The appropriate solutions of Equation 47 are given in terms of what called ordinary Mathieu functions which, indeed, are periodic in time *t* with period π and 2π.

Corresponding to extremely small values of

*q*, the first region of instability is bounded by the curves

$a=\pm \phantom{\rule{2.77695pt}{0ex}}q+1$

(49)

The conditions for oscillation lead to the problem of the boundary regions of Mathieu functions where Melaclan [

11] gives the condition of stability as

${\left|\mathrm{\Delta}\left(0\right){\mathrm{sin}}^{2}\left(\frac{\pi a}{2}\right)\right|}^{\frac{1}{2}}\le 1$

(50)

where Δ(0) is the Hill's determinant.

An approximation criterion for the stability near the neighborhood of the first stable domains of the Mathieu stability domains given by Morse and Feshbach [12] which is valid only for small values of *h*^{2} or *q*, i.e., the frequency *ω* of the electric field is very large.

This criterion, under the present circumstances, states that the model is ordinary stable if the restriction

${h}^{4}-16\left(1-b\right){h}^{2}+32b\left(1-b\right)\ge 0$

(51)

is satisfied where the equality is corresponding to the marginal stability state. The inequality (51) is a quadratic relation in

*h*^{2} and could be written as

$\left({h}^{2}-{\alpha}_{1}\right)\left({h}^{2}-{\alpha}_{2}\right)\ge 0$

(52)

where

*α*_{1 and}*α*_{2} are, the two roots of the equality of the relation (51), being

${\alpha}_{1}=8\left(1-b\right)-\mathrm{\Delta}$

(53)

${\alpha}_{2}=8\left(1-b\right)+\mathrm{\Delta}$

(54)

with

${\mathrm{\Delta}}^{2}=32\left(1-b\right)\left(2-3b\right)$

(55)

The electrogravitational stability and instability investigations analysis should be carried out in the following two cases

(i). 0 < *b* < 2/3

In this case Δ

^{2} is positive and therefore the two roots

*α*_{1 and}*α*_{2} of the equality (51) are real. Now, we will show that both

*α*_{1 and}*α*_{2} are positive. If

*α*_{1} α +

*ve* then

*α*_{1} must be negative and this means that

$8\left(1-b\right)\le b$

(56)

or alternatively

$64{\left(1-b\right)}^{2}\le 32\left(1-b\right)\left(2-3b\right)$

and this is contradiction, so *α*_{1} must be positive and consequently *α*_{2} ≥ 0 as well (noting that *α*_{2} > *α*_{1}). This means that both the quantities (*h*^{2} -*α*_{1}) and (*h*^{2} -*α*_{2}) are negative and that in turn show that the inequality (51) is identically satisfied.

(ii). 2/3 < *b* < 1

In this case, in which *b* < 1 and simultaneously 3*b* > 2, it is found that Δ^{2} is negative, i.e., Δ is imaginary; therefore, the two roots *α*_{1} and *α*_{2} are complex. We may prove that the inequality (51) is satisfied as follows.

Let

*h*^{2} -

*c* and

*α*_{1,2} =

*c*_{1} -

*ic*_{2} where

*c*,

*c*_{1}, and

*c*_{2} are real, so

$\begin{array}{cc}\hfill \left({h}^{2}-{\alpha}_{1}\right)\left({h}^{2}-{\alpha}_{2}\right)& =\left[-c-\left({c}_{1}+i{c}_{2}\right)\right]\left[-c-\left({c}_{1}-i{c}_{2}\right)\right]\hfill \\ ={c}^{2}+2c{c}_{2}+{c}_{1}^{2}+{c}_{2}^{2}\hfill \\ ={\left(c+{c}_{1}\right)}^{2}+{c}_{2}^{2}=+\mathsf{\text{ve}}\hfill \end{array}$

(58)

which is positive definite.

By an appeal to the cases (i) and (ii), we deduce that the model is stable under the restrictions

This means that the model is stable if there exists a critical value

*ω*_{0} of the electric field frequency

*ω* such that

*ω* >

*ω*_{0} where

*ω*_{0} is given by

$\pi G{\rho}^{\left(i\right)}\left(\frac{x{{I}^{\prime}}_{0}\left(x\right)}{{I}_{0}\left(x\right)}\right)\left({I}_{0}\left(x\right){K}_{0}\left(x\right)-\frac{1}{2}\right)>0$

(60)

One has to mention here that if

*ω* = 0,

*β* = 0, and

*E*_{0} = 0 and we suppose that

$\gamma \left(t\right)=\left(\mathsf{\text{const}}\right)exp\left(\sigma t\right)$

(61)

The second-order integro-differential equation of Mathieu equation (

41) yields

${\sigma}^{2}=4\pi G{\rho}^{\left(i\right)}\left(\frac{x{{I}^{\prime}}_{0}\left(x\right)}{{I}_{0}\left(x\right)}\right)\left({I}_{0}\left(x\right){K}_{0}\left(x\right)-\frac{1}{2}\right)$

(62)

where σ is the temporal amplification and note by the way that ${\left(4\pi G{\rho}^{i}\right)}^{-\frac{1}{2}}$ has a unit of time. The relation (62) is identical to the gravitational dispersion relation derived for the first time by Chandrasekhar and Fermi [1]. In fact, they [1] have used a totally different technique rather than that used here. They have used the method of representing the solenoidal vectors in terms of poloidal and toroidal vector fields for axisymmetric perturbation.

To determine the effect of *ω*, it is found more convenient to investigate the eigenvalue relation (62) since the right side of it is the same the middle side of (60).

Taking into account the recurrence relation of the modified Bessel's functions and their derivatives, we see, for

*x* α 0, that

$\left(\frac{x{{I}^{\prime}}_{0}\left(x\right)}{{I}_{0}\left(x\right)}\right)>0$

(63)

and

$\left({I}_{0}\left(x\right){K}_{0}\left(x\right)\right)>\frac{1}{2},\mathsf{\text{or}}\left({I}_{0}\left(x\right){K}_{0}\left(x\right)\right)\frac{1}{2}$

(64)

based on the values of *x*.

Now, returning to the relation (62), we deduce that the determining of the sign σ

^{2}/(4

*πGρ*^{
i
} ) is identified if the sign of the quantity

${Q}_{\mathsf{\text{o}}}\left(x\right)=\left({I}_{0}\left(x\right){K}_{0}\left(x\right)-\frac{1}{2}\right)$

(65)

is identified.

Here, it is found that the quantity *Q*_{0} (*x*) may be positive or negative depending on *x* α 0 values. Numerical investigations and analysis of the relation (62) reveal that σ^{2} is positive for small values of *x* while it is negative in all other values of *x*. In more details, it is unstable in the domain 0 < *x* < 1.0667 while it is stable in the domains 1.0667 ≤ *x* < ∞ where the equality is corresponding to the marginal stability state.

From the foregoing discussions, investigations, and analysis, we conclude (on using (65) for (62)) that the quantity

${L}^{2}=\left(\frac{x{{I}^{\prime}}_{0}\left(x\right)}{{I}_{0}\left(x\right)}\right)\left({I}_{0}\left(x\right){K}_{0}\left(x\right)-\frac{1}{2}\right),\phantom{\rule{1em}{0ex}}L=\frac{\sigma}{{\left(4\pi G\rho \right)}^{\frac{1}{2}}}$

(66)

has the following properties

$\begin{array}{ll}{L}^{2}\le 0\hfill & \text{in}\phantom{\rule{0.25em}{0ex}}\text{the}\phantom{\rule{0.25em}{0ex}}\text{ranges}\phantom{\rule{0.25em}{0ex}}1.0667\le x<\infty \hfill \\ {L}^{2}>0\hfill & \text{in}\phantom{\rule{0.25em}{0ex}}\text{the}\phantom{\rule{0.25em}{0ex}}\text{range}\phantom{\rule{0.25em}{0ex}}0<x<1.0667\hfill \end{array}\}$

(67)

Now, returning to the relation (60) concerning the frequency

*ω*_{0} of the periodic electric field

$\frac{{\omega}^{2}}{\left(4\pi G\rho \right)}>\left[\left(\frac{x{{I}^{\prime}}_{0}\left(x\right)}{{I}_{0}\left(x\right)}\right)\left(\frac{1}{2}-{I}_{0}\left(x\right){K}_{0}\left(x\right)\right)\right]>0.$

(68)

Therefore, we deduce that the electrodynamic force (with a periodic time electric field) has stabilizing influence and could predominate and overcoming the self-gravitating destabilizing influence of the dielectric fluid cylinder dispersed in a dielectric medium of negligible motion.

However, the self-gravitating destabilizing influence could not be suppressed whatever is the greatest value of the magnitude and frequency of the periodic electric field because the gravitational destabilizing influence will persist.