(R_t+R_{\mathit{ea}}\right)a^2{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}\theta \mathit{dz}}+T_a^{\frac{1}{2}}{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\zeta \mathit{dz}}+R_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\Phi \mathit{dz}}\hfill \end{array}$$

Utilising Eqs (12)–(14) and the boundary conditions (15), we can write

(18)

$$\begin{array}{l}\left(R_t+R_{\mathit{ea}}\right)a^2{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}\theta \mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-\left(R_t+R_{\mathit{ea}}\right)a^2{\displaystyle \underset{0}{\overset{1}{\int }}\theta \left(D^2-a^2-{\mathit{An}}^{*}\right){\theta }^{*}\mathit{dz},}\hfill \end{array}$$

(19)

$$\begin{array}{l}{T}_a^{\frac{1}{2}}{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\zeta \mathit{dz}}=-T_a^{\frac{1}{2}}{\displaystyle \underset{0}{\overset{1}{\int }}{\mathit{Dw}}^{*}\zeta \mathit{dz}}\hfill \\ \hspace{0.17em}={\displaystyle \underset{0}{\overset{1}{\int }}\zeta \left(\Lambda \left(D^2-a^2\right)-D_a^{-1}-\frac{n^{*}}{P_r}\right){\zeta }^{*}\mathit{dz},}\hfill \end{array}$$

(20)

$$\begin{array}{l}{R}_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\Phi \mathit{dz}}=-R_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}D\Phi \left(D^2-a^2-{\mathit{An}}^{*}\right){\theta }^{*}\mathit{dz}}\hfill \\ =-R_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}\mathit{dz}}+R_{\mathit{ea}}{a}^2\left(a^2+{\mathit{An}}^{*}\right){\displaystyle \underset{0}{\overset{1}{\int }}{\theta }^{*}D\Phi \mathit{dz}}\hfill \\ =-R_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}\mathit{dz}}-R_{\mathit{ea}}{a}^2\left(a^2+{\mathit{An}}^{*}\right){\displaystyle \underset{0}{\overset{1}{\int }}\Phi D{\theta }^{*}\mathit{dz}}\hfill \\ =-R_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}\mathit{dz}}+R_{\mathit{ea}}{a}^2\left(a^2+{\mathit{An}}^{*}\right){\displaystyle \underset{0}{\overset{1}{\int }}\Phi \left(D^2-a^2\right){\Phi }^{*}\mathit{dz}.}\hfill \end{array}$$

Eqs (17)–(20), on combining, yield

(21)

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}\left(D^2-a^2\right)\left(\Lambda \left(D^2-a^2\right)-D_a^{-1}-\frac{n}{P_r}\right)\mathit{wdz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-\left(R_t+R_{\mathit{ea}}\right)a^2{\displaystyle \underset{0}{\overset{1}{\int }}\theta \left(D^2-a^2-{\mathit{An}}^{*}\right){\theta }^{*}\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \underset{0}{\overset{1}{\int }}\zeta \left(\Lambda \left(D^2-a^2\right)-D_a^{-1}-\frac{n^{*}}{P_r}\right){\zeta }^{*}\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-R_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+R_{\mathit{ea}}{a}^2\left(a^2+\mathit{An}*\right){\displaystyle \underset{0}{\overset{1}{\int }}\Phi \left(D^2-a^2\right)\Phi *\mathit{dz}}\hfill \end{array}$$

Integrating the various terms of Eq. (21), by parts, for a suitable number of times from z = 0 to z = 1, we get

(22)

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}\Lambda \left({\left.D^{2}w\right|}^2+2a^2{\left.\mathit{Dw}\right|}^2+a^4{\left.w\right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(D_a^{-1}+\frac{n}{P_r}\right){\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.\mathit{Dw}\right|}^2+a^2{\left.w\right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\left(R_t+R_{\mathit{ea}}\right)a^2{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\theta \right|}^2+a^2{\left.\theta \right|}^2+{\mathit{An}}^{*}{\left.\theta \right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \underset{0}{\overset{1}{\int }}\left(\Lambda \left({\left.D\zeta \right|}^2+a^2{\left.\zeta \right|}^2\right)+D_a^{-1}{\left.\zeta \right|}^2+\frac{n^{*}}{P_r}{\left.\zeta \right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-R_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-R_{\mathit{ea}}{a}^2\left(a^2+{\mathit{An}}^{*}\right){\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\Phi \right|}^2+a^2{\left.\Phi \right|}^2\right)\mathit{dz}.}\hfill \end{array}$$

Now multiplying Eq. (14) by Dθ^* and integrating the resulting equation from z = 0 to z = 1, we obtain

(23)

$${\displaystyle \underset{0}{\overset{1}{\int }}{D}^2}{\theta }^{*}D\Phi \mathit{dz}+a^2{\displaystyle \underset{0}{\overset{1}{\int }}\Phi D{\theta }^{*}\mathit{dz}}={\displaystyle \underset{0}{\overset{1}{\int }}{\left.D\theta \right|}^2\mathit{dz}}.$$

Also, when we multiply the complex conjugate of Eq. (14) by Φ and integrate over the vertical range of z, we obtain

(24)

$${\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\Phi \right|}^2+a^2{\left.\Phi \right|}^2\right)\mathit{dz}}={\displaystyle \underset{0}{\overset{1}{\int }}\Phi D{\theta }^{*}\mathit{dz}}.$$

It is obvious from Eqs (23) and (24) that ${\displaystyle \underset{0}{\overset{1}{\int }}{D}^2{\theta }^{*}D\Phi \mathit{dz}}$ is real.

Equating imaginary parts of both sides of Eq. (22), we get

(25)

$$n_i\left(\begin{array}{c}{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.\mathit{Dw}\right|}^2+a^2\left.w^2\right|\right)\mathit{dz}}+a^{2}A\left(R_t+R_{\mathit{ea}}\right)P_r{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}-{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\zeta \right|}^2\mathit{dz}}\\ -R_{\mathit{ea}}{a}^2{\mathit{AP}}_r{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\Phi \right|}^2+a^2{\left.\Phi \right|}^2\right)\mathit{dz}}\end{array}\right)=0.$$

It is evident from Eq. (25) that one cannot conclude from it, as in Pellew and Southwell’s [20] case that n_r = 0 implies n_i = 0 for all a². This is due to the fact that the last two terms in the left-hand side of this equation misbehave as regards their signs. Thus, oscillatory instability may occur in certain parameter regime for the present problem. Hence, we derive upper limits for the complex growth rate of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude, which are important, especially when at least one of the bounding surfaces is rigid, so that exact solutions are not obtainable. We proceed in the following manner:

Multiplying Eq. (14) by Φ^* and integrating, by parts, we have

(26)

$$\begin{array}{c}{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\Phi \right|}^2+a^2{\left.\Phi \right|}^2\right)\mathit{dz}}=-{\displaystyle \underset{0}{\overset{1}{\int }}\theta D{\Phi }^{*}\mathit{dz}}\\ \le \left.{\displaystyle \underset{0}{\overset{1}{\int }}\theta D{\Phi }^{*\mathit{dz}}}\right|\\ \le {\displaystyle \underset{0}{\overset{1}{\int }}\left.\theta \right|\left.D\Phi \right|\mathit{dz}}\\ \le {\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}\right)}^{\frac{1}{2}}{\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.D\Phi \right|}^2\mathit{dz}}\right)}^{\frac{1}{2}},\end{array}$$

(Utilising the Schwartz inequality) which implies that

$${\displaystyle \underset{0}{\overset{1}{\int }}{\left.D\Phi \right|}^2\mathit{dz}\le {\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}\right)}^{\frac{1}{2}}{\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.D\Phi \right|}^2\mathit{dz}}\right)}^{\frac{1}{2}}}.$$

Thus

(27)

$${\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.D\Phi \right|}^2\mathit{dz}}\right)}^{\frac{1}{2}}\le {\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}\right)}^{\frac{1}{2}}.$$

Combining inequalities (26) and (27), we get

(28)

$${\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\Phi \right|}^2+a^2{\left.\Phi \right|}^2\right)\mathit{dz}}\le {\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}.}$$

Multiplying Eq. (12) by its complex conjugate and integrating the resulting equation, by parts, for an appropriate number of times and making use of boundary conditions (15), we obtain

(29)

$${\displaystyle \underset{0}{\overset{1}{\int }}\left(\begin{array}{l}{\left.D^2\theta \right|}^2+2a^2{\left.D\theta \right|}^2+a^4{\left.\theta \right|}^2\hfill \\ +2{\mathit{An}}_r\left({\left.D\theta \right|}^2+a^2{\left.\theta \right|}^2\right)+A^2{\left.n\right|}^2{\left.\theta \right|}^2\hfill \end{array}\right)\mathit{dz}={\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}},$$

Since n_r ≥ 0, Eq. (29) implies that

(30)

$${\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}\le \frac{1}{A^2{\left.n\right|}^2}{\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}.$$

Using Eq. (30) in Eq. (28), we get

(31)

$${\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\Phi \right|}^2+a^2{\left.\Phi \right|}^2\right)\mathit{dz}}\le \frac{1}{A^2{\left.n\right|}^2}{\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}.$$

Multiplying Eq. (13) by its complex conjugate and integrating the resulting equation, by parts, for an appropriate number of times and making use of boundary conditions (15), we obtain

(32)

$${\displaystyle \underset{0}{\overset{1}{\int }}\left(\begin{array}{l}{\Lambda }^2\left({\left.D^2\zeta \right|}^2+2a^2{\left.\zeta \right|}^2+a^4{\left.\zeta \right|}^2\right)+2\Lambda D_a^{-1}\left({\left.D\zeta \right|}^2+a^2{\left.\zeta \right|}^2\right)\hfill \\ +\frac{2\Lambda n_r}{P_r}\left({\left.D\zeta \right|}^2+a^2{\left.\zeta \right|}^2\right)+D_a^{-1}{\left.\zeta \right|}^2+\frac{2D_a^{-1}{n}_r}{P_r}{\left.\zeta \right|}^2+\frac{{\left.n\right|}^2}{P_r^2}{\left.\zeta \right|}^2\hfill \end{array}\right)\mathit{dz}}=T_a{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\mathit{Dw}\right|}^2\mathit{dz}.}$$

Since, n_r ≥ 0, from Eq. (32), we can write

(33)

$${\displaystyle \underset{0}{\overset{1}{\int }}{\left.\zeta \right|}^2\mathit{dz}}\le \frac{T_a{P}_r^2}{{\left.n\right|}^2}{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\mathit{Dw}\right|}^2\mathit{dz}}.$$

Using inequalities (31) and (33) in Eq. (25), we get

(34)

$$\begin{array}{l}\left(1-\frac{T_a{P}_r^2}{{\left.n\right|}^2}\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left.\mathit{Dw}\right|}^2\mathit{dz}}+a^2\left(1-\frac{R_{\mathit{ea}}{P}_r}{A{\left.n\right|}^2}\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+a^2\left(R_t+R_{\mathit{ea}}\right)P_{r}A{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}\le 0,}\hfill \end{array}$$

which implies

$${\left.n\right|}^2<max\left(T_a{P}_r^2,\frac{R_{\mathit{ea}}{P}_r}{A}\right).$$

From the physical standpoint, Theorem 1 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 a semicircle in the right half of the n_rn_i − plane whose centre is at the origin and radius equals $\sqrt{max\left(T_a{P}_r^2,\frac{R_{\mathit{ea}}{P}_r}{A}\right)}$ (Fig. 2).

Figure 2:

Subcase (ii): Rigid boundaries

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 Dθ^* and integrating the resulting equation from z = 0 to z = 1, we get

(35)

$${\left.D{\theta }^{*}D\Phi \right]}_0^1-{\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}\mathit{dz}}-a^2{\displaystyle \underset{0}{\overset{1}{\int }}\Phi D{\theta }^{*}\mathit{dz}}=-{\displaystyle \underset{0}{\overset{1}{\int }}{\left.D\theta \right|}^2\mathit{dz}}.$$

Since ${\displaystyle \underset{0}{\overset{1}{\int }}{\left.D\theta \right|}^2\mathit{dz}}$ is real and the integral ${\displaystyle \underset{0}{\overset{1}{\int }}\Phi D{\theta }^{*}\mathit{dz}}$ is also real by Eq. (24), therefore from Eq. (35), two cases arise:

• (a) When ${\left.D{\theta }^{*}D\Phi \right]}_0^1$ and ${\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}\mathit{dz}}$ are both real, then on equating the imaginary parts of both sides of Eq. (22) and cancelling n_i (≠ 0) both sides, we obtain an equation that is exactly the same as Eq. (25) and the proof follows as for the case of free boundaries to obtain the same result.

• (b) When imaginary part of ${\left.D{\theta }^{*}D\Phi \right]}_0^1$ = imaginary part of ${\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}dz}$ we proceed as follows:

Theorem 2

If R_t > 0, R_ea > 0, P_r > 0, T_a > 0, A > 0, D_a > 0, Λ > 0, n = n_r + in_i , n_r ≥ 0 and n_i ≠ 0 , then a necessary condition for the existence of a non-trivial solution (w, θ, ζ, Φ, n) of Eqs (11)–(14) together with the boundary conditions (16) is

$$n^2{n}_i^2<max\left(T_a^2{P}_r^4,{\left(\frac{R_{\mathit{ea}}{P}_r}{A}\right)}^2\right).$$

Proof

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,

(36)

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}\Lambda \left({\left.D^{2}w\right|}^2+2a^2{\left.\mathit{Dw}\right|}^2+a^4{\left.w\right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(D_a^{-1}+\frac{n}{P_r}\right){\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.\mathit{Dw}\right|}^2+a^2{\left.w\right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\left(R_t+R_{\mathit{ea}}\right)a^2{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\theta \right|}^2+a^2{\left.\theta \right|}^2+{\mathit{An}}^{*}{\left.\theta \right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \underset{0}{\overset{1}{\int }}\left(\Lambda \left({\left.D\zeta \right|}^2+a^2{\left.\zeta \right|}^2\right)+\frac{n^{*}}{P_r}{\left.\zeta \right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+R_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\Phi \mathit{dz}}.\hfill \end{array}$$

Equating imaginary parts of both sides of Eq. (36) and cancelling n_i (≠ 0) throughout, we get

(37)

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.\mathit{Dw}\right|}^2+a^2{\left.w\right|}^2\right)\mathit{dz}}\hfill \\ =-\left(R_t+R_{\mathit{ea}}\right)a^2{\mathit{AP}}_r{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}+{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\zeta \right|}^2\mathit{dz}}\hfill \\ +\frac{R_{\mathit{ea}}{a}^2{P}_r}{n_i}\mathit{imaginary}\hspace{0.17em}\mathit{part}\hspace{0.17em}\mathit{of}\hspace{0.17em}{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\Phi \mathit{dz}}.\hfill \end{array}$$

Now

(38)

$$\begin{array}{c}\frac{R_{\mathit{ea}}{a}^2{P}_r}{n_i}\mathit{imaginary}\hspace{0.17em}\mathit{part}\hspace{0.17em}\mathit{of}\hspace{0.17em}{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\varphi \mathit{dz}}\\ \le \left.\frac{R_{\mathit{ea}}{a}^2{P}_r}{n_i}{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\varphi \mathit{dz}}\right|\\ \le \frac{R_{\mathit{ea}}{a}^2{P}_r}{n_i}\left.{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\varphi \mathit{dz}}\right|\\ \le \frac{R_{\mathit{ea}}{a}^2{P}_r}{\left.n_i\right|}{\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}\right)}^{\frac{1}{2}}{\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.D\varphi \right|}^2\mathit{dz}}\right)}^{\frac{1}{2}}.\\ \left(\text{Using}\hspace{0.17em}\text{Schwartz}\hspace{0.17em}\text{inequality}\right)\end{array}$$

Using inequality (31) in inequality (38), we have $\frac{R_{\mathit{ea}}{a}^2{P}_r}{n_i}$ imaginary part of ${\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\varphi \mathit{dz}}$

(39)

$$\le \frac{R_{\mathit{ea}}{a}^2{P}_r}{A\left.n\right|\left.n_i\right|}{\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}.$$

Now multiplying Eq. (13) by ζ ^* and integrating from z = 0 to z = 1, we have

(40)

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}\left(\Lambda \left({\left.D\zeta \right|}^2+a^2{\left.\zeta \right|}^2\right)+D_a^{-1}{\left.\zeta \right|}^2+\frac{n}{P_r}{\left.\zeta \right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=T_a^{\frac{1}{2}}{\displaystyle \underset{0}{\overset{1}{\int }}{\zeta }^{*}\mathit{Dwdz}}.\hfill \end{array}$$

Equating the imaginary part of Eq. (40) on both sides, we obtain

$$\begin{array}{c}{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\zeta \right|}^2\mathit{dz}=\frac{T_a^{\frac{1}{2}}{P}_r}{n_i}\mathit{imaginary}\hspace{0.17em}\mathit{part}\hspace{0.17em}\mathit{of}\hspace{0.17em}{\displaystyle \underset{0}{\overset{1}{\int }}{\zeta }^{*}\mathit{Dwdz}}}\\ \le \left.\frac{T_a^{\frac{1}{2}}{P}_r}{n_i}{\displaystyle \underset{0}{\overset{1}{\int }}{\zeta }^{*}\mathit{Dwdz}}\right|\\ \le \frac{T_a^{\frac{1}{2}}{P}_r}{\left.n_i\right|}{\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.\zeta \right|}^2\mathit{dz}}\right)}^{\frac{1}{2}}{\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.\mathit{Dw}\right|}^2\mathit{dz}}\right)}^{\frac{1}{2}}.\\ \left(\text{Using}\hspace{0.17em}\text{Schwartz}\hspace{0.17em}\text{inequality}\right)\end{array}$$

Using inequality (33) in the above inequality, to get

(41)

$${\displaystyle \underset{0}{\overset{1}{\int }}{\left.\zeta \right|}^2\mathit{dz}\le \frac{T_a{P}_r^2}{\left.n_i\right|\left.n\right|}{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\mathit{Dw}\right|}^2\mathit{dz}}}.$$

Utilizing inequalities (39) and (41) in Eq. (37), we get

(42)

$$\begin{array}{r}\hfill \left(1-\frac{T_a{P}_r^2}{\left.n\right|\left.n_i\right|}\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left.\mathit{Dw}\right|}^2\mathit{dz}}+\left(1-\frac{R_{\mathit{ea}}{P}_r}{A\left.n\right|\left.n_i\right|}\right)a^2{\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}\\ \hfill +\left(R_t+R_{\mathit{ea}}\right)P_r{\mathit{Aa}}^2{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}\le 0,}\end{array}$$

which clearly implies that

(43)

$$\begin{array}{r}\hfill n^2{n}_i^2<max\left(T_a^2{P}_r^4,{\left(\frac{R_{\mathit{ea}}{P}_r}{A}\right)}^2\right)\\ \hfill \text{or}\hspace{0.17em}\left(n_r^2+n_i^2\right)n_i^2<max\left(T_a^2{P}_r^4,{\left(\frac{R_{\mathit{ea}}{P}_r}{A}\right)}^2\right).\end{array}$$

This proves the theorem.

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 n_rn_i − plane) given by inequality (43) (Fig. 3).

Figure 3:

3.2. Case II: When the fluid layer is heated from above

Subcase (i): Free boundaries

We prove the following theorem:

Theorem 3

If R_t < 0, R_ea > 0, P_r > 0, T_a > 0, Λ > 0, D_a > 0, A > 0, n = n_r + in_i, n_r ≥ 0 and n_i ≠ 0, then a necessary condition for the existence of a non-trivial solution (w, θ, ζ, Φ, n) of of Eqs (11)–(14) together with the boundary conditions (15) is

$${\left.n\right|}^2<max\left(T_a{P}_r^2,\frac{\left(\left.R_t\right|+R_{\mathit{ea}}\right)P_r}{A}\right).$$

Proof

In the present case, R_t < 0. Thus, with R_t = −|R_t|, Eq.(11) becomes

(44)

$$\begin{array}{l}\left(D^2-a^2\right)\left(\Lambda \left(D^2-a^2\right)-D_a^{-1}-\frac{n}{P_r}\right)w\hfill \\ =-\left.R_t\right|a^2\theta +T_a^{\frac{1}{2}}D\zeta +R_{\mathit{ea}}{a}^2\left(\theta +D\Phi \right),\hfill \end{array}$$

Multiplying Eq. (44) by w^* and integrating the resulting equation from z = 0 to z = 1 and adopting the same procedure that was used to prove Theorem 1, we get

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}\Lambda \left({\left.D^{2}w\right|}^2+2a^2{\left.\mathit{Dw}\right|}^2+a^4{\left.w\right|}^2\right)\mathit{dz}}\hfill \\ +\left(D_a^{-1}+\frac{n}{P_r}\right){\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.\mathit{Dw}\right|}^2+a^2{\left.w\right|}^2\right)\mathit{dz}}\hfill \\ =\left(-\left.R_t\right|+R_{\mathit{ea}}\right)a^2{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\theta \right|}^2+a^2{\left.\theta \right|}^2+{\mathit{An}}^{*}{\left.\theta \right|}^2\right)\mathit{dz}}\hfill \\ -{\displaystyle \underset{0}{\overset{1}{\int }}\left(\Lambda \left({\left.D\zeta \right|}^2+a^2{\left.\zeta \right|}^2\right)+D_a^{-1}{\left.\zeta \right|}^2+\frac{n^{*}}{P_r}{\left.\zeta \right|}^2\right)}\mathit{dz}\hfill \\ -R_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}\mathit{dz}-R_{\mathit{ea}}{a}^2\left(a^2+{\mathit{An}}^{*}\right)}{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\Phi \right|}^2+a^2{\left.\Phi \right|}^2\right)\mathit{dz}}.\hfill \end{array}$$

The imaginary part of the above equation is given by

(45)

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.\mathit{Dw}\right|}^2+a^2{\left.w\right|}^2\right)\mathit{dz}}=\left.R_t\right|a^2{\mathit{AP}}_r{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}\hfill \\ -R_{\mathit{ea}}{a}^2{\mathit{AP}}_r{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}+{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\zeta \right|}^2\mathit{dz}}\hfill \\ +R_{\mathit{ea}}{a}^2{\mathit{AP}}_r{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\Phi \right|}^2+a^2{\left.\Phi \right|}^2\right)\mathit{dz}}.\hfill \end{array}$$

Using inequalities (30), (31) and (33) in Eq. (45), we get

(46)

$$\begin{array}{l}\left(1-\frac{T_a{P}_r^2}{{\left.n\right|}^2}\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left.\mathit{Dw}\right|}^2\mathit{dz}+a^2\left(1-\frac{\left.R_t\right|P_r}{A{\left.n\right|}^2}-\frac{R_{\mathit{ea}}{P}_r}{A{\left.n\right|}^2}\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+a^2{R}_{\mathit{ea}}{P}_{r}A{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}\le 0,}\hfill \end{array}$$

which implies that

(47)

$${\left.n\right|}^2<max\left(T_a{P}_r^2,\frac{\left(\left.R_t\right|+R_{\mathit{ea}}\right)P_r}{A}\right).$$

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 n_rn_i − plane whose centre is at the origin and radius equals $\sqrt{max\left(T_a{P}_r^2,\frac{\left(\left.R_t\right|+R_{\mathit{ea}}\right)P_r}{A}\right)}$ (Fig. 4).

Figure 4:

Subcase (ii): Rigid boundaries

Similar arguments hold for the present case as were used in subcase (ii) of case I, that is, when ${\left.D{\theta }^{*}D\Phi \right]}_0^1$ and ${\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}\mathit{dz}}$ are both real, then we have the same result as obtained in Theorem 2 and when the imaginary part of ${\left.D{\theta }^{*}D\Phi \right]}_0^1$ = the imaginary part of ${\displaystyle \underset{0}{\overset{1}{\int }}D\Phi D^2{\theta }^{*}\mathit{dz}}$ , then we proceed as follows:

Theorem 4

If R_t < 0, R_ea > 0, P_r > 0, T_a > 0, Λ > 0, D_a > 0, A > 0, n = n_r + in_i, n_r ≥ 0 and n_i ≠ 0, then a necessary condition for the existence of a non-trivial solution (w, θ, ζ, Φ, n) of Eqs (11)–(14) together with the boundary conditions (16) is

$$n^2{n}_i^2<max\left(T_a^2{P}_r^4,{\left(\frac{\left(\left.R_t\right|+R_{\mathit{ea}}\right)P_r}{A}\right)}^2\right).$$

Proof

Multiplying Eq. (44) by w^* throughout, integrating the various terms of the resulting equation, by parts, for an appropriate number of times, by using Eqs (12), (13) and the boundary conditions (16), we get

(48)

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}\Lambda \left({\left.D^{2}w\right|}^2+2a^2{\left.\mathit{Dw}\right|}^2+a^4{\left.w\right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(D_a^{-1}+\frac{n}{P_r}\right){\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.\mathit{Dw}\right|}^2+a^2{\left.w\right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\left(-\left.R_t\right|+R_{\mathit{ea}}\right)a^2{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\theta \right|}^2+a^2{\left.\theta \right|}^2+{\mathit{An}}^{*}{\left.\theta \right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \underset{0}{\overset{1}{\int }}\left(\Lambda \left({\left.D\zeta \right|}^2+a^2{\left.\zeta \right|}^2\right)+\frac{n^{*}}{P_r}{\left.\zeta \right|}^2\right)\mathit{dz}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+R_{\mathit{ea}}{a}^2{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\Phi \mathit{dz}}.\hfill \end{array}$$

Imaginary part of Eq. (48) can be written as

(49)

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.\mathit{Dw}\right|}^2+a^2{\left.w\right|}^2\right)\mathit{dz}}\hfill \\ =\left.R_t\right|a^2{\mathit{AP}}_r{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}-R_{\mathit{ea}}{a}^2{\mathit{AP}}_r{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}+{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\zeta \right|}^2\mathit{dz}}\hfill \\ +\frac{R_{\mathit{ea}}{a}^2{P}_r}{n_i}\mathit{imaginary}\hspace{0.17em}\mathit{part}\hspace{0.17em}\mathit{of}\hspace{0.17em}{\displaystyle \underset{0}{\overset{1}{\int }}{w}^{*}D\Phi \mathit{dz}}.\hfill \end{array}$$

Now, multiplying Eq. (12) by θ^* and integrating, we have

$${\displaystyle \underset{0}{\overset{1}{\int }}\left({\left.D\theta \right|}^2+a^2{\left.\theta \right|}^2+\mathit{An}{\left.\theta \right|}^2\right)\mathit{dz}}={\displaystyle \underset{0}{\overset{1}{\int }}{\theta }^{*}\mathit{wdz}}.$$

Imaginary part of the above equation is given by

$${\mathit{An}}_i{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}\hspace{0.17em}}=\mathit{imaginary}\hspace{0.17em}\mathit{part}\hspace{0.17em}\mathit{of}{\displaystyle \underset{0}{\overset{1}{\int }}{\theta }^{*}\mathit{wdz}},$$

which implies that

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}\le \frac{1}{A\left.n_i\right|}\left.{\displaystyle \underset{0}{\overset{1}{\int }}{\theta }^{*}\mathit{wdz}}\right|}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le \frac{1}{A\left.n_i\right|}{\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}\right)}^{\frac{1}{2}}{\left({\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}\right)}^{\frac{1}{2}},\hfill \end{array}$$

(utilising the Schwartz inequality)

which on using inequality (30) yields

(50)

$${\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}}\le \frac{1}{A^2\left.n_i\right|\left.n\right|}{\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}.$$

Utilizing inequalities (39), (41) and (50) in Eq. (49), we get

$$\begin{array}{l}\left(1-\frac{T_a{P}_r^2}{\left.n\right|\left.n_i\right|}\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left.\mathit{Dw}\right|}^2\mathit{dz}+\left(1-\frac{\left.R_t\right|P_r}{A\left.n\right|\left.n_i\right|}-\frac{R_{\mathit{ea}}{P}_r}{A\left.n\right|\left.n_i\right|}\right)a^2{\displaystyle \underset{0}{\overset{1}{\int }}{\left.w\right|}^2\mathit{dz}}}\hfill \\ +R_{\mathit{ea}}{R}_r{\mathit{Aa}}^2{\displaystyle \underset{0}{\overset{1}{\int }}{\left.\theta \right|}^2\mathit{dz}\le 0,}\hfill \end{array}$$

and consequently, we have

(51)

$$n^2{n}_i^2<max\left(T_a^2{P}_r^4,{\left(\frac{\left(\left.R_t\right|+R_{\mathit{ea}}\right)P_r}{A}\right)}^2\right).$$

From the physical standpoint, Theorem 4 can be stated as: the complex growth rate of an arbitrary oscillatory disturbance of growing amplitude, in a rotating dielectric fluid layer saturating sparsely distributed porous medium heated from above, lies inside the region (in the right half of the n_rn_i – plane) given by inequality (51) (Fig. 5).

Figure 5:

4. Conclusions

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: