Electrodynamic instability of a self-gravitating dielectric fluid cylinder under a radial varying electric field

Journal of Magnetism and Magnetic Materials 87 (1990) 193-198 North-Holland

193

ELECTRODYNAMiC INSTABILITY OF A SELF-GRAVITATING DIELECTRIC FLUID CYLINDER UNDER A RADIAL VARYING ELECTRIC FIELD A h m e d E. R A D W A N Department of Mathematics, Faculty of Science, Ain-Shams Universtt); Abbassia, Cairo, Egypt

Received 5 October 1989

The electrodynamJc instability of a dielectric self-gravitating fluid cylinder pervaded by a uniform electric field ambient with a self-gravitating vacuum penetrated by radial yawing field is developed. The stability criterion valid for all possible modes of perturbation in the three dimensions space is derived and studied. The analytical results are confirmed numerically and interpreted physically. Gravitationally the model is unstable for small axisymmetric disturbances and stable in all the rest. The radial varying field has a strong stabifizing influence for all wavelengths whether the perturbation is sDrtmetric or/and asymmetric and the interior uniform one as weU. The bending and twisting of the lines of force due to the strong stabilizing influence of the electrodynamic force causes the decreasing of the gravitational instability. Moreover, above a certain value of the basic applied electric field it is found that the gravitational instability is completely elL,mnated and stability arises.

1. Introduction

2. Fundamental state

Instability fluctuation of a fluid cylinder or a layer ambient with a different fiuid or with a v a c u u m acting upon the electrodynamic, pressure gradient, curvature pressure o r / a n d other forces has recently received attention in the context of n u m e r o u s and varied investigators [1-13]. Chandraseldaar and Fermi [14] investigated the stability o f a pure self-gravitating fluid cyfinder to small axisymmetric disturbances by utilizing the energy principle. R a d w a n [15] examined the influence of the eleetrodynamic force (with longitudinal electric uniform fields) on the self-gravitating dielectric fluid cylinder in the limiting case o f zero surface charges at the interface of the equilibrium basic state. In his work [15] he did attract and draw attention for investigating such kind of instability for its crucial application in the astrophysics domain e.g. in understanding the dynamical behaviour of the spiral arms o f galaxies (cf. ref. [141). T h e endeavors of the present work lie in elaborating the electrodynamic stability o f a selfgravitating dielectric fluid cylinder embedded in a sdf-gravitating vacuum pervaded by a radially varying electric field.

Consider a self-gravitational incompressible, inviscid fluid cylinder of radius a with density 0 i and dielectric constant c i submerged in a vacuum. The latter is self-gravitating with dielectric constant ~ . The dielectric fluid cylinder is pervaded by the uniform electric field ( E o, 0, 0) where the components are taken along the used cylindrical polar coordinates (r, 4', z ) s y s t e m with the z-axis coinciding with the ~'ds of the cylinder. The fluid is acting u p o n the self-gravitating, pressure gradient and electrodynamic forces. The surrounding gravitational medium o f the cylinder is acting upon the electrodynamic force only. It is assumed that, initially there are no surface charges at the fluid-vacuum interface and therefore the surface charge density will be considered to be zero during the perturbation [10]. The quasi-static approximations are assumed to be valid for the problem under consideration as in the related problems (cf. refs. [7-13]). The fundamental equations, under the present circum'~:ances, are the vector equation of motion, continuity equation, the equations satisfying the gravitational potentials and those of electric fields interior and exterior to the fluid cylinder. For the

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194

A.E. Radwan / Electrodynamic instability of a sdf-gravgating dielectricfluid cylinder

problem at hand

with

i[ Oui P + ("'" v),,'] :=-iVPi.+oiiVvi+

= 8 ( t ) a exp[i(kz + mq,)]. ~1 tV( ¢i ( E i • El)),

(1)

V'u j =0,

(2)

IVzVi = - 4('n0iG),

(3)

IV2V~ = 0,

(4)

IV.(,e)"

= 0,

W × E i'e = O.

(5)

(6)

Here pi is the kinetic pressure of the fluid, u i is the velocity vector; V i and Ve are the gravitational potentials interior and exterior to the fluid, E ~ and E * are the electric field intensities acting inside and outside the fluid and G is the gravitational constant. The basic state is studied and described by

(13)

Here k (any real number) is a longitudinal wavenumber, m (an integer) is the azimuthal wavenumber and the amplitude 8 ( 0 of the perturbed wave is described by

8(t) = 8 exp(ot),

(14)

where 8o is the initial amplitude and o is the growth rate at time t. The second term T/ at the right hand side of (12) is the elevation of the surface wave normalized with respect to a and measured from the equilibrium position. By an appeal to the foregoing expansions, the basic equations (1)-(5) are linearized and the following equations are obtained p'o,,~ = - V U i,

(15)

H i = P¢ + piv,i - ½,i(Ei" E i ) , ,

(16)

v-,,i

(17)

E; = ( E0, 0, 0),

(7)

E ~ = ( 7Eo, a O, 0) ,

(8)

v2v~ ~'e = 0,

(18)

Vo' = -~rp'Gr 2,

(9)

v-(,e,) °=o,

(19) (20)

= o,

V(~= - 2 v o i G a 2 [ l o g ( a ) + ½],

(10)

W × Eli "~= O.

Po=~rG(pi)2(a2-r2) - (Eg/2)(C - ,i),

(11)

Taking the divergence of (15) and combining the resulting equation with eq. (17), one obtains

where the index 0 recognizes the basic quantities and, later, quantities with index 1 are the fluctuation quantities.

3. Perturbation and solution

For small departures from the fundamental state, every physical quantity Q(r, q,, z; t) can be expressed as its equilibrium part plus a fluctuation part

Q(r, e~, z; t) = a o ( r ) + 8 ( t ) Q , ( r , ep, z), where Q stands for u, P, V, E and the radial distance of the fluid cylinder. The latter, in considering a small wave propagating in the positive z-direction, is

r = a + TI, 71<
(12)

IV2H~ = 0.

(21)

The circulation equation (20) concerning the relevant electric field intensities E~"¢ means that E~"e can be derived by means of scalar functions, tk~.e say, .e = -

.°.

(22)

Combining eqs. (19) and (22) we get V 2 ~ 'e = 0.

(23)

Therefore, the soludon of the relevant perturbation equations (15)-(20) can be determined by identifying the solution of the Laplace's equations (18), (21) and (23). By a resort to the expansions (12)-(14) and based on the linearized perturbation technique, the relevant perturbed scalar quantities //~, V~.~

A.E. Radar,an / Electrodynamic instability of a self-gravitating dielectric fluid cylinder

and ~/,~e may be written as exp[i(kz + mqO + ot] times an amplitude function of r

n~(r, ~, z; t)

= 8o//~(r ) exp[ot + i(kz + mq,)],

(24)

V,i'e(r, ~b, z; t) = 8oVli'e(r) exp[ot+i(kz+mq~)],

(25)

,7,'e(r, ,~, z; t) = 8oq,~'¢(r) exp[ ot + i(kz + m.6)].

(26)

Substituting from (24)-(26) into (18), (21) mad (23), after some manipulation, a total differential equation of second order is obtained for the variables H~(r), V1i'¢(r) and q,ii*(r ). It is found that the solutions of these differential equations are given in terms of the ordinary Bessel functions with an imaginary argument. By excluding the infinite solution, the non-singular solutions are given by

V~(r) = 6 ( t ) a A K m ( k r ) exp[i(kz + r n , ) ] ,

(27)

Vii(r) = 8 ( t ) a B l m ( k r ) exp[i(kz + mq,)],

(28)

H~(r) = 8(t)aCIm(kr

(29)

) exp[i(kz +md?)],

ffil(r ) = 6(t)aflm(kr) exp[i(kz + rnq0],

(30)

q~](r) = 6 ( t ) a g K m ( k r ) exp[i(kz + mq,)],

(31)

where l,,(kr) and Km(kr ) are the modified Bessel functions of the first and second kind of the order m while A, B, C, f and g are unspecified constants of integration to be determined by making use of appropriate boundary conditions.

195

(ii) The electric potential ff must be continuous across the interface r = a + 71 at r = a. (iii) The normal component of the electric displacement is also continuous across the interface i.e. N - [ c i E i - - c e E e] = 0 across r = a + r/ at r = a where N is a unit vector outward normal to the fluid-vacuum interface. (iv) The normal (radial) component of the velocity must be compatible with the velocity of the disturbed fluid interface at r = a. (v) The normal component of the total stress tensor must be continuous across the perturbed interface (12) at r = a. By applying the foregoing boundary conditions (i)-(iv) and with the aid of the solutions (24)-(31) and utilizing the features of the modified Bessel functions (cf. reL [17]) we get the following. The conditions (i) lead to

A K , , ( x ) = BIm(x ),

x = ka

(32)

and , x [ A ./ (,,,( , x ) - BI~(x)]

=

-4~oiaG,

(33)

where the prime over the Bessel functions denotes the derivative with respect to the argument and x( = ka) is the non-dimensional longitudinal wave number. Solving (32) with (33) we have

A = 4 ~rpiaGI, n ( x

(34)

),

B = 4~roiaGKm( x ) .

(35)

Applying the conditions (ii) and (iii) at r = a across the interface, (12) degenerates to

x[ ,'fI~,( x ) - CgK~ ( x )] = ,eE o 4. Boundary conditions

(36)

and

t i m ( x ) = gK,,(x ), The solution of the relevant perturbation equations (15)-(20), which describe the disturbance of the self-gravitating fluid cylinder ambient with a self-gravitating vacuum presented by (24)-(31), must satisfy certain boundary conditions. For the problem at hand, the appropriate boundary conditions are the following. (i) The gravitational potential V(= V0 + 8V~) and its derivative are continuous across the deformed interface (12) at r = a.

(37)

from which we get

f = g( K,,( x ) / l ~ , ( x ) )

(38)

with

g=¢eEoI,,(x) i

•

t

'

(39)

196

A.E. Radwan / Electrodynamic instability of a self-gravitating dielectric fluid cylinder

Also condition (iv) gives

c = -.2o2/xt

(x).

(40)

Finally, the continuity of the normal component of the total stress tensor reads •

OPd

1 i,Ei

= -~C(EC'E¢),-½'e'O

Or

'

(41)

where we have from (16)

pli

- l,i(Ei..Ei),__~ O, U~i + p'V,'.•

(42)

Substituting from (7), (8), (11), (24)-(31) with the aid of (32)-(40~ and (42) into the condition (41) at r = a, the following eigenvalue relation is derived 02 = 4 ~ p i G [ ( x : ~ , ( x ) / I , , , ( x ) ) ( 1 , , , ( x ) K , ~ ( x ) -- ('eE2°/pia2)

5.1. Self-gravitating instability In the absence of the electric fields influences, the eigenvalae relation (43) reduces to

- ½)]

02 = 4 rp G[xI"

{ x12,(x)

-

I.,(x)K,.(x) -

(44)

I m( x )

xI'(x)K'(x)

}

'

(43)

where ( ==

o r / a n d ~0 with the natural quantity (4"rrffG) -W2 as well as (£eEg/pia2)-l/2 as a unit of time. It is worthwhile to mention here that the relation (43) is a simple linear combination of the eigenvalue relations of a full fluid cylinder acting upon the electrodynamic force (with radial varying electric field) only and that one acting upon the self-gravitating force only. This simple additivity is not only due to the linearization of the fundamental basic equations but also because no volume and no surface charges are assumed to be present in the bulk and at the interface of the fluid cylinder.

£i/£c,

5. General discussions Eq. (43) is the required eigenvalue relation of an electrodynamic self-gravitating dielectric fluid cylinder embedded in a self-gravitating vacuum pervaded by radial varying electric field. It contains the most information of the (in)stability of the model under consideration. The marginal stability is identified from (43) by just putting o = 0. The relation (43) relates the growth rate of instability o or rather the oscillation frequency oJ (that if o ( = io~) is imaginary) with the dielectric constants (i and ( ~, the modified Bessel functions lm(x), K ~ ( x ) and their derivatives, the wave numbers m and x; and with the parameters G, pi, a and E 0 of the problem. Moreover (43) relates o

From the analytical and numerical discussions of (44), we can deduce the following conclusions. The fluid cylinder is gravitationally stable to all purely asymmetric modes m ___1 of perturbation for all wavelengths. To the sausage mode m = 0, the fluid cylinder is gravitationally unstable as long as x < 1.0668 while it is stable if x > 1.0668 where the equality corresponds to the marginal (neutral) stability. For more detmis concerning the (in)stability investigations of such a case we may refer to ref.

[141. 5.2. Electrodynamic instabifity If the electrodynamic force influence is superior to the self-gravitating force, the eigenvalue relation (43) degenerates to

xr(x)rZ(x) o2 = (,OEo /pi.2) [ , r ( x ) K = ( x ) - tm(x)KZ(x)]

xr(x) 1re(x) }

(451

A.E. Rad~,an / Electrodynamic instability of a setf-graeitating dielectric fluid cylinder

By a resort to the recurrence relations of the modified Bessel functions (see ref. [17]) 21~,(x) = I,,,_,(x) +/rn+l(X),

(46)

2K,~(x) = - K m _ , ( x ) - Km+,(x)

(47)

t97

more convenient to rewrite (43) in the dimensionless form 0, 2

4~rGp i 2

and for every non-zer,3 real value of x that l,fix) > 0 and monotonic :nereasing while K , , ( x ) > 0 but monotonic decreasing, one can deduce that I ' ( x ) is always positive while K ' ( x ) is never positive. Therefore for x 4=0 whether m = 0 o r / a n d m > 1 we have

Ira(x) [Im(x)K'(x

-

t t" (52)

where

l ' ( x ) > O,

tm(x ) > O,

(48)

Km(x ) < O, K,,,(x) > 0

(49)

from which for c 4:0 we get

[,l'(x)X,(x)

- 1,,,(x)K,~(x)]

> 0

(50)

and

L7 =

× [,I,~(x)Km(x) - Zm(x)K,~(x)] -' < 0 . (51) Now, returning to criterion (45): the influence of the electric field pervaded interior to the cylinder is presented by the term ( - - x I ' ( x ) K ' ( x ) ) . Since the latter is negative, see the inequalities (48) and (49), it has a stabilising influence on the dielectric cylinder. The influence of the varying electric field penetrating in the region surrounding the cylinder is represented b y the term L~ (see the inequalities (49)-(51) and it has a ztrong stabilizing effect for all (short and long) wavelengths. Therefore we conclude that a dielectric fluid cylinder pervaded by a uniform radial electric field and surrounded by a radial varying electric •field is stable for all (short and long) wavelengths whether the disturbance is axisymmetric m = 0 o r / a n d non-axisyrranetric ones m >_ 1.

5.3. Electro-gravitodynamic (EGD) stability In order to investigate the (in)stability states of such a general case in which the fluid cylinder is acting upon the combined effect of the electrodynamic and ~If-gravitating forces, it is found

Ea = 2fla( ,¢/crG) -,/z

(53)

By an appeal to the results of sections (5.1) and (5.2) and utilizing their conclusions, the E G D instability states and characteristics of the present model can be identified. From the analytical discussions of the results of the stability criterion (52) we may document the following conclusions. The dielectric fluid cylinder is E G D stable in the pure non-axisymmetric disturbances m >_ 1 for all x and (Eo/Ea) values and the stability character is increasing with increasing electric field intensities. As the disturbance ~5 of a ~ ymmetric type m = 0, the serf-gravitating eyli~zd~r is unstable as tong as x < 1.0668 while it is gra~-tationally stable if x > 1.0668 and the critical points at which a transition from instability to stability states are occurred at x -- 1.0668. The largest gravitational domain of instability is 0 < x < 1.0668 is shrinking if one takes into account the strong stabilizing influence of the electrodynamic force. Since each of the interior and exterior electric fields is stabilizing for all m >_ 0 and x ~ 0 values, one expects that the gravitational instability of the cylinder in the states m = 0 with 0 _< x < 1.0668 will be completely suppressed above a certain value of Eo/E ~ and then the model will be purely E G D stable. For all possiNe modes of perturbations m >__0, the self-gravitating dielectric cylinder is E G D stable if the restrictions

(eo/E ) >_

- }] [1

(54)

198

A.E. Radwan / Electrodynamicinstabilityof a self-gravitatingdielectricfluid cylinder

are satisfied and vice versa where the equality holds for the marginal (neutral) stability. In establishing the restrictions (54) we have assumed that ( ~ = c ~ and use has been made of the inequality ( x I ' , ( x ) / I , ~ ( x ) ) > 0 a n d also the Wronskian expression I,,(x)K,~(x) -

I,~,(x)K,,,(x) = - x -1. The strong stabilizing features of the d e c trodynamic force can be interpreted physically as follows. The electrodynarnic force allows the electric field to give a measure of rigidity to the dielectric fluid. Moreover, the electrodynamic force exerts a strong influence not only in the axisymmetric mode that causes the bending of lines of force but also in the non-axisymmetric that causes also the twisting lines of force. In particular in our case in which the surface charges are neglected at the interface of the fluid and also the volume charges at the bulk of the fluid as well: the electrodynamic pressure is superior while the electrodynamic tension is completely negligible. Consequently the measure of rigidity in the fluid due to the electrodynamic force is extremely large and that causes a high stabilizing influence whatever the pervaded electric field is non-uniform or not.

5,4. Nwnerical discussions In order to confirm the previous analytical resuits of the (in)stability states of the present model, the E G D eigenvalue relation (52) would be analysed numerically. This has already been carried out for the most dangerous (sausage) m o d e m = 0 of perturbation where use has been made of the relations Io(x ) = It(x ) and K~(x) = - K t ( x ) . The functions

* ( x ) ----[xl~(x)/Io(x)] [ l o ( x ) K o ( x ) - ½] and X ( C = 1)

_ xI,(x) xI,(x)K,(x) ~Io(x + [ l o ( x ) K i ( x ) + d , ( x ) K o ( x ) ] are calculated numerically in the crucial domain 0 _< x < 1.1 which is the largest gravitational domain of instability. For different values of ( E 0 / E a) the values of the growth rate e / 4 ~ p i G corre-

sponding to the instability states as well as the values of the oscillation frequency t 0 / ~ p i G corresponding to the stability states are collected and tabulated in that domain. It is found that at ( E 0 / E a ~ ~ 2.9791 no gravitational instability exists any more and stability arises. Moreover, with increasing electric field intensity E 0 relative to Ea, the instability d o m a i n s are decreasing while those o f stability are increasing. In fact it is found that both the critical value of the longitudinal wavenumber and the m a x i m u m temporal amplification are simultaneously decreasing with increasing ( E o / E c ) values. This shows how much is the strong stabilizing influence of the electrodynamic force and h o w fast the shrinking o f the instability domains with increasing applied basic electric field.

References [1] L. Baker, Phys. Fluids 26 (1983) 391. [2] P.M. Bisch, H. Wendel and D. Gallez, J. Colloid Inter. Sci. 92 (1983) 105. [3] M. Boiti, J. Leon and F. Pempinelli, J. Math. Phys. 25 (1984) 1725. [4] H. Brenner and L.G. Leal, J. Colloid Inter. Sci. 88 (1982) 136. [5] D. Censor, Phys. Rev. A25 (1982) 437. [6] R.C. Diprima and J.T. Stuart, J. Appl. Mech. 50 (1983) 983. [7] E.F. Elsehaway et al., Quart. Appt. Math. 43 (1986) 483; Nuovo Cimento 6D (1985) 291. [8] J.F. Hoburg, J. Fltdd Mech. 132 (1983) 231. [9] G.P. Kiaassen and W.R. Peltier, J. Fluid Mech. 155 (1985) i. [t0] A. Larraza and S. Putterrnan, J. Fluid Mech. 148 (1984) 443. [11] A.A. Mohamed et al., J. Math. Phys. 26 (1985) 2072; Phys. Scripta 31 (1985) 193; J. Chem. Phys. (USA) 85 (1986) 445; J. Phys. A 3 (1970) 296. [12] D.F. Parker, Physica D 16 (1985) 385. B.K. Shivamoggi and L. Debnath, Nucvo Cimento 94B (1986) 140. [13] M. Tajiri, J. Phys. Soc. Japan 53 (1984) 3759. H.H, Yeh, J. Fluid Mech. 152 (1985) 479. [14] S. Chandrasekhar and E, Fermi, Astrophys. J. 118 (1953) 116. [15] A.E. Radwan, J. Phys. Soc. Japan 58 (1989) in press. [16] H.H. Woodson and J.R. Meleher, Electromechanical Dynamics (John Wiley, New York, 1968). [171 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).