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Non-axisymmetric magnetodynamic instability of a fluid cylinder subject to varying fields
33
Journal of Magnetism and Magnetic Materials 79 (1989) 33-41 North-Holland, Amsterdam
NON-AXISYMMETRIC MAGNETODYNAMIC INSTABILITY OF A FLUID CYLINDER SUBJECT TO VARYING FIELDS Ahmed E. RADWAN Department
of Mathematics,
Faculty of Science, Ain-Shams
University, Abbassia,
Cairo, Egypt
and Samia S. ELAZAB Department
of Mathematics,
Women’s University College, Ain-Shams
University, Heliopolis,
Cairo, Egypt
Received 18 May 1988; in revised form 2 January 1989
The magnetodynamic instability of a fluid cylinder subject to the inertia, capillary and electromagnetic (the fluid is pervaded by uniform field and surrounded by a field which is generally varying) forces is presented to all (axisymmetric and non-axisymmetric) modes of perturbation. A general dispersion relation is derived, studied analytically and the results are confirmed numerically for different values of the vacuum magnetic field. The surface tension has the same influence as in the absence of the magnetic fields; however, sometimes for certain values of the vacuum field parameters the capillary instability is completely suppressed. The axial magnetic fields inside and outside the cylinder always have stabilizing effects to all modes of perturbation, whatever the values of the problem parameters are, as is expected. The azimuthal vacuum varying magnetic field has a destabilizing or stabilizing effect according to certain restrictions. This is confirmed by studying the dispersion relation asymptotically and numerically. It is found that there are unstable states in the non-axisymmetric modes, this is due to the fact that the vacuum magnetic field is a varying field and not uniform. In contrast to the case in which just uniform fields are acting, the model is completely stable in all non-axisymmetric modes.
1. Introduction The hydro-, gravito- and magnetohydromagnetic stability of a liquid cylinder has received considerable attention. It was started by the experimental and theoretical work of Plateau [l], and Rayleigh [2] has documented the previous work and his own results. Chandrasekhar [3] also summarized and made several extensions for that model and others subjected to different forces. The purpose of the present work is to investigate the magnetodynamic instability of a fluid cylinder subject to the inertia, capillary and electromagnetic forces. The fluid cylinder (radius R,) is endowed with surface tension (coefficient T), pervaded by a uniform magnetic field (intensity H,) and surrounded by the vacuum varying magnetic field Hovac (= (0, BH,,R,/r, aH,,)). Here (Y
and /3 are parameters which satisfy some restrictions (see the inequality (11)); r is the radial distance of a cylindrical polar coordinate (r, 9, t) system with the z-axis coinciding with the axis of the cylinder. In fact the present work is an extension to that of Chandrasekhar (ref. [3], p. 537), by considering the disturbances with all possible modes of perturbation and that the cylinder is surrounded by the non-uniform field H,‘aC. The present results with (Y= 1, p = 0 and m = 0 (where m is the azimuthal wave number) correspond to those of Chandrasekha (ref. [3], p. 542). The arrangement of the paper is as follows. The basic equations and equilibrium state are considered in section 2. In section 3 the dispersion relation is derived. Section 4 is devoted to the general discussions of the dispersion relation, the surface tension influence, magnetic field effects on
0304~8853/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A. E. Radwan, S. S. Elazab / Instability
34
the instability of the perfectly conducting fluid cylinder and the influence of the magnetic field on the capillary instability. The asymptotic behaviour of the dispersion relation for very long wavelengths to all modes of perturbation and its numerical discussions are considered in sections 5 and 6.
2. Basic equations and equilibrium state For an incompressible, inviscid and perfectly conducting fluid, the basic equations are p i (
+ (u*V))u-p(curl
H) XH=
-vp,
div u=O,
(1) (2)
i3H =curl(uXH),
at
The equilibrium state is studied and the equilibrium pressure distribution is given by PO = ($5,2/2)(
a2 + P* - 1) + (T/R,)
with the restrictions (p. > 0) a* + p* > 1 - (~T//LR~H;).
div H vat = 0
(5)
and
The last term in the right side of (10) is the contribution due to the capillary force influence. As a limiting case with p = 0 and (Y= 1 (the case which was considered by Chandrasekhar [3], p. 537), eq. (10) indicates that the magnetic fields have no influence on the equilibrium kinetic pressure po. The latter is due to the fact that the applied fields inside and outside the fluid are originally taken as the same.
For a small departure from the equilibrium state, every perturbed quantity Q(r, +, z; t) can be expressed as Q(r,
$3 z; t) = Q,(t)
+EQ,(~,
(6) interface we also have (7)
i.e. the jump of the total pressure [4] (the sum of the kinetic and magnetic pressures P = p + (p/2)( H H) must be balanced by the surface tension. R, (= of/ 1vf I) is the outward unit vector normal to the cylinder surface, where l
+, z; t) =0
(8)
is the equation of the boundary surface at time t and f(r, cp, z; t) = 0 satisfy %+(..v)f=O.
(9)
9, z) + . * *. (12)
Here Q stands for p, u, H, Hvac and the radial distance of the cylinder I = R, + rR, cos( kz + WI+),
curl H’“” = 0.
(P> = T div ri,,
(11)
(4
Here p, u and p are the fluid mass density, velocity vector and kinetic pressure; H and p are, respectively, the applied magnetic field and coefficient of the magnetic permeability. In the vacuum region, since there is no current, we have
Along the fluid-vacuum
(10)
3. Perturbation analysis and dispersion relation
div H = 0.
f(r,
of a fluid cylinder
(13)
where k and m are the longitudinal and azimuthal wave numbers. The subscripts 0 and 1 characterize the equilibrium and perturbed quantities, respectively. The second term in the right hand side of (13) is the elevation of the surface wave normalized with respect to R, and measured from the equilibrium position. E is the amplitude of the perturbation, see Chandrasekhar [3], being e = e. exp( at),
(14)
where co is the initial (at time t = 0) amplitude and u is the growing rate of instability; if u is imaginary u = io where w/2n is the oscillation frequency. By the use of the expansions (12)-(14) and following the same steps as the linear perturbation
A. E. Radwan, S. S. Elazab
technique used in refs. [5-g], the basic equations are linearized, solved and the constants of integration are determined by applying appropriate boundary conditions. Apart from the singular solutions we finally have ui = (aR,,/xZ~(x))
grad(Z,(kr)
cos(kz + m+)), (15)
H1 = (-ZZ,,/Z~(x))
grad(Z,,,(kr)
sin(kz+m+)),
Instability
of a fluid cylinder
35
and the Wronskian W(Z,(x),
K,(x))
= Z,,,(x)K;(x)
- Z;(x)K,(x)
= -x--l
(for the mode m = 0); eq. (21) corresponds to that of Chandrasekhar [3], p. 545 in studying the influence of a uniform axial magnetic field on the capillary instability of a liquid jet.
(16) Hi==
4. Discussions
- [(m/3 + ox)/xKXx)l xgrad(K,,,(kr)
(PI> = - (T/&)(1
sin(kz+m+)),
(17)
- m* - x2) cos( Zcz+ m+) (18)
and P,=
-[(o’+B;)/xZ,(x)]Z&r) Xcos(kz
+ m$).
(19)
Here x (= kR,) is the longitudinal dimensionless wave number; Z, and K,,, are the modified Bessel functions of the first and second kind of the order m and s2, (= (~ZZ~k*/p)‘/*) is the Alfvtn wave frequency defined in terms of ZZ,. Moreover, if we apply the continuity of the normal component of the total stress tensor across the boundary surface r = R,; we will end up, after some manipulation, with the relation c* = (U&)[
xZ~(x)/Z,(x)](l
+(ILZ&%R;){
-m*-x2)
- x2 + P”(xZ;(x)/Z,(x))
+(mp+ax)* x[Z;(x)Km(x)/Z,,z(x)K,&)l}.
(20)
If we put m = 0 with cy= 1 and p = 0 in eq. (20), we obtain
(1 -
c* = (V&)[xZ;(x)/Z&)] +&T&R;){
-x2
x2)
+ CY*X*Z,&)K,(X)
/Z,(x)K;(x)}.
(21)
By virtue of the relations (cf. ref. [lo]) Z;(x)
=Z,(x)
and
K;(x)
= -K,(x)
of the dispersion relation
4.1. General discussions The relation (20) is the stability criterion of a fluid cylinder endowed with surface tension, pervaded by a homogeneous axial magnetic field and surrounded by a vacuum varying magnetic field. It contains the most information about the instability of the problem at hand. It is a simple linear combination of the dispersion relations of a liquid cylinder subject to the capillarity force only and that subject to the electromagnetic force only. It is remarkable that (in the axisymmetric mode m = 0) for a liquid jet pervaded and surrounded by uniform axial magnetic fields and subject to a capillary force, Chandrasekhar [3] (p. 545, eq. (165) there) also obtained just such a simple combination of separate dispersion relations. One may be inclined to think that this is a general feature due to the linearization. However, this argument is not sufficient as is clear from the dispersion relation obtained in the presence of viscosity and resistivity. A detailed analysis shows that the simple linear addition in the dispersion relation is to be expected whether the perturbation is axisymmetric or not and whether the acting magnetic field is uniform or non-uniform, but usually not if the force added depends directly or indirectly on the velocity as in the case of a magnetic field. Indeed the velocity occurs in eq. (3) of evolution of the magnetic field thus influencing indirectly the electromagnetic force. In fact due to the vectorial product (U x If) there occurs a mixing of the velocity components i.e. the ith component occurring in the jth component of (u X H). This mixing, however, is undone when
A.E. Radwan, S.S. Elazab / A‘nstability of a fluid cylinder
36
taking the circulation of (u x H ); the i th component of H only depends on the i th component of u (the simple form of the magnetic field plays here an important role). That, together of course with the linearization, is the main feature leading to the simple additivity of the dispersion relation. The dispersion relation (20) relates the growing rate of instability u or rather the oscillation frequency w (for stability states) with the quantities (T/pRi) and (pHt/pRt) as units of (time)-*, the parameters OLand p of the vacuum magnetic field, the dimensionless wave number x and the modified Bessel functions Z, and K, of the order m. By means of this relation the stability as well as instability states can be identified together with the stabilizing or destabilizing influence of both the capillary and electromagnetic forces. 4.2. Capillaary force injluence If the liquid cylinder is subject to the capillary force only, its dispersion relation can be obtained from eq. (20) by setting Ho = 0: u2=
(T/~R~,)(xZL(X)/Z~(X))(~
-m*-x2). (22)
This corresponds to the classical dispersion relation of a capillary liquid jet derived by Rayleigh [2]. It is found that the cylindrical model is stable in the purely non-axisymmetric deformations; but it is unstable in the axisymmetric varicose deformations with wavelengths exceeding the circumference of the cylinder. For more analytical and numerical investigations we may refer to Chandrasekbar [3], p. 537. 4.3. Electromagnetic
force influence
If we study the perturbation of the liquid cylinder under the influence of the magnetic fields described above and neglect the capillary force influence (T = 0), the corresponding dispersion relation is obtained from (20) as 2 (I
2_ --
l-4
xz’(x) -x2
PR”O
+(mj?+ax)
+
p2.L_
L(x)
(23)
The influence of the uniform axial magnetic field pervades the interior of the fluid cylinder is represented by the term -x2 in the brackets, followed by the fundamental unit (pHz/pRt) in eq. (23). It always has a stabilizing influence. Similarly the influence of the homogeneous axial vacuum field is represented by the term including cy; it always has a stabilizing influence. Henceforth we may conclude that as was expected, as in the case of Chandrasekhar [3], p. 542, a fluid cylinder acting upon axial fields is stable to all modes of perturbation whatever the intensity of the applied magnetic fields. The influence of the azimuthal vacuum field is represented by the term including /3. It always has a destabilizing influence in the term /3’(xZA(x)/ Z,,,(x)) since this term is always positive. In the other term it always has stabilizing influence. However, if rnxpa < 0, its contribution is smaller than if it is positive. We have the choice in the sign of m; if we take m so that m/h c 0, that means if we change the perturbed vacuum field to have the opposite direction in the boundary surface, the stabilization due to this term is decreased. Since stability is determined by the least stable mode we have essentially to deal with the case rnpcx < 0. Concluding we may say that the transverse vacuum varying magnetic field always has a destabilizing influence. Due to that, the instability character of the magnetic field arises not only in the axisymmetric mode m = 0 but also in the non-axisymmetric modes m >, 1. Consequently, in the present problem, there are magnetodynamic instability states in the non-axisymmetric modes (see also sections 5 and 6). The destabilizing influence of the transverse magnetic field will be the greatest when rnp + ax = 0 i.e. when the perturbed vacuum magnetic field vanishes (see eq. (17)). It is important to observe that when we omit this term containing (mp + cyx), the dispersion relation significantly becomes an even function of m, p and independent of (Yand a complicated term disappears. Since this term always has a stabilizing effect, the omission leads to an upper limit of u2. Clearly if this upper limit allows only stability, the real case is stable also.
A.E. Radwan, S.S. Elarab / Instability
4.4. Effect tension
of a fluid cylinder
of the magnetic field on the surface
Now, returning to the general relation (20). The surface tension always has a stabilizing influence in all non-axisymmetric modes m > 1 for all wavelengths and also in the axisymmetric mode m = 0 if the wavelength is shorter than 2nR,. Meanwhile it has a destabilizing effect only in the axisymmettic disturbances whose wavelength is longer than the circumference of the cylinder. If the disturbance wavelength is equal to 21rR, then there will be a marginal stability. Combining these results with those of the electromagnetic force, one may conclude that there are instability domains whether the disturbance is axisymmetric or non-axisymmetric. In contrast to Chandrasekhar’s results (ref. [3], p. 542) where there is no destabilizing effect to the uniform field used there and even above a certain value of the equilibrium magnetic field, he found [3] that the
0
1
2
3 Fig. 2.
Fig. 1.
5
x
38
A.E. Radwan. S.S. Eiazab
Instability
capillary instability is completely suppressed and then stability sets in. In the present work there is no complete suppression of the capillary instability except for some unexpected rare values of p as we will see in the asymptotic and numerical discussions, see sections 5 and 6.
of a fluid cylinder
For axisymmetric reduces to
behaviour of the dispersion relation
f *. ),
i&f, = +x2( p’ - 2) + (g/3’ - a2 In fyx)x2
(25) + ..., (26)
The asymptotic behaviour of the general dispersion relation (20) is studied for very long wavelengths i.e. for very small values of X. This has already been carried out not only in the axisymmetric mode m = 0 but also in the lowest nonaxisymmetric mode m = 1 and in the higher nonaxisymmetric modes m 2 2. On utilizing the asymptotic behaviour of the modified Bessel functions Z,(x) and K,(x) in the modes m = 0, m = 1 and m a 2 for x -SC1 (cf. ref. [lo]), we have the following different classes.
where In y is the Eulerian constant. The asymptotic relation (24) with the aid of (25) and (26) shows, for very long wavelengths, that the MHD fluid jet is stable if
T+pH,2R,(~2- 2)<0,
(27)
where the equality holds for the marginal stability. Moreover, if g2 c 2 then a larger ZZ, increases the stabilizing influence.
Y=Y ---_-_ Y:SH, ----.-.-
m = 0, eq. (20)
(24)
L, = $X2(1 - gx2-t 5. Asymptotic
perturbation
H,=X)H, b = BH,
Fig. 3.
A.E. Radwan, S.S. Elazab / Instability
For the lowest non-axisymmetric m = 1, the relation (20) yields c2 = (VP@)
perturbation
L, + (P@/&)MI
(28)
with L, = 4x2(4 + x2 + . * * ),
(29)
M,=x{(-2a/3)+[:/32-(l+fx2) - +/I(1 + 4 In +rx)] x - [ $@(l
+ 4 In +rx)] x2 + 0(x3)
+ . . - }. (30)
From which it is clear that the magnetohydrodynamic fluid cylinder is unstable if 2Qx is negative for very long wavelengths. Since we restrict here the sign of m to be positive we have to allow x to be positive and negative, thus there is always instability for very low wavelengths whatever the sign of Cup. This confirms the analytical results of the magnetodynamic dispersion relation (20), as has been discussed in section 4.4. For higher non-axisymmetric perturbations m > 1, the stability criterion (20) generates to e2 = (W@)L,
+ (P&WG)%
(31)
of a fluid cylinder
39
influence of the electromagnetic force on the capillary instability of the MHD fluid cylinder. The numerical calculations are performed in order to obtain the values of u or/and those of w normalized with respect to the fundamental unit (T/&)‘12 for regular values of x in the wide range 0 G x G 5. This is carried out for the rotationally axisymmetric mode m = 0 for several values of I-I,, relative to H,( = (T/~R0)‘/2) for the pair values (cx,~) = (l,l), (1,2), (l,lO), (2,2) and (5,2). The numerical results are tabulated and plotted for u/(T/&)~/~ (for instability) or/and ~/U’/P%) 3 ‘I2 (for oscillation) against x in the range 0 6 x ( 5, see figs. l-5. There are many features of interest in these figures.
6i
sL
with L,=m(l-m2)+
(l-m-3m2)x (2m + 1)
2+
.
..)
4(32)
Mm = -2ma/3
[
(1+ c?) +
(2m2+3m-1) 2(m + 1)(2m + 1)’
+ -*-.
3-
2 x2 (33)
The leading term is independent of x for very long wavelengths as m -c 1. Since it is negative, the fluid cylinder is MHD stable for perturbations with very long wavelengths in the modes m > 1.
6. Numerical discussions
-_____ =
I
of the dispersion relation
The stability criterion (20) is computed in the computer simulation to determine and identify the
H,
2-1
H,
i i Ii,
H,
= 5H,
-.-.-
H, =lOl-&,
--
r, = ZOH,
I: II I’ L O0-
1
2 Fig. 4.
3
I
~
4x
40
of a fluid cylinder
A.E. Radwan, S. S. Elazab / Instability
-
(Ho=H,)
=-
&
__-__ = &
(H,=SHsI P
-.-.-
q
-&
(H,=lOH,) P.
a= 5 p=2
Fig. 5.
For the same pair value of (or,@) = (1,2), (2,2) and ($2); it is found that the maximum growing rate of instability is regularly increasing (the area under the instability curves is increasing) with increasing J&,/H, values; while for the same values of H,,/H, the area under the stability curves is decreasing. See figs. 2, 4 and 5. For the case (a,/3) = (1,l); the capillary instability is completely suppressed and the cylindrical interface of the MHD fluid cylinder is stable for all wavelengths (whether the x value is less or greater than unity) for all values of the applied equilibrium field relative to Z!Z,.This is completely different from the other case (cr,/3) = (1,lO). In the latter case in which p = 10 ((Y = 1 also), the cylindrical surface of the fluid jet is absolutely MHD unstable. This confirms the results of the analytical discussions of the dispersion relation (20): that the high values of p may cause a high destabilizing influence which overcomes the stabilizing effects of the surface tension (as 1 Q x < 5) together with that of the axial fields for all x values. See fig. 3. However, it is worthwhile to mention here that in the previous case (in which (a,p) = (l,l)), the
destabilizing effect of the azimuthal vacuum magnetic field (for all values of x) in addition to that of surface tension as 0 < x < 1 are not so strong as to overcome the stabilizing effect of the surface tension as 1 < x < cc. Therefore, the model is stable for all (short and long) wavelengths whatever the values of E&/H, (see fig. 1). Acknowledgements The authors would like to thank the Editor and the referee for the improvements and Professor D. Callebaut for fruitful discussions and encouragement. References (1) J. Plateau, Statique Experimentale et Thtorique des Liquides Soumis aux Seules Forces Mol&ulaires, ~01s. 1 and 2 (Gauthier-Villars, Paris, 1873). 121 J. Rayleigh, The Theory of Sound, ~01s. 1 and 2 (Dover, New York, 1945). [3] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981). [4] P. Roberts, An Introduction to Magnetohydrodynamics (Longman, London, New York, 1967).
A.E. Radwan, S.S. Elazab / Instability (51 A.E. Radwan and D. Callebaut, Proc. Belg. Phys. Sot. (Brussels) (1986) 21. [6] A.E. Radwan and D. Callebaut, Proc. Intern. Conf. on Computing in Plasma Physics (C.P.P.), Garmisch, Fed. Rep. Germany (1986) 6. [7] A.E. Radwan and S. Elazab, Simon Stevin 61 (1987) 293.
of a fluid cylinder
41
[8] A.E. Radwan, J. Magn. Magn. Mat. 72 (1988) 219. [9] D. Callebaut and A.E. Radwan, Proc. Intern. Conf. on Magnetic Fluids, Tokyo and Sendai, Japan, IV-19 (1986) 76. [lo] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
Journal of Magnetism and Magnetic Materials 79 (1989) 33-41 North-Holland, Amsterdam
NON-AXISYMMETRIC MAGNETODYNAMIC INSTABILITY OF A FLUID CYLINDER SUBJECT TO VARYING FIELDS Ahmed E. RADWAN Department
of Mathematics,
Faculty of Science, Ain-Shams
University, Abbassia,
Cairo, Egypt
and Samia S. ELAZAB Department
of Mathematics,
Women’s University College, Ain-Shams
University, Heliopolis,
Cairo, Egypt
Received 18 May 1988; in revised form 2 January 1989
The magnetodynamic instability of a fluid cylinder subject to the inertia, capillary and electromagnetic (the fluid is pervaded by uniform field and surrounded by a field which is generally varying) forces is presented to all (axisymmetric and non-axisymmetric) modes of perturbation. A general dispersion relation is derived, studied analytically and the results are confirmed numerically for different values of the vacuum magnetic field. The surface tension has the same influence as in the absence of the magnetic fields; however, sometimes for certain values of the vacuum field parameters the capillary instability is completely suppressed. The axial magnetic fields inside and outside the cylinder always have stabilizing effects to all modes of perturbation, whatever the values of the problem parameters are, as is expected. The azimuthal vacuum varying magnetic field has a destabilizing or stabilizing effect according to certain restrictions. This is confirmed by studying the dispersion relation asymptotically and numerically. It is found that there are unstable states in the non-axisymmetric modes, this is due to the fact that the vacuum magnetic field is a varying field and not uniform. In contrast to the case in which just uniform fields are acting, the model is completely stable in all non-axisymmetric modes.
1. Introduction The hydro-, gravito- and magnetohydromagnetic stability of a liquid cylinder has received considerable attention. It was started by the experimental and theoretical work of Plateau [l], and Rayleigh [2] has documented the previous work and his own results. Chandrasekhar [3] also summarized and made several extensions for that model and others subjected to different forces. The purpose of the present work is to investigate the magnetodynamic instability of a fluid cylinder subject to the inertia, capillary and electromagnetic forces. The fluid cylinder (radius R,) is endowed with surface tension (coefficient T), pervaded by a uniform magnetic field (intensity H,) and surrounded by the vacuum varying magnetic field Hovac (= (0, BH,,R,/r, aH,,)). Here (Y
and /3 are parameters which satisfy some restrictions (see the inequality (11)); r is the radial distance of a cylindrical polar coordinate (r, 9, t) system with the z-axis coinciding with the axis of the cylinder. In fact the present work is an extension to that of Chandrasekhar (ref. [3], p. 537), by considering the disturbances with all possible modes of perturbation and that the cylinder is surrounded by the non-uniform field H,‘aC. The present results with (Y= 1, p = 0 and m = 0 (where m is the azimuthal wave number) correspond to those of Chandrasekha (ref. [3], p. 542). The arrangement of the paper is as follows. The basic equations and equilibrium state are considered in section 2. In section 3 the dispersion relation is derived. Section 4 is devoted to the general discussions of the dispersion relation, the surface tension influence, magnetic field effects on
0304~8853/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A. E. Radwan, S. S. Elazab / Instability
34
the instability of the perfectly conducting fluid cylinder and the influence of the magnetic field on the capillary instability. The asymptotic behaviour of the dispersion relation for very long wavelengths to all modes of perturbation and its numerical discussions are considered in sections 5 and 6.
2. Basic equations and equilibrium state For an incompressible, inviscid and perfectly conducting fluid, the basic equations are p i (
+ (u*V))u-p(curl
H) XH=
-vp,
div u=O,
(1) (2)
i3H =curl(uXH),
at
The equilibrium state is studied and the equilibrium pressure distribution is given by PO = ($5,2/2)(
a2 + P* - 1) + (T/R,)
with the restrictions (p. > 0) a* + p* > 1 - (~T//LR~H;).
div H vat = 0
(5)
and
The last term in the right side of (10) is the contribution due to the capillary force influence. As a limiting case with p = 0 and (Y= 1 (the case which was considered by Chandrasekhar [3], p. 537), eq. (10) indicates that the magnetic fields have no influence on the equilibrium kinetic pressure po. The latter is due to the fact that the applied fields inside and outside the fluid are originally taken as the same.
For a small departure from the equilibrium state, every perturbed quantity Q(r, +, z; t) can be expressed as Q(r,
$3 z; t) = Q,(t)
+EQ,(~,
(6) interface we also have (7)
i.e. the jump of the total pressure [4] (the sum of the kinetic and magnetic pressures P = p + (p/2)( H H) must be balanced by the surface tension. R, (= of/ 1vf I) is the outward unit vector normal to the cylinder surface, where l
+, z; t) =0
(8)
is the equation of the boundary surface at time t and f(r, cp, z; t) = 0 satisfy %+(..v)f=O.
(9)
9, z) + . * *. (12)
Here Q stands for p, u, H, Hvac and the radial distance of the cylinder I = R, + rR, cos( kz + WI+),
curl H’“” = 0.
(P> = T div ri,,
(11)
(4
Here p, u and p are the fluid mass density, velocity vector and kinetic pressure; H and p are, respectively, the applied magnetic field and coefficient of the magnetic permeability. In the vacuum region, since there is no current, we have
Along the fluid-vacuum
(10)
3. Perturbation analysis and dispersion relation
div H = 0.
f(r,
of a fluid cylinder
(13)
where k and m are the longitudinal and azimuthal wave numbers. The subscripts 0 and 1 characterize the equilibrium and perturbed quantities, respectively. The second term in the right hand side of (13) is the elevation of the surface wave normalized with respect to R, and measured from the equilibrium position. E is the amplitude of the perturbation, see Chandrasekhar [3], being e = e. exp( at),
(14)
where co is the initial (at time t = 0) amplitude and u is the growing rate of instability; if u is imaginary u = io where w/2n is the oscillation frequency. By the use of the expansions (12)-(14) and following the same steps as the linear perturbation
A. E. Radwan, S. S. Elazab
technique used in refs. [5-g], the basic equations are linearized, solved and the constants of integration are determined by applying appropriate boundary conditions. Apart from the singular solutions we finally have ui = (aR,,/xZ~(x))
grad(Z,(kr)
cos(kz + m+)), (15)
H1 = (-ZZ,,/Z~(x))
grad(Z,,,(kr)
sin(kz+m+)),
Instability
of a fluid cylinder
35
and the Wronskian W(Z,(x),
K,(x))
= Z,,,(x)K;(x)
- Z;(x)K,(x)
= -x--l
(for the mode m = 0); eq. (21) corresponds to that of Chandrasekhar [3], p. 545 in studying the influence of a uniform axial magnetic field on the capillary instability of a liquid jet.
(16) Hi==
4. Discussions
- [(m/3 + ox)/xKXx)l xgrad(K,,,(kr)
(PI> = - (T/&)(1
sin(kz+m+)),
(17)
- m* - x2) cos( Zcz+ m+) (18)
and P,=
-[(o’+B;)/xZ,(x)]Z&r) Xcos(kz
+ m$).
(19)
Here x (= kR,) is the longitudinal dimensionless wave number; Z, and K,,, are the modified Bessel functions of the first and second kind of the order m and s2, (= (~ZZ~k*/p)‘/*) is the Alfvtn wave frequency defined in terms of ZZ,. Moreover, if we apply the continuity of the normal component of the total stress tensor across the boundary surface r = R,; we will end up, after some manipulation, with the relation c* = (U&)[
xZ~(x)/Z,(x)](l
+(ILZ&%R;){
-m*-x2)
- x2 + P”(xZ;(x)/Z,(x))
+(mp+ax)* x[Z;(x)Km(x)/Z,,z(x)K,&)l}.
(20)
If we put m = 0 with cy= 1 and p = 0 in eq. (20), we obtain
(1 -
c* = (V&)[xZ;(x)/Z&)] +&T&R;){
-x2
x2)
+ CY*X*Z,&)K,(X)
/Z,(x)K;(x)}.
(21)
By virtue of the relations (cf. ref. [lo]) Z;(x)
=Z,(x)
and
K;(x)
= -K,(x)
of the dispersion relation
4.1. General discussions The relation (20) is the stability criterion of a fluid cylinder endowed with surface tension, pervaded by a homogeneous axial magnetic field and surrounded by a vacuum varying magnetic field. It contains the most information about the instability of the problem at hand. It is a simple linear combination of the dispersion relations of a liquid cylinder subject to the capillarity force only and that subject to the electromagnetic force only. It is remarkable that (in the axisymmetric mode m = 0) for a liquid jet pervaded and surrounded by uniform axial magnetic fields and subject to a capillary force, Chandrasekhar [3] (p. 545, eq. (165) there) also obtained just such a simple combination of separate dispersion relations. One may be inclined to think that this is a general feature due to the linearization. However, this argument is not sufficient as is clear from the dispersion relation obtained in the presence of viscosity and resistivity. A detailed analysis shows that the simple linear addition in the dispersion relation is to be expected whether the perturbation is axisymmetric or not and whether the acting magnetic field is uniform or non-uniform, but usually not if the force added depends directly or indirectly on the velocity as in the case of a magnetic field. Indeed the velocity occurs in eq. (3) of evolution of the magnetic field thus influencing indirectly the electromagnetic force. In fact due to the vectorial product (U x If) there occurs a mixing of the velocity components i.e. the ith component occurring in the jth component of (u X H). This mixing, however, is undone when
A.E. Radwan, S.S. Elazab / A‘nstability of a fluid cylinder
36
taking the circulation of (u x H ); the i th component of H only depends on the i th component of u (the simple form of the magnetic field plays here an important role). That, together of course with the linearization, is the main feature leading to the simple additivity of the dispersion relation. The dispersion relation (20) relates the growing rate of instability u or rather the oscillation frequency w (for stability states) with the quantities (T/pRi) and (pHt/pRt) as units of (time)-*, the parameters OLand p of the vacuum magnetic field, the dimensionless wave number x and the modified Bessel functions Z, and K, of the order m. By means of this relation the stability as well as instability states can be identified together with the stabilizing or destabilizing influence of both the capillary and electromagnetic forces. 4.2. Capillaary force injluence If the liquid cylinder is subject to the capillary force only, its dispersion relation can be obtained from eq. (20) by setting Ho = 0: u2=
(T/~R~,)(xZL(X)/Z~(X))(~
-m*-x2). (22)
This corresponds to the classical dispersion relation of a capillary liquid jet derived by Rayleigh [2]. It is found that the cylindrical model is stable in the purely non-axisymmetric deformations; but it is unstable in the axisymmetric varicose deformations with wavelengths exceeding the circumference of the cylinder. For more analytical and numerical investigations we may refer to Chandrasekbar [3], p. 537. 4.3. Electromagnetic
force influence
If we study the perturbation of the liquid cylinder under the influence of the magnetic fields described above and neglect the capillary force influence (T = 0), the corresponding dispersion relation is obtained from (20) as 2 (I
2_ --
l-4
xz’(x) -x2
PR”O
+(mj?+ax)
+
p2.L_
L(x)
(23)
The influence of the uniform axial magnetic field pervades the interior of the fluid cylinder is represented by the term -x2 in the brackets, followed by the fundamental unit (pHz/pRt) in eq. (23). It always has a stabilizing influence. Similarly the influence of the homogeneous axial vacuum field is represented by the term including cy; it always has a stabilizing influence. Henceforth we may conclude that as was expected, as in the case of Chandrasekhar [3], p. 542, a fluid cylinder acting upon axial fields is stable to all modes of perturbation whatever the intensity of the applied magnetic fields. The influence of the azimuthal vacuum field is represented by the term including /3. It always has a destabilizing influence in the term /3’(xZA(x)/ Z,,,(x)) since this term is always positive. In the other term it always has stabilizing influence. However, if rnxpa < 0, its contribution is smaller than if it is positive. We have the choice in the sign of m; if we take m so that m/h c 0, that means if we change the perturbed vacuum field to have the opposite direction in the boundary surface, the stabilization due to this term is decreased. Since stability is determined by the least stable mode we have essentially to deal with the case rnpcx < 0. Concluding we may say that the transverse vacuum varying magnetic field always has a destabilizing influence. Due to that, the instability character of the magnetic field arises not only in the axisymmetric mode m = 0 but also in the non-axisymmetric modes m >, 1. Consequently, in the present problem, there are magnetodynamic instability states in the non-axisymmetric modes (see also sections 5 and 6). The destabilizing influence of the transverse magnetic field will be the greatest when rnp + ax = 0 i.e. when the perturbed vacuum magnetic field vanishes (see eq. (17)). It is important to observe that when we omit this term containing (mp + cyx), the dispersion relation significantly becomes an even function of m, p and independent of (Yand a complicated term disappears. Since this term always has a stabilizing effect, the omission leads to an upper limit of u2. Clearly if this upper limit allows only stability, the real case is stable also.
A.E. Radwan, S.S. Elarab / Instability
4.4. Effect tension
of a fluid cylinder
of the magnetic field on the surface
Now, returning to the general relation (20). The surface tension always has a stabilizing influence in all non-axisymmetric modes m > 1 for all wavelengths and also in the axisymmetric mode m = 0 if the wavelength is shorter than 2nR,. Meanwhile it has a destabilizing effect only in the axisymmettic disturbances whose wavelength is longer than the circumference of the cylinder. If the disturbance wavelength is equal to 21rR, then there will be a marginal stability. Combining these results with those of the electromagnetic force, one may conclude that there are instability domains whether the disturbance is axisymmetric or non-axisymmetric. In contrast to Chandrasekhar’s results (ref. [3], p. 542) where there is no destabilizing effect to the uniform field used there and even above a certain value of the equilibrium magnetic field, he found [3] that the
0
1
2
3 Fig. 2.
Fig. 1.
5
x
38
A.E. Radwan. S.S. Eiazab
Instability
capillary instability is completely suppressed and then stability sets in. In the present work there is no complete suppression of the capillary instability except for some unexpected rare values of p as we will see in the asymptotic and numerical discussions, see sections 5 and 6.
of a fluid cylinder
For axisymmetric reduces to
behaviour of the dispersion relation
f *. ),
i&f, = +x2( p’ - 2) + (g/3’ - a2 In fyx)x2
(25) + ..., (26)
The asymptotic behaviour of the general dispersion relation (20) is studied for very long wavelengths i.e. for very small values of X. This has already been carried out not only in the axisymmetric mode m = 0 but also in the lowest nonaxisymmetric mode m = 1 and in the higher nonaxisymmetric modes m 2 2. On utilizing the asymptotic behaviour of the modified Bessel functions Z,(x) and K,(x) in the modes m = 0, m = 1 and m a 2 for x -SC1 (cf. ref. [lo]), we have the following different classes.
where In y is the Eulerian constant. The asymptotic relation (24) with the aid of (25) and (26) shows, for very long wavelengths, that the MHD fluid jet is stable if
T+pH,2R,(~2- 2)<0,
(27)
where the equality holds for the marginal stability. Moreover, if g2 c 2 then a larger ZZ, increases the stabilizing influence.
Y=Y ---_-_ Y:SH, ----.-.-
m = 0, eq. (20)
(24)
L, = $X2(1 - gx2-t 5. Asymptotic
perturbation
H,=X)H, b = BH,
Fig. 3.
A.E. Radwan, S.S. Elazab / Instability
For the lowest non-axisymmetric m = 1, the relation (20) yields c2 = (VP@)
perturbation
L, + (P@/&)MI
(28)
with L, = 4x2(4 + x2 + . * * ),
(29)
M,=x{(-2a/3)+[:/32-(l+fx2) - +/I(1 + 4 In +rx)] x - [ $@(l
+ 4 In +rx)] x2 + 0(x3)
+ . . - }. (30)
From which it is clear that the magnetohydrodynamic fluid cylinder is unstable if 2Qx is negative for very long wavelengths. Since we restrict here the sign of m to be positive we have to allow x to be positive and negative, thus there is always instability for very low wavelengths whatever the sign of Cup. This confirms the analytical results of the magnetodynamic dispersion relation (20), as has been discussed in section 4.4. For higher non-axisymmetric perturbations m > 1, the stability criterion (20) generates to e2 = (W@)L,
+ (P&WG)%
(31)
of a fluid cylinder
39
influence of the electromagnetic force on the capillary instability of the MHD fluid cylinder. The numerical calculations are performed in order to obtain the values of u or/and those of w normalized with respect to the fundamental unit (T/&)‘12 for regular values of x in the wide range 0 G x G 5. This is carried out for the rotationally axisymmetric mode m = 0 for several values of I-I,, relative to H,( = (T/~R0)‘/2) for the pair values (cx,~) = (l,l), (1,2), (l,lO), (2,2) and (5,2). The numerical results are tabulated and plotted for u/(T/&)~/~ (for instability) or/and ~/U’/P%) 3 ‘I2 (for oscillation) against x in the range 0 6 x ( 5, see figs. l-5. There are many features of interest in these figures.
6i
sL
with L,=m(l-m2)+
(l-m-3m2)x (2m + 1)
2+
.
..)
4(32)
Mm = -2ma/3
[
(1+ c?) +
(2m2+3m-1) 2(m + 1)(2m + 1)’
+ -*-.
3-
2 x2 (33)
The leading term is independent of x for very long wavelengths as m -c 1. Since it is negative, the fluid cylinder is MHD stable for perturbations with very long wavelengths in the modes m > 1.
6. Numerical discussions
-_____ =
I
of the dispersion relation
The stability criterion (20) is computed in the computer simulation to determine and identify the
H,
2-1
H,
i i Ii,
H,
= 5H,
-.-.-
H, =lOl-&,
--
r, = ZOH,
I: II I’ L O0-
1
2 Fig. 4.
3
I
~
4x
40
of a fluid cylinder
A.E. Radwan, S. S. Elazab / Instability
-
(Ho=H,)
=-
&
__-__ = &
(H,=SHsI P
-.-.-
q
-&
(H,=lOH,) P.
a= 5 p=2
Fig. 5.
For the same pair value of (or,@) = (1,2), (2,2) and ($2); it is found that the maximum growing rate of instability is regularly increasing (the area under the instability curves is increasing) with increasing J&,/H, values; while for the same values of H,,/H, the area under the stability curves is decreasing. See figs. 2, 4 and 5. For the case (a,/3) = (1,l); the capillary instability is completely suppressed and the cylindrical interface of the MHD fluid cylinder is stable for all wavelengths (whether the x value is less or greater than unity) for all values of the applied equilibrium field relative to Z!Z,.This is completely different from the other case (cr,/3) = (1,lO). In the latter case in which p = 10 ((Y = 1 also), the cylindrical surface of the fluid jet is absolutely MHD unstable. This confirms the results of the analytical discussions of the dispersion relation (20): that the high values of p may cause a high destabilizing influence which overcomes the stabilizing effects of the surface tension (as 1 Q x < 5) together with that of the axial fields for all x values. See fig. 3. However, it is worthwhile to mention here that in the previous case (in which (a,p) = (l,l)), the
destabilizing effect of the azimuthal vacuum magnetic field (for all values of x) in addition to that of surface tension as 0 < x < 1 are not so strong as to overcome the stabilizing effect of the surface tension as 1 < x < cc. Therefore, the model is stable for all (short and long) wavelengths whatever the values of E&/H, (see fig. 1). Acknowledgements The authors would like to thank the Editor and the referee for the improvements and Professor D. Callebaut for fruitful discussions and encouragement. References (1) J. Plateau, Statique Experimentale et Thtorique des Liquides Soumis aux Seules Forces Mol&ulaires, ~01s. 1 and 2 (Gauthier-Villars, Paris, 1873). 121 J. Rayleigh, The Theory of Sound, ~01s. 1 and 2 (Dover, New York, 1945). [3] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981). [4] P. Roberts, An Introduction to Magnetohydrodynamics (Longman, London, New York, 1967).
A.E. Radwan, S.S. Elazab / Instability (51 A.E. Radwan and D. Callebaut, Proc. Belg. Phys. Sot. (Brussels) (1986) 21. [6] A.E. Radwan and D. Callebaut, Proc. Intern. Conf. on Computing in Plasma Physics (C.P.P.), Garmisch, Fed. Rep. Germany (1986) 6. [7] A.E. Radwan and S. Elazab, Simon Stevin 61 (1987) 293.
of a fluid cylinder
41
[8] A.E. Radwan, J. Magn. Magn. Mat. 72 (1988) 219. [9] D. Callebaut and A.E. Radwan, Proc. Intern. Conf. on Magnetic Fluids, Tokyo and Sendai, Japan, IV-19 (1986) 76. [lo] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).