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Magnetohydrodynamic stability of a compound liquid jet
299
Journal of Magnetism and Magnetic Materials 81 (1989) 299-312 North-Holland, Amsterdam
MAGNETOHYDRODYNAMIC
STABILITY
OF A COMPOUND
LIQUID JET
.Ahmed E. RADWAN Department
of Mathematics,
Faculty of Science, Ain-Shams
University,
Cairo, Egypt
Received 17 April 1989
The magnetohydrodynamics (MHD) stability of a compound nonmiscible fluid jet is discussed. A general eigenvalue relation, for that model which involves the fluid inertia, capillarity and electromagnetic forces, is derived. The model is capillary unstable only for small axisymmetric disturbances and stable for the rest. The magnetic fields interior and exterior to the gas-mantle jet have always a stabilizing influence. The radii ratio of the concentric jets plays an important role in the (instability) stability states and are (decreasing) increasing with increasing magnetic field intensity as the exterior radius is much larger than the interior radius; under some restrictions of the radii ratio and above a certain value of the magnetic field the capillary instability is omitted and completely suppressed and then stability sets in. The latter result is verified analytically and confirmed numerically in the case in which the cylindrical surface of the outer jet is sited at infinity.
1. Introdnction It has been well known, since the pioneering and miscellaneous scientific works of Rayleigh [l] and Chandrasekhar [2] that the most involved works concerning stability problems have involved that of a full liquid jet ambient with vacuum. In the present era, in particular the last decade the scientific province has been concerned with the stability of more complicated models than the naive one earlier. The study of two phase jets is of great importance in designing pulverized fuel furnaces and in pollution control. The principle and basic physics of the new type of liquid-in-air jet are described by Hertz and Hermanrud [3]. The experimental study of the propagation of a gas jet in a liquid has been performed by Surin et al. [4]. A theoretical model is developed to determine the flow characteristics and dynamics of vertical annular liquid jets by Esser and Abdel-Khalik [5]. The hydrodynamic instability of a gas jet submerged in a liquid of different density is analysed by Cheng [6], as a first step in studying the formation of bubbles caused by the injection of air into water. However it is worthwhile to mention here that the
result given by Cheng [6], in eqs. (4) and (5) there, are incorrect in the third term. In fact the quantity (1 - s2 - k2a2) must be in the numerator as is clear from eq. (3) there, see also eqs. (39) and (41) in the present work and see also Chandrasekhar (ref. [2], pp. 538, 540, eqs. (147), (155)). Kendall [7] devised and studied experimentally the capillary instability of an annular liquid jet where the liquid inertia force is predominant over that of the gas jet. Around the realm of this field of studies comprehensive account of hydrodynamic instability has recently been documented by Monin [8]. Radwan and Elazab [9], following Kendall’s restrictions, have investigated the capillary instability of a viscous hollow cylinder for small axisymmetric disturbances. The effect of magnetic field on the capillary instability of a hollow liquid jet has recently been elaborated in a definitive account for all modes of perturbation, by Callebaut and Radwan [lo]. They also elaborated the stability of an annular liquid jet having a solid axis as a mantle in ref. [ll]. Their results can be obtained as limiting cases from the present more general work. We studied the magnetohydrodynamic stability of a compound fluid jet of double interfaces and
0304-8853/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
300
A. E. Radwan
/ Magnetohydrodynamic
its relation to the capillarity and electromagnetic forces taking into account the influence of the inertia of the nonmiscible fluids.
2. Basic state We consider a gas cylinder, of density p and radius a, concentric and coaxial with a liquid cylinder of radius b and density p’ (where b > a). The fluids are assumed to be incompressible, inviscid and perfectly conducting. We shall use a cylindrical polar coordinates system (r, cp, z) with the axis of the coaxial cylinders coinciding with the z-axis. The effect of the surrounding fluid on the compound fluid jet is neglected. Each of the nonmiscible fluids as well as the surrounding vacuum is pervaded by the uniform magnetic field (0, 0, H,,). Each Of these fluids is subject to inertia, capillarity and electromagnetic forces. The surface tension effect along the gas-liquid inter-
stability
face is proposed to be greater than that acting along the liquid-vacuum interface. The basic equations of a fluid are those of motion, conservation of mass, equation of motion of the boundary surface, evolution equation of the magnetic field, conservation of magnetic flux and that of surface tension. For the problem under consideration they are $+(..
V)U-%(H-V)H=
-vlI,
div u=o, ;rcr,
(2) cp, z; t) = 0,
(3)
aH =curl(uXH), at
-
divH=O,
(5)
P,= Tdiv ii,,
(6)
where
pII(
=
P + :p(H*
H))
is the magnetohy-
Table1 p//p = 1.0 Ho/H,
= 0.0
x
o/ FxT/pa
0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0.0
0.45 0.50 0.55 0.60 0.65 0.677 0.70 0.75 0.80 0.85 0.90 0.95
1.00
0.035236 0.069883 0.0103512 0.135762 0.166309 0.194850 0.221091 0.244739 0265490 0.283015 0.296948 0.306864 0.312242 0.313034 0.312419 0.306496 0.293167 0.270342 0.234137 0.175012 0.0
Ho/H,
= 0.1 o/J9
0.0 0.05 0.034880 0.10 0.06964 0.15 0.102420 0.20 0.134281 0.25 0.164420 0.30 0.192527 0.35 0.218304 0.40 0.241448 0.45 0.261648 0.50 0.278563 0.55 0.291810 0.60 0.300941 0.65 0.305401 0.667 0.0305733 0.70 0.304476 0.75 0.297178 0.80 0.282041 0.85 0.256632 0.90 0.216148 0.95 0.146985 0.985 0.013885 0.0
H,,/H,
(1)
= 0.3
x
o/ 7xT/pa
0.0
0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.589 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.865 0.866
0.031884 0.063113 0.093219 0.121784 0.148438 0.172820 0.194567 0.213301 0.228604 0.239994 0.246887 0.248664 0.248526 0.243865 0.231313 0.208122 0.168366 0.089778 0.057596 0.029270 0.018906
H, /H,
= 0.5
Ho/H,
= 0.7
a/m
x
a/m
0.0
0.0
0.0
0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.425 0.45 0.50 0.55 0.60 0.602 0.604 0.606 0.608 0.610 0.612 0.614 0.618
0.024832 0.048823 0.071343 0.091823 0.109699 0.124365 0.135117 0.141058 0.141853 0.140925 0.132655 0.112042 0.064540 0.061257 0.057733 0.053921 0.049754 0.045134 0.039902 0.033777 0.014790
0.050 0.051 0.053 0.055 0.057 0.059 0.061 0.064 0.065 0.067 0.069 0.071 0.073 0.075 0.077 0.079 0.08 0.083 0.087 0.089 0.091
0.004078 0.004121 0.004200 0.004268 0.004325 0.004369 0.004400 0.004420 0.004419 0.004404 0.004371 0.004317 0.004214 0.004139 0.004008 0.003844 0.003639 0.003384 0.002654 0.002096 0.001196
A. E. Radwan
/ Magnetohydrodynamic
301
stabiliry
Table 2 p’/p = 1 .o H,/H,
H,,/H,
= 0.0
x
o/ VXT/pa
1.00 1.05 1.15 1.25 1.35 1.45 1.55 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0 0.1938 0.3680 0.5166 0.6599 0.8030 0.9477 1.2450 1.3982 1.5545 1.7140 1.8766 2.0424 2.2112 2.3831 2.5580 2.7358 2.9165 3.1001 3.2865 3.4758 3.8625 4.2601 4.6682 5.0868 5.5154 5.9541 6.4025 6.8606 7.3281 7.6848
0.986 1.00 1.05 1.15 1.25 1.35 1.45 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.65 4.85 5.00
= 0.1
H, /H,
= 0.3
Q/W
x
o/ TXT/pa
0.0216 0.1000 0.2204 0.3856 0.5315 0.6736 0.8160 1.1073 1.2573 1.4104 1.5667 1.7262 1.8889 2.0547 2.2237 2.3957 2.5706 2.7486 2.9294 3.1132 3.2998 3.4891 3.8762 4.2740 4.6825 5.1013 5.5303 5.9692 6.8763 7.3441 7.7011
0.867 0.90 0.95 1.05 1.15 1.25 1.35 1.55 1.65 1.75 1.85 1.94 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.25 3.45 3.65 3.85 4.05 4.25 4.65 4.85 5.00
0.0120 0.1344 0.2249 0.3698 0.5044 0.6384 0.7743 1.0556 1.2016 1.3512 1.5043 1.6610 1.8210 1.9844 2.1510 2.3209 2.4939 2.6699 2.8489 3.0309 3.2158 3.4036 3.9837 4.3840 4.7949 5.2162 5.6477 6.0891 7.0010 7.4711 7.8298
drodynamic pressure of the fluid. This is the sum of the kinetic and magnetic pressures, u is the fluid velocity, W and I-( are the magnetic field excitation and magnetic field excitation, respectively, P, is the curvature pressure of the cylindrical surface and T is the coefficient of the surface tension. ri, is the outward normal vector to the interface being n, = Wr,
cp, r; t)/l
vf(r,
H, /H,
cp, r; t> I,
(7)
where f(r, cp, z; t) = 0 is the equation of the gas-liquid interface at time t. Similar systems of equations like the system (l)-(5) can be written down with dashes over the variables to represent the outer liquid jet.
H,/H,
= 0.5 a/ 7xT/pa
0.619 0.65 0.75 0.85 0.95 1.05 1.15 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.65 2.85 3.05 3.25 3.45 3.65 2.85 4.05 4.25 4.65 4.85 5.00
0.0039 0.0901 0.2160 0.3279 0.4415 0.5596 0.8109 0.9440 1.0818 1.2242 1.3709 1.5217 1.6765 1.8350 1.9971 2.1627 2.3317 2.5040 2.6795 3.0397 3.4119 3.7956 4.1904 4.5961 5.0123 5.4388 5.8754 6.3219 7.2438 7.7189 8.0812
= 0.7 a/ 7 T/pa
0.092 0.10 0.15 0.25 0.35 0.45 0.65 0.75 0.85 0.94 1.05 1.15 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.65 4.85 5.00
0.0004 0.0040 0.0176 0.0544 0.1055 0.1695 0.3309 0.4262 0.5300 0.6415 0.7601 0.8851 1.0161 1.2942 1.5915 1.9058 2.2354 2.5791 2.9360 3.3054 3.6865 4.0791 4.4827 4.8970 5.3217 5.7566 6.2014 7.6936 7.5936 8.0763 8.4443
In the vacuum surrounding the compound jet, there is no current; the basic equations are div HVBC= 0,
(8)
curl H”= = 0,
(9)
where H vac is the magnetic field intensity in the vacuum region. The equilibrium state is studied, from which II, = constant,
l7; = constant
(10)
and the balance of the pressure at r = u gives P,=P;+(T/a),
(11)
where the subscript 0 refers to equilibrium quanti-
302
A.E. Radwan / Magnetohydrodynamic
stability
Table 3 p’/p = 1 .o x
Ho/H, w/m
Q/JT/pa3
a/ 7 T/pa
a/ 7 T/pa
w/ 7 T/pa
0.0 0.50 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00
0.0 0.0073 0.0308 0.0688 0.1209 0.1857 0.2619 0.3486 0.4445 0.5490 0.6612 0.7804 0.9061 1.0377 1.1748 1.3171 1.4641 1.6156 1.7713 1.9311 2.0947 2.2619 2.6069 3.0564 3.5251 4.0119 4.5160 5.0369 5.5738 6.1265 6.6944 7.2772 7.8746 8.4862
0.0 0.0189 0.0607 0.1110 0.1718 0.2431 0.3246 0.4158 0.5158 0.6239 0.7395 0.8620 0.9909 1.1255 1.2656 1.4108 1.5607 1.7150 1.8735 2.0360 2.2023 2.3722 2.7224 3.1783 3.6532 4.1461 4.6561 5.1826 5.7252 6.2832 6.8565 7.4445 8.0469 8.6635
0.0 0.0354 0.1085 0.1866 0.2713 0.3633 0.4629 0.5700 0.6845 0.8058 0.9337 1.0677 1.2074 1.3525 1.5026 1.6575 1.8167 1.9802 2.1477 2.3189 2.4938 2.6721 3.0387 3.5145 4.0085 4.5199 5.0479 5.5919 6.1514 6.7260 7.3153 7.9190 8.5367 9.1682
0.0 0.0935 0.2815 0.4715 0.6641 0.8599 1.0591 1.2619 1.4683 1.6783 1.8919 2.1089 2.3292 2.5528 2.7794 3.0091 3.2416 3.4769 3.7148 3.9553 4.1984 4.4438 4.9418 5.5769 6.2254 6.8869 7.5610 8.2474 8.9458 9.6560 10.3779 11.1112 11.8559 12.6117
0.0 0.2475 0.7428 1.2388 1.7359 2.2342 2.7339 3.2349 3.7374 4.2413 4.7467 5.2535 5.7617 6.2713 6.7821 7.2943 7.8077 8.3223 8.8381 9.3550 9.8731 10.3923 11.4339 12.7417 14.0559 15.3762 16.7027 18.0352 19.3737 20.7181 22.0684 23.4245 24.7866 26.1544
= 0.72
H,, /H,
= 0.8
H,/H,
ties. The magnetic field has no contribution in the equilibrium restriction (11) because we have taken the intensity of the magnetic field in the two fluids to be originally the same.
3. Perturbation analysis If we allow a small departure from the equilibrium state, every physical quantity Q( r, cp, z; t) can be expressed as its equilibrium part, plus a
= 1.0
H, /H,
= 2.0
H,/H,
= 5.0
fluctuation part
Q(r, cp, z; t> = Q,(r)
+ c(t)Q,(r,
cp, z>,
04
where E is the amplitude of the perturbation at time t and where Q(r, q,, z) stands for each of P', P, II', II,u',H’, u, H and the radial distance of each fluid jet. The latter are, if we consider a single Fourier term of a sinusoidal wave, given by r, = a + eOa exp[ at + i( kz +
mq)] ,
r, = b + e,,b exp[ at + i(kz + m(p)],
(13) 04
303
A. E. Radwan / Magnetohydrodynamic stability
Table4 p’/p = 2.0 Ho/H,=0
Ho/H,
= 0.1
x
o/w
x
o/ 7XT/pa
0.0 0.05
0.0
0.0
0.0
0.035168 0.069463 0.102347 0.133432 0.162414 0.189036 0.213075 0.234315 0.252541 0.267519 0.278980 0.286607 0.289990 0.290114 0.288591 0.281665 0.268096 0.246070 0.212172 0.157927 0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.652 0.70 0.75 0.80 0.85 0.90 0.95 0.985
0.034812 0.068748 0.101267 0.131977 0.160568 01.86782 0.210388 0.231164 0.248887 0.263311 0.274155 0.281075 0.283636 0.283641 0.281254 0.273102 0.257921 0.233591 0.195871 0.132635 0.012494
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.661 0.70 0.75 0.80 0.85 0.90 0.95 1.00
H,,/H,
= 0.3
HI = [ -ikH,/(a2
0.0
0.0
0.0
0.0
0.0
0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.577 0.60 0.65 0.70 0.75 0.80 0.85 0.855 0.860 0.866
0.031822 0.062736 0.092170 0.119694 0.144961 0.167663 0.187512 0.2044216 0.217454 0.226954 0.231950 0.232702 0.232121 0.226485 0.213671 0.191261 0.153968 0.081717 0.075619 0.057596 0.017184
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.418 0.45 0.50 0.55 0.60 0.601 0.603 0.605
0.024783 0.048529 0.070540 0.090247 0.107129 0.120654 0.130217 0.135050 0.135436 0.134052 0.125392 0.105263 0.060280 0.058766 0.055579 0.052149 0.048423 0.044326 0.034485 0.028170 0.013785
0.05 0.051 0.053 0.055 0.057 0.059 0.061 0.064 0.065 0.067 0.069 0.071 0.073 0.075 0.077 0.079 0.081 0.083 0.085 0.087 0.091
0.004070 0.004113 0.004191 0.004258 0.004314 0.004358 0.004388 0.004407 0.004406 0.004390 0.004356 0.004302 0.004226 0.004124 0.003993 0.003828 0.003623 0.003369 0.003050 0.002641 0.001190
+ S;ri)] grad II,,
(16)
mcp)],
0.609 0.613 0.615 0.618
II; = [ -u/( u2 +
clip)] grad
~~ = [ -ikH,/(
a2 + ~?a)]
II; = [Bl,(kr)
II;. grad III;,
(19) (20)
- CK,(kr)]
x exp[ at + i( kz + mcp)]
(21)
and Hr==
v{[Km(kr)] exp[ut+i(kz+mg,)]}. (22)
Here A, B, C and E are constants to be determined, I,( kr) and K,( kr) are the modified Bessel functions of the first and second kind of the order m; s2, and aa are the Alfven wave frequencies of the gas and liquid jets 0, = ( pH;k2/p)1’2,
(23)
ii!; = ( p’H;k2/p’)1’2,
(24)
(17)
P,,=(-T/a)(l-m2-k2a2) xexp[at+i(kz+mcp)],
o/ 7 T/pa
o/ TXT/pa
(15)
II, = AI,,,( kr) exp[ at + i( kz +
= 0.1
x
grad II,,
+ Q:)]
H,,/H,
= 0.5
44%
where cc, is the initial amplitude at time t = 0, u is the temporal amplification, k is (real) the longitudinal wavenumber and m is (integer) the azimuthal wavenumber. By inserting the expansions (12)-(14) into the basic equations (l)-(9), the relevant perturbation equations are identified. Based on the linear perturbation technique and the (cp, z)-dependence, every relevant perturbed quantity can be written as a function of r times exp[ot + i(kz + mcp)]. Consequently by following the same procedure as in refs. [2,10], the linearized system of the basic equations are solved and the non-singular solutions are given by ur = [-o/(02
H, /H,
(18)
where CL’is the coefficient of the magnetic permeability of the liquid of the outer jet.
304
A.E. Radwan / Magnetohydrodynamic stability
Table 5 p’/p = 2.0 Ho/H,=0
H,/H,
x
a/ J----x T/pa
1.00 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0 0.1735 0.3272 0.4564 0.5798 0.7019 0.8247 0.9489 1.0750 1.2032 1.3337 1.4665 1.6017 1.7393 1.8792 2.0214 2.1660 2.3129 2.4622 2.6136 2.7674 2.9234 3.2420 3.5693 3.9051 4.2494 4.6018 4.9623 5.3308 5.7070 6.0909 6.3838
= 0.1
H,/H,
= 0.3
a/ 7xT/pa 0.986 1.00 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0194 0.0898 0.1973 0.3428 0.4696 0.5918 0.7133 0.8356 0.9596 1.0855 1.2137 1.3442 1.4770 1.6122 1.7498 1.8897 2.0321 2.1768 2.2328 2.4731 2.6247 2.7785 3.2535 3.5810 3.9171 4.2615 4.6142 4.9749 5.3436 5.7201 6.1042 6.3973
0.861 0.90 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
= 0.5
H&H,
o/JT/;;;Is
x
a/ FxT/pa
0.0109 0.1218 0.2029 0.3311 0.4485 0.5640 0.6803 0.7983 0.9186 1.0413 1.1666 1.2945 1.4250 1.5581 1.6937 1.8318 1.9724 2.1154 2.2608 2.5588 2.8660 3.0230 3.3437 3.6731 4.0111 4.3575 4.7121 5.0748 5.4454 5.8238 6.2098 6.5043
0.619 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.35 2.55 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0037 0.0837 0.1985 0.2984 0.3984 0.5010 0.6070 0.7164 0.8294 0.9457 1.0653 1.1881 1.3139 1.4427 1.5743 1.7088 1.8459 2.1280 2.4202 2.7221 3.0333 3.1924 3.5172 3.8508 4.1930 4.5435 4.9021 5.2689 5.6435 6.0258 6.4157 6.7131
4.Boundary conditions The solution given by eqs. (15)-(22) must satisfy the following boundary conditions at the interfaces r = a and r = b. (9 The continuity of the normal components of the velocities of the fluids across the interface, eq. (13), at r = a and simultaneously they must be compatible with the velocity of the deformed interface (13) at r = a. The normal component of the velocity (n u’) ’ of the (outer) liquid jet must be compatible with the velocity of the deformed interface, eq. (14), at r = b. l
Ho/H,
= 0.7 a/ F T/pa
0.092 0.10 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.15 1.35 1.55 1.75 1.95 2.15 2.35 2.55 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0004 0.0040 0.0174 0.0531 0.1017 0.1612 0.2302 0.3073 0.3917 0.4824 0.5789 0.7870 1.0126 1.2536 1.5081 1.7749 2.0532 2.3421 2.6413 2.9501 3.2683 3.4308 3.7625 4.1029 4.4518 4.8089 5.1741 5.5473 5.9282 6.3168 6.7128 7.0147
(iii) The normal component of the magnetic fields H and H ’ must be continuous across the interface, eq. (13), at r = a. (iv) The normal component of the magnetic fields H’ and HVaC must be continuous across the interface, eq. (14), at r = b. (v) The normal component of the total stress tensor must be continuous across the (gas-liquid) interface, eq. (13), at r = a. The foregoing boundary conditions, (i)-(iv), are satisfied such that A = --c()+2+ B = [a’yK;((y)
sz~)/xI;(x), - b’&(x)]
(2% G(x,
Y),
(26)
A. E. Radwan / Magnetohydrodynamic stability
305
Table6 p’/p = 2.0 H,/H,
x
0.0 0.05
0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
= 0.72
Ho/H,
= 0.8
H,/H,=I.O
H,/H,
o/w
a/ 7 T/pa
a/ 7 T/pa
a/ 7 T/pa
0.0
0.0
0.0
0.0073 0.0304 0.0672 0.1165 0.1766 0.2461 0.3237 0.4085 0.4997 0.5966 0.6987 0.9169 1.1513 1.4001 1.6618 1.9354 2.2199 2.5150 2.8199 3.1343 3.4580 3.7905 4.1317 4.4814 4.8393 5.2054 5.5793 5.9610 6.3503 6.7471 7.0495
0.0188 0.0600 0.1084 0.1655 0.2312 0.3050 0.3861 0.4740 0.5679 0.6673 0.7718 0.9945 1.2332 1.4862 1.7521 2.2097 2.3183 2.6173 2.9261 3.2444 3.5718 3.9081 4.2529 4.6052 4.9676 5.3371 5.7144 6.0994 6.4920 6.8920 7.1968
0.0354 0.1073 0.1822 0.2614 0.3456 0.4349 0.5294 0.6290 0.7335 0.8425 0.9560 1.1950 1.4489 1.7161 1.9955 2.2863 2.5877 2.8991 3.2201 3.5503 3.8893 4.2370 4.5930 4.9571 5.3292 5.7092 6.0968 6.4919 6.8944 7.3041 7.6161
(27)
= 2.0
0.0
Ho/H,
a/ 7 T/pa 0.0
0.0934 0.2784 0.4604 0.6400 0.8180 0.9950 1.1720 1.3493 1.5276 1.7072 1.8882 2.2555 2.6303 3.0131 3.4037 3.8022 4.2083 4.6218 5.0426 5.4703 5.9050 6.3463 6.7942 7.2486 7.7092 8.1762 8.6492 9.1284 9.6135 10.1045 10.4766
0.2470 0.7344 1.2098 1.6730 2.1253 2.5685 3.0044 3.4346 3.8605 4.2833 4.7038 5.5410 6.3762 7.2122 8.0504 8.8918 9.7369 10.5857 11.4383 12.2950 13.1553 14.0194 14.8871 15.7583 16.6329 17.5110 18.3924 19.2771 20.1650 21.0562 21.7266
normal stress tensor (condition (vi) reads)
3%
II, + q,a exp[ at + i( kz + m(p)] ar
(28) (2% t p’
qy
=
chw;(x)
- MawY)~
(30)
where x( = ku) and y( = kb) are the longitudinal dimensionless wavenumbers. The continuity of the
= 5.0
=p,,.
II,l +
~,,a
1 an;,
exp[ at + i( kz + m(p)] ar
) (31)
On utilizing eqs. (lo), (17), (18), (21), (23)-(27), (29) and (30) for the condition (31), the following
A. E. Radwan / Magnetohydro&namic
306
stability
Table7 p’/p = 5.0 H,,/H,
= 0
Ho/H,
H,/H,
= 0.1
= 0.3
Q/ F*T/pa
H, /H, = 0.5
a/ 7xT/pa
Ho/H,
x
o/ 7.xT/pa
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.629 -0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.034965 0.068245 0.099074 0.127102 0.151187 0.174291 0.193415 0.209567 0.222739 0.232888 0.239928 0.243712 0.244329 0.244011 0.240480 0.232598 0.219547 0.199951 0.171169 0.126557 0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.621 -0.65 0.670 0.75 0.80 0.85 0.90 0.95 0.985
0.034611 0.067542 0.098028 0.125715 0.150458 0.172213 0.190976 0.206749 0.219516 0.229225 0.235777 0.239008 0.239318 0.238665 0.234366 0.225527 0.211214 0.189811 0.158018 0.106289 0.009968
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.551 -0.60 0.65 0.70 0.75 0.80 0.85 0.855 0.865 0.866
0.031639 0.061636 0.089222 0.114016 0.135834 0.154585 0.170211 0.182647 0.191793 0.197487 0.199480 0.199483 0.197380 0.190576 0.178050 0.157943 0.126086 0.066402 0.055862 0.021571 0.013930
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.402 0.45 0.50 0.55 0.60 0.601 0.603 0.605 0.607 0.611 0.615 0.617 0.618
0.024640 0.047678 0.068284 0.085966 0.100384 0.111243 0.118203 0.120787 0.120790 0.118232 0.109160 0.090527 0.051258 0.049960 0.47230 0.044296 0.041113 0.033716 0.023876 0.016785 0.011676
0.050 0.051 0.053 0.055 0.057 0.059 0.061 0.064 -0.065 0.067 0.069 0.071 0.073 0.075 0.077 0.079 0.081 0.083 0.087 0.089 0.091
0.004046 0.004088 0.004164 0.004229 0.004283 0.004325 0.004353 0.004369 0.004367 0.004349 0.004314 0.004258 0.004180 0.004078 0.003946 0.003781 0.003577 0.003324 0.002603 0.002055 0.001172
eigenvalue relation is derived a2 = (T/pa3)(1
- m2 - ~‘>W;:,+,“,G,(x~
-(~~H,Z/pa~)x~yl,(x)L,“,G,(x,
Y) b2 2 -YL,”
-(y’H,Z/p’u2)(p’/p)x21~(x) [
xG,&
Y>,
Y>
1
(32)
with G,_‘k
Y)
5. General discussions
Eq. (32) is the stability criterion of a liquid jet having a gas jet (of different density) acting as a
o/ 7.xT/pa
= 0.7 o/ J--T/pa
mantle upon the inertia, capillarity and electromagnetic forces. It is a quadratic equation in the temporal amplification (I. The eigenvalue relation (32) relates (I to the fundamental quantities ~;~:~;$1’2 as well as ( ~‘u~/~_L’H,)~‘~ and as a unit of time, the two kinds of modified Bessel functions and their derivatives, different combinations (L,mY,Lr and G,(x, y)) of the Bessel functions, the densities and radii ratio (p’/p) and (b/u) and with the wavenumbers x, y and m. By means of the dispersion relation (32) the characteristics and the (in-)stability states can be identified. The marginal stability can be obtained by setting u = 0. It is worthwhile to mention here that the relation (32) is somewhat complicated, to some extent, because we are dealing with double interfaces and consequently two kinds of perturbation. That kind of disturbance is totally different from those which are considered in refs. [1,2,8-lo], where only one single interface is considered.
A.E. Radwan / Magnetohydrodynamic
This relation, for small axisymmetric disturbances m = 0, coincides with our recent result [12], if we neglect the basic flow velocity there. For the stability discussions of the relation (35) we may refer to ref. [12] or/and [9] in neglecting the streaming effect or/and viscosity influence. If p’ = 0, Ho = 0 and at the same time y tends to infinity; the model is a full liquid jet (submerged in a vacuum) subject to inertia and capillary forces. The corresponding dispersion relation can be determined from (32) as
6. Stability discussions Since the problem under consideration is somewhat more general, several eigenvalues of different hydrodynamic and hydromagnetic problems can be recovered as limiting cases from the general eigenvalue relation (32). If p = 0, H,, = 0 and simultaneously y tends to infinity; the model is a hollow cylinder endowed with surface tension and is acted upon by the liquid inertia force. The dispersion relation of this case can be obtained from (32) as cJ2= (T/pV)[
307
stability
a*=
(T/pa3)[xI;((x)/lm(x)](l
-rn*-X2).
- xKx-4/Kn(41
(36)
X(1-d-x*).
(35)
This is the classical
Ho/H,= 0.3
H,, /H,
dispersion
relation of a cir-
Table8 p'/p= 5.0 Ho/H,=0
Ho/H,= 0.1
x
w/JT/pn3
x
m'/7x T/pa
1.00
0.0
1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.1374 0.2564 0.3545 0.4467 0.5371 0.6273 0.7179 0.8095 0.9022 0.9964 1.0919 1.1890 1.2876 1.3878 1.4896 1.5930 1.6979 1.8044 1.9125 2.0221 2.1334 2.3604 2.5935 2.8326 3.0776 3.3283 3.5868 3.8468 4.1143 4.3872 4.5954
0.986 1.00 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0155 0.0715 0.1562 0.2686 0.3647 0.4560 0.5458 0.6356 0.7260 0.8174 0.9101 1.0042 1.0997 1.1968 1.2954 1.3957 1.4975 1.6009 1.7058 1.8124 2.0303 2.1416 2.3687 2.6020 2.8412 3.0864 3.3373 3.5939 3.8561 4.1238 4.3968 4.6051
a/
0.867 0.90 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
7x T/pa
0.0088 0.0982 0.1626 0.2622 0.3515 0.4380 0.5242 0.6109 0.6987 0.7878 0.8785 0.9707 1.0646 1.1601 1.2573 1.3561 1.4567 1.5588 1.6627 1.8752 2.0942 2.2060 2.4344 2.6689 2.9095 3.1559 3.4081 3.6661 3.9295 4.1985 4.4729 4.6821
= 0.5 a/
0.619 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.35 2.55 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
H,,/H,= 0.7
47xT/pa
0.0031 0.0704 0.1639 0.2425 0.3193 0.3967 0.4756 0.5564 0.6390 0.7237 0.8103 0.8989 0.9894 1.0818 1.1761 1.2723 1.3703 1.5716 1.7799 1.9949 2.2165 2.3297 2.5608 2.7981 3.0414 3.2906 3.5356 3.8063 4.0725 4.3442 4.6212 4.8325
u/
0.091 0.10 0.15 0.25 0.35 0.45 0.55 0.75 0.75 0.85 0.95 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
7 T/pa
0.0004 0.0039 0.0168 0.0498 0.0923 0.1422 0.1979 0.3234 0.3234 0.3920 0.4639 0.5389 0.6972 0.8657 1.0435 1.2298 1.4241 1.6260 1.8352 2.0514 2.2743 2.5037 2.7394 2.9812 3.2291 3.4828 3.7423 4.0074 4.2779 4.5539 4.8352 5.0496
A.E.Radwan / Magnetohydrodynamicstability
308 Table9 p'/p= 5.0 x
0.0 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
H,/H, = 0.12
Ho/H,= 0.8
H&H,
o/ F T/pa
w/ 7 T/pa 0.0 0.0187 0.0581 0.1016 0.1503 0.2039 0.2623 0.3249 0.3914 0.4614 0.5348 0.6111 0.7723 0.9437 1.1244 1.3138 1.5115 1.7164 1.9287 2.1480 2.3740 2.6065 2.8453 3.0902 3.3411 3.5978 3.8601 4.1281 4.4015 4.6802 4.9642 5.1806
a/ F T/pa 0.0 0.0325 0.1039 0.1708 0.2373 0.3048 0.3740 0.4455 0.5194 0.5960 0.6752 0.7569 0.9281 1.1087 1.2983 1.4964 1.7023 1.9158 2.1364 2.3638 2.5979 2.8383 3.0848 3.3373 3.5957 3.8597 4.1293 4.4043 4.6847 4.9703 5.2611 5.4825
0.0 0.0073 0.0294 0.0630 0.1057 0.1558 0.2116 0.2724 0.3373 0.4060 0.4781 0.5533 0.6313 0.8810 0.1059 0.1246 0.1441 0.1643 0.1853 0.2070 0.2293 0.2523 0.2759 0.3002 0.3250 0.3504 0.3764 0.4030 0.4301 0.4578 0.4859 0.5074
cular liquid column. Regarding its (in-)&ability we may refer to Chandrasekhar (ref. [2], p. 537). If p # 0, p’ # 0, ZZ, = 0 and y tends to infinity, then the model is a gas-core liquid jet. The driving forces in such a case, are capillarity and the inertia of the nonmiscible fluids. The corresponding dispersion relation is established from (32) in the form
=l.O
Ho/H,= 2.0
Ho/H,= 5.0
a/ F T/pa 0.0 0.0928 0.2695 0.4314 0.5810 0.7214 0.8557 0.9861 1.1143 1.2413 1.3681 1.4951 1.7516 2.0129 2.2796 2.5524 2.8311 3.1156 3.4059 3.7017 4.0029 4.3092 4.6205 4.9368 5.2578 5.5834 5.9136 6.2482 6.5873 6.9306 7.2782 7.5416
w/ F T/pa 0.0 0.2455 0.7109 1.1336 1.5186 1.8745 2.2089 2.5280 2.8363 3.1370 3.4325 3.7246 4.3032 4.8794 5.4566 6.0368 6.6207 7.2087 7.8008 8.3968 8.9967 9.6002 10.2071 10.8172 11.4303 12.0464 12.6652 13.2868 13.9109 14.5375 15.1666 15.6400
By resort to the recurrence relations [13] of the modified Bessel functions and the each non-zero real value of x that Z,,,(x) is positive and monotonic increasing while K,,,(x) is monotonic decreasing but never negative. One can show that Z;(x) is positive definite while K:(x) is negative for each non-zero real value of x. By the aid of these arguments we can prove, for each non-zero real value of x, that
Using this inequality
for the criterion (37) we
A. E. Radwan
/ Magnetohydrodynamic
stability
309
Table10 p’/p = 10.0 H,,/H,
=0
H,,/H,
= 0.1
H,, /H,
= 0.3
Ho/H,
= 0.5
H,/H,
= 0.7
x
a/FxT/pa
o/m
x
o/m
x
o/Jvz
x
o/ lf-- T/pa
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.550 0.598 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.034635 0.066350 0.094253 0.118298 0.138738 0.155899 0.170071 0.181482 0.190288 0.196575 0.200361 0.201592 0.201591 0.200180 0.195740 0.188038 0.176403 0.159772 0.136091 0.100167 0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.598 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.985
0.034284 0.065667 0.093259 0.117007 0.137162 0.154040 0.167926 0.179041 0.187534 0.193483 0.196894 0.197752 0.197700 0.195745 0.190764 0.182321 0.169708 0.151669 0.125635 0.084125 0.007866
0.05 0.10 0.15 0.20 0.25 0.30 0.35' 0.40 0.45 0.50 0.523 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.855 0.865 0.866
0.031340 0.059925 0.084881 0.106118 0.123830 0.138272 0.149668 0.158169 0.163850 0.166694 0.167022 0.166583 0.163267 0.156304 0.144925 0.127685 0.101308 0.053059 0.044614 0.017210 0.011112
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.384 0.40 0.45 0.50 0.60 0.601 0.603 0.605 0.607 0.609 0.611 0.613 0.615 0.618
0.024408 0.046354 0.064962 0.080011 0.095134 0.099504 0.103936 0.104819 0.104599 0.101007 0.092139 0.042399 0.041317 0.039046 0.036607 0.033965 0.031067 0.027834 0.024132 0.019697 0.009628
0.050 0.051 0.053 0.055 0.059 0.063 0.064 0.065 0.067 0.069 0.071 0.073 0.075 0.077 0.079 0.081 0.083 0.085 0.087 0.089 0.091
0.004008 0.004048 0.004121 0.004183 0.004271 0.004308 0.004309 0.004304 0.004284 0.004246 0.004188 0.004108 0.004004 0.003872 0.003708 0.003505 0.003254 0.002943 0.002545 0.002007 0.001143
conclude, through its analytical discussions, that the model of a gas-core liquid jet is unstable only in the axisymmetric disturbances m = 0 for certain wavelengths which are longer than the circumference of the gas jet. If p = 0, p’ # 0, H,, f 0 and y tends to infinity, then the model is a hollow jet which acts upon the capillarity liquid inertia and electromagnetic forces. The eigenvalue relation (32) in such a case degenerates to a2 = (T/p’a3)[
-xKk(x)/K,(x)](l
-m*
X(~‘H,2/p’a2)[-x/(I~(x)K,(x))l
- x2)
9 (39)
hollow jet in the domain m = 0, 0 G x -C 1 and increases the stability of the surface tension in the domains (m = 0, x > l), and (m a 1 for all x values). Moreover, one can show that the capillary instability of the hollow cylinder is completely suppressed if EC, > (fiH,/2) (where H,( = (7’/ pR,,)“*) has a unit of magnetic field) and then stability sets in. Now returning to the general case of compound MHD liquid jets where the stability criterion is given by eq. (32) with eqs. (33) and (34). If the surface tension influence is paramount over that of the magnetic field, the relation (32) degenerates to
where use has been made of the Wronskian w,(Ll(X)~
u*= (T/pa’)
L(x))
= I,(x)KA(x)
-IA(x)&(x)
= -x-l.
(40)
As H,, = 0, eq. (39) reduces to (35) which is the capillary stability criterion of a hollow cylinder. The magnetic field has a stabilizing influence and therefore reduces the capillary instability of the
VqJA (x >
X
YLc4qJ + (d/P) X(1-m*-x2).
[
5 -.Yq
1
4x4 (41)
310
A.E. Radwan / Magnetohydrodynamic
stability
Table11 p’/p = 10.0 Ho/H,=0
Ho/H,
x
w/ 7xT/pa
1.00
0.0
1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.15 2.85 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.1079 0.2000 0.2750 0.3449 0.4131 0.4807 0.5486 0.6169 0.6861 0.7561 0.8272 0.8993 0.9725 1.0468 1.1223 1.1989 1.2167 1.3557 1.4358 1.5170 1.5994 1.7676 1.9402 2.1173 2.2987 2.4844 2.6142 2.8682 3.0662 3.2683 2.4223
= 0.1
H,/H,
= 0.3
a/ 7xT/pa
0.986 1.00 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0122 0.0564 0.1227 0.2096 0.2829 0.3521 0.4198 0.4871 0.5547 0.6230 0.6920 0.7620 0.8331 0.9052 0.9784 1.0527 1.1282 1.2049 1.2827 1.3617 1.4418 1.6056 1.7738 1.9466 2.1238 2.3053 2.4911 2.6811 2.8751 3.0733 3.2754 3.4296
= 0.5
w/ 7xT/pa
0.867 0.90 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0070 0.0781 0.1287 0.2059 0.2742 0.3398 0.4047 0.4698 0.5355 0.6020 0.6695 0.7381 0.8079 0.8788 0.9509 1.0242 1.0987 1.1745 1.2514 1.3296 1.4894 1.6539 1.8230 1.9967 2.1748 2.3572 2.5440 2.7349 2.9299 3.1290 3.3321 2.4869
Utilizing the recurrence relations [13] of the modified Bessel functions together with their properties, we can prove for every non-zero real value of x, Y and (p’/p) that the following combinations of Bessel functions satisfy the inequalities. L,“>O,
H,/H,
Lx”y< 0,
{ XYL,~,M~NL(X~ Y)} ‘0 and never change sign for all axisymmetric (m = 0) and non-axisymmetric (m f 0) modes provided
H, /H,
a/ J-‘x T/pa
0.619 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0025 0.0577 0.1325 0.1938 0.2527 0.3115 0.3710 0.4316 0.4934 0.5565 0.6210 0.6868 0.7540 0.8226 0.8925 0.9638 1.0364 1.1103 1.2619 1.4186 1.5802 1.7466 1.9177 2.0933 2.2134 2.4578 2.6465 2.8395 3.0365 3.2315 3.4426 3.5989
= 0.1 a/ 7 T/pa
0.092 0.10 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 4.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0003 0.0038 0.0160 0.0454 0.0812 0.1215 0.1653 0.2121 0.2615 0.3132 0.3671 0.4231 0.5409 0.6658 0.7973 0.9351 1.0788 1.2281 1.3827 1.5426 1.7074 2.5014 2.0514 2.2303 2.4137 2.6014 2.1934 2.9895 3.1897 3.3939 3.6020 3.7606
that YLY”’ (b/a
)’
(43)
is fulfilled. From the viewpoint of the restriction (43) and the inequalities (42), the stability discussions of the relation (41) predict the following results. The concentric liquid jets are only unstable in the axisymmetric mode m = 0 if the perturbed wavelength X is longer than the circumference of the gas-mantle jet 2~~2. While the model is stable in the axisymmetric disturbance m = 0 as long as
A.E. Radwan / Magnetohydrodynamic
X G (2~) and in all non-axisymmetric modes m 2 1 for all (short and long) wavelengths. In order to examine the magnetic field influence and its effect on the capillary (in-)stability it is found convenient to write the general dispersion relation (32) in the dimensionless form
Wn(x~ Y>,
(40
311
stability
where and
H, = (WV$‘*
H,’ = ( T/p’p’ )1’2.
The last two terms (of (44)) which include Ho represent the effects of the magnetic fields interior and exterior to the gas jet. For one single interface it is proved (cf., refs. [2,10]) that such a homogeneous magnetic field always has a stabilizing influence, whatever the kind of models considered. From the analytical discussions of (44) it is also found that the magnetic fields inside and outside the compound jet have stabilizing effects for all values of (p//p), x (f 0) and y ( f 0) provided that condition (43) is satisfied. The stabilized influence of the magnetic field increases with in-
Table 12 p’/p = 10.0 x
0.0 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
Ho/H,
= 0.12
H,, /H,
= 0.8
H,,/H,
=l.O
Ho/H,
= 2.0
H,,/H,
= 5.0
w/ 7 T/pa
a/ 7 T/pa
w/ 7 T/pa
w/ 7 T/pa
w/ 7 T/pa
0.0 0.0072 0.0280 0.0574 0.0930 0.1331 0.1767 0.2234 0.2727 0.3244 0.3784 0.4345 0.5524 0.6776 0.8094 0.9475 1.0916 1.2413 1.3963 1.5566 1.7218 1.8919 2.0667 2.2460 2.4298 2.6179 2.8102 3.0067 3.2073 3.4119 3.6204 3.7792
0.0 0.0186 0.0552 0.0926 0.1321 0.1742 0.2190 0.2665 0.3164 0.3687 0.4232 0.4799 0.5991 0.7258 0.8592 0.9990 1.1448 1.2963 1.4531 1.6152 1.7823 1.9542 2.1207 2.3119 2.4974 2.6873 2.8813 3.0795 3.2818 3.4880 3.6981 3.8582
0.0 0.0348 0.0988 0.1557 0.2087 0.2604 0.3123 0.3653 0.4199 0.4762 0.5344 0.5944 0.7200 0.8527 0.9921 1.1378 1.2895 1.4469 1.6096 1.7775 1.9503 2.1279 2.3101 2.4967 2.6877 2.8829 3.0822 3.2856 3.4930 3.7042 3.9192 4.0830
0.0 0.0919 0.2563 0.3933 0.5108 0.6163 0.7146 0.8088 0.9008 0.9919 1.0828 1.1741 1.3589 1.5480 1.7419 1.9408 2.1446 2.3531 2.6933 2.9301 3.0051 3.2307 3.4601 3.6933 3.9301 4.1704 4.4141 4.6612 4.9116 6.1651 5.4219 5.6165
0.0 0.2432 0.6763 1.0335 1.3353 1.6014 1.8446 2.0734 2.2929 2.5066 2.7167 2.9248 3.3384 3.7526 4.1696 4.5904 5.0153 5.4444 5.8774 6.3141 6.7542 7.1974 7.6437 8.0926 8.5440 8.9978 9.4538 9.9119 10.3721 10.8343 11.2984 11.6476
A.E. Radwan
312
/ Magnetohydrodynamic
creasing magnetic field intensity. Therefore as is expected under the restriction (43) the capillary instability will be omitted and completely suppressed above a certain value of ZZ,,relative to H, and H,’ and then the stability sets in. This means that, in the unstable domain 0 d x < 1 for m = 0, either the growth rate u/( T/p a 3)1’2 or the critical wavenumber x, or both are decreasing with increasing intensity of the magnetic field until it completely shrinks and the model will be stable for all problem parameters in all modes of perturbations m z 0 under the restriction (43). Finally we intend to investigate the MHD stability of a gas core liquid jet which is a special case arising from our present general case, as the cylindrical surface of the outer jet is cited at infinity. In such a case the stability criterion is given from (32) in the dimensionless form a2
= { (1 - m2 - x2)
(VW3) +(H~/H,)2[Z~(x)K~(x)l
-‘} (4%
w?(x), where Z%(x) = xZ~(x)K~(x)[Z,(x)K~,(x) -
Z:(x) K;(x)
= :(Z,&) = -f@,-,(x)
+ Z?n+*(x)), + L+,(x)),
(46) relations
>O
111Lord Rayleigh,
(cf., PI (47) (48)
and to each non-zero real value of x that Zm(x), Z:(x) and K,(x) are always positive while KA(x,) is negative one can show that El(x)
axisymmetric modes m > 0 for all x, (p’/p) and (Ho/H,) values. If one considers the combined effect of the capilarity and electromagnetic forces simultaneously, the capillary instability is decreasing by increasing the magnetic field intensity. Moreover, it is found that above the critical value ,/m of Ho, the capillary instability is completely suppressed and stability sets in. Noting that in identifying that critical value of H,, the inequality Z,(x)K,(x) G : for x > 0. These analytical results are confirmed numerically for different values of (p’/p) = 1.0, 2.0, 5.0 and 10.0, then for each value of (p’/p) the capillary instability of the gas-core liquid jet is examined by considering several values of (Ho/H,) = 0, 0.1, 0.3 . . . 5.0. It is found that the critical values of (Ho/H,) above which the capillary instability is completely suppressed is H,,/H, = 0.702. . . for all values of (p//p), see tables 1-12. It is also remarkable that however much larger the density of the outer liquid is than that of the gas jet, there are always instability domains. This shows that the ratio (p’/p) plays a secondary role in the stability of a compound liquid jet.
References
b’/d~~~x)~m(x)l -l.
By an appeal to the recurrence ref. [13]):
stability
(49)
for all m and (p//p) values. In view of the inequality (49) the gas-core liquid jet at Ho = 0 is capillary unstable for the axisymmetric mode m = 0 if the wavenumber x is less than unity, while it is stable for m = 0 if x > 1 and for m 2 1 for all x values. In neglecting the surface tension influence, eq. (45) (by the aid of the relations (47) and (48)) indicates that the magnetic field has a strong stabilizing effect for all axisymmetric and non-
[31 [41 PI [61 171 PI [91 PO1
1111 WI [I31
The Theory of Sound (Dover, New York, 1945); Scientific papers Vols. I, II and III (Cambridge, 1902). S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981). C.H. Hertz and B. Hermanrud, J. Fluid Mech. 131 (1983) 271. V.A. Surin, V.N. Evchenko and V.M. Rubin, J. Eng. Phys. (USA) 45 (1983) 1091. P.D. Esser and S.I. Abdel-Khahk, J. Fluids Eng. (USA) 106 (1984) 45. L.Y. Cheng, Phys. Fluids 28 (1985) 2614. J.M. Kendall, Phys. Fluids 29 (1986) 2086. A.S. Monin, Sov. Phys. USP (USA) 29 (1986) 843; Phys. Abst., Sci. Abst., series A (1987) 1104. A.E. Radwan and S.S. Elazab, Simon Stevin 61 (1987) 293. D.K. Cahebaut and A.E. Radwan, Europ. Phys. Sot., 10D (1986) 11; A.E. Radwan and CaIlebaut, Proc. Int. Conf. on Computing in Plasma Physics, Garmisch, Fed. Rep. of Germany (1986) p. 6. A.E. Radwan and D.K. CaIlebaut, Proc. Belg. Phys. Sot. P.P. (1985) 21. A.E. Radwan, J. Phys. Sot. Japan 58 (1989) 1225. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
Journal of Magnetism and Magnetic Materials 81 (1989) 299-312 North-Holland, Amsterdam
MAGNETOHYDRODYNAMIC
STABILITY
OF A COMPOUND
LIQUID JET
.Ahmed E. RADWAN Department
of Mathematics,
Faculty of Science, Ain-Shams
University,
Cairo, Egypt
Received 17 April 1989
The magnetohydrodynamics (MHD) stability of a compound nonmiscible fluid jet is discussed. A general eigenvalue relation, for that model which involves the fluid inertia, capillarity and electromagnetic forces, is derived. The model is capillary unstable only for small axisymmetric disturbances and stable for the rest. The magnetic fields interior and exterior to the gas-mantle jet have always a stabilizing influence. The radii ratio of the concentric jets plays an important role in the (instability) stability states and are (decreasing) increasing with increasing magnetic field intensity as the exterior radius is much larger than the interior radius; under some restrictions of the radii ratio and above a certain value of the magnetic field the capillary instability is omitted and completely suppressed and then stability sets in. The latter result is verified analytically and confirmed numerically in the case in which the cylindrical surface of the outer jet is sited at infinity.
1. Introdnction It has been well known, since the pioneering and miscellaneous scientific works of Rayleigh [l] and Chandrasekhar [2] that the most involved works concerning stability problems have involved that of a full liquid jet ambient with vacuum. In the present era, in particular the last decade the scientific province has been concerned with the stability of more complicated models than the naive one earlier. The study of two phase jets is of great importance in designing pulverized fuel furnaces and in pollution control. The principle and basic physics of the new type of liquid-in-air jet are described by Hertz and Hermanrud [3]. The experimental study of the propagation of a gas jet in a liquid has been performed by Surin et al. [4]. A theoretical model is developed to determine the flow characteristics and dynamics of vertical annular liquid jets by Esser and Abdel-Khalik [5]. The hydrodynamic instability of a gas jet submerged in a liquid of different density is analysed by Cheng [6], as a first step in studying the formation of bubbles caused by the injection of air into water. However it is worthwhile to mention here that the
result given by Cheng [6], in eqs. (4) and (5) there, are incorrect in the third term. In fact the quantity (1 - s2 - k2a2) must be in the numerator as is clear from eq. (3) there, see also eqs. (39) and (41) in the present work and see also Chandrasekhar (ref. [2], pp. 538, 540, eqs. (147), (155)). Kendall [7] devised and studied experimentally the capillary instability of an annular liquid jet where the liquid inertia force is predominant over that of the gas jet. Around the realm of this field of studies comprehensive account of hydrodynamic instability has recently been documented by Monin [8]. Radwan and Elazab [9], following Kendall’s restrictions, have investigated the capillary instability of a viscous hollow cylinder for small axisymmetric disturbances. The effect of magnetic field on the capillary instability of a hollow liquid jet has recently been elaborated in a definitive account for all modes of perturbation, by Callebaut and Radwan [lo]. They also elaborated the stability of an annular liquid jet having a solid axis as a mantle in ref. [ll]. Their results can be obtained as limiting cases from the present more general work. We studied the magnetohydrodynamic stability of a compound fluid jet of double interfaces and
0304-8853/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
300
A. E. Radwan
/ Magnetohydrodynamic
its relation to the capillarity and electromagnetic forces taking into account the influence of the inertia of the nonmiscible fluids.
2. Basic state We consider a gas cylinder, of density p and radius a, concentric and coaxial with a liquid cylinder of radius b and density p’ (where b > a). The fluids are assumed to be incompressible, inviscid and perfectly conducting. We shall use a cylindrical polar coordinates system (r, cp, z) with the axis of the coaxial cylinders coinciding with the z-axis. The effect of the surrounding fluid on the compound fluid jet is neglected. Each of the nonmiscible fluids as well as the surrounding vacuum is pervaded by the uniform magnetic field (0, 0, H,,). Each Of these fluids is subject to inertia, capillarity and electromagnetic forces. The surface tension effect along the gas-liquid inter-
stability
face is proposed to be greater than that acting along the liquid-vacuum interface. The basic equations of a fluid are those of motion, conservation of mass, equation of motion of the boundary surface, evolution equation of the magnetic field, conservation of magnetic flux and that of surface tension. For the problem under consideration they are $+(..
V)U-%(H-V)H=
-vlI,
div u=o, ;rcr,
(2) cp, z; t) = 0,
(3)
aH =curl(uXH), at
-
divH=O,
(5)
P,= Tdiv ii,,
(6)
where
pII(
=
P + :p(H*
H))
is the magnetohy-
Table1 p//p = 1.0 Ho/H,
= 0.0
x
o/ FxT/pa
0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0.0
0.45 0.50 0.55 0.60 0.65 0.677 0.70 0.75 0.80 0.85 0.90 0.95
1.00
0.035236 0.069883 0.0103512 0.135762 0.166309 0.194850 0.221091 0.244739 0265490 0.283015 0.296948 0.306864 0.312242 0.313034 0.312419 0.306496 0.293167 0.270342 0.234137 0.175012 0.0
Ho/H,
= 0.1 o/J9
0.0 0.05 0.034880 0.10 0.06964 0.15 0.102420 0.20 0.134281 0.25 0.164420 0.30 0.192527 0.35 0.218304 0.40 0.241448 0.45 0.261648 0.50 0.278563 0.55 0.291810 0.60 0.300941 0.65 0.305401 0.667 0.0305733 0.70 0.304476 0.75 0.297178 0.80 0.282041 0.85 0.256632 0.90 0.216148 0.95 0.146985 0.985 0.013885 0.0
H,,/H,
(1)
= 0.3
x
o/ 7xT/pa
0.0
0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.589 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.865 0.866
0.031884 0.063113 0.093219 0.121784 0.148438 0.172820 0.194567 0.213301 0.228604 0.239994 0.246887 0.248664 0.248526 0.243865 0.231313 0.208122 0.168366 0.089778 0.057596 0.029270 0.018906
H, /H,
= 0.5
Ho/H,
= 0.7
a/m
x
a/m
0.0
0.0
0.0
0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.425 0.45 0.50 0.55 0.60 0.602 0.604 0.606 0.608 0.610 0.612 0.614 0.618
0.024832 0.048823 0.071343 0.091823 0.109699 0.124365 0.135117 0.141058 0.141853 0.140925 0.132655 0.112042 0.064540 0.061257 0.057733 0.053921 0.049754 0.045134 0.039902 0.033777 0.014790
0.050 0.051 0.053 0.055 0.057 0.059 0.061 0.064 0.065 0.067 0.069 0.071 0.073 0.075 0.077 0.079 0.08 0.083 0.087 0.089 0.091
0.004078 0.004121 0.004200 0.004268 0.004325 0.004369 0.004400 0.004420 0.004419 0.004404 0.004371 0.004317 0.004214 0.004139 0.004008 0.003844 0.003639 0.003384 0.002654 0.002096 0.001196
A. E. Radwan
/ Magnetohydrodynamic
301
stabiliry
Table 2 p’/p = 1 .o H,/H,
H,,/H,
= 0.0
x
o/ VXT/pa
1.00 1.05 1.15 1.25 1.35 1.45 1.55 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0 0.1938 0.3680 0.5166 0.6599 0.8030 0.9477 1.2450 1.3982 1.5545 1.7140 1.8766 2.0424 2.2112 2.3831 2.5580 2.7358 2.9165 3.1001 3.2865 3.4758 3.8625 4.2601 4.6682 5.0868 5.5154 5.9541 6.4025 6.8606 7.3281 7.6848
0.986 1.00 1.05 1.15 1.25 1.35 1.45 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.65 4.85 5.00
= 0.1
H, /H,
= 0.3
Q/W
x
o/ TXT/pa
0.0216 0.1000 0.2204 0.3856 0.5315 0.6736 0.8160 1.1073 1.2573 1.4104 1.5667 1.7262 1.8889 2.0547 2.2237 2.3957 2.5706 2.7486 2.9294 3.1132 3.2998 3.4891 3.8762 4.2740 4.6825 5.1013 5.5303 5.9692 6.8763 7.3441 7.7011
0.867 0.90 0.95 1.05 1.15 1.25 1.35 1.55 1.65 1.75 1.85 1.94 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.25 3.45 3.65 3.85 4.05 4.25 4.65 4.85 5.00
0.0120 0.1344 0.2249 0.3698 0.5044 0.6384 0.7743 1.0556 1.2016 1.3512 1.5043 1.6610 1.8210 1.9844 2.1510 2.3209 2.4939 2.6699 2.8489 3.0309 3.2158 3.4036 3.9837 4.3840 4.7949 5.2162 5.6477 6.0891 7.0010 7.4711 7.8298
drodynamic pressure of the fluid. This is the sum of the kinetic and magnetic pressures, u is the fluid velocity, W and I-( are the magnetic field excitation and magnetic field excitation, respectively, P, is the curvature pressure of the cylindrical surface and T is the coefficient of the surface tension. ri, is the outward normal vector to the interface being n, = Wr,
cp, r; t)/l
vf(r,
H, /H,
cp, r; t> I,
(7)
where f(r, cp, z; t) = 0 is the equation of the gas-liquid interface at time t. Similar systems of equations like the system (l)-(5) can be written down with dashes over the variables to represent the outer liquid jet.
H,/H,
= 0.5 a/ 7xT/pa
0.619 0.65 0.75 0.85 0.95 1.05 1.15 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.65 2.85 3.05 3.25 3.45 3.65 2.85 4.05 4.25 4.65 4.85 5.00
0.0039 0.0901 0.2160 0.3279 0.4415 0.5596 0.8109 0.9440 1.0818 1.2242 1.3709 1.5217 1.6765 1.8350 1.9971 2.1627 2.3317 2.5040 2.6795 3.0397 3.4119 3.7956 4.1904 4.5961 5.0123 5.4388 5.8754 6.3219 7.2438 7.7189 8.0812
= 0.7 a/ 7 T/pa
0.092 0.10 0.15 0.25 0.35 0.45 0.65 0.75 0.85 0.94 1.05 1.15 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.65 4.85 5.00
0.0004 0.0040 0.0176 0.0544 0.1055 0.1695 0.3309 0.4262 0.5300 0.6415 0.7601 0.8851 1.0161 1.2942 1.5915 1.9058 2.2354 2.5791 2.9360 3.3054 3.6865 4.0791 4.4827 4.8970 5.3217 5.7566 6.2014 7.6936 7.5936 8.0763 8.4443
In the vacuum surrounding the compound jet, there is no current; the basic equations are div HVBC= 0,
(8)
curl H”= = 0,
(9)
where H vac is the magnetic field intensity in the vacuum region. The equilibrium state is studied, from which II, = constant,
l7; = constant
(10)
and the balance of the pressure at r = u gives P,=P;+(T/a),
(11)
where the subscript 0 refers to equilibrium quanti-
302
A.E. Radwan / Magnetohydrodynamic
stability
Table 3 p’/p = 1 .o x
Ho/H, w/m
Q/JT/pa3
a/ 7 T/pa
a/ 7 T/pa
w/ 7 T/pa
0.0 0.50 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00
0.0 0.0073 0.0308 0.0688 0.1209 0.1857 0.2619 0.3486 0.4445 0.5490 0.6612 0.7804 0.9061 1.0377 1.1748 1.3171 1.4641 1.6156 1.7713 1.9311 2.0947 2.2619 2.6069 3.0564 3.5251 4.0119 4.5160 5.0369 5.5738 6.1265 6.6944 7.2772 7.8746 8.4862
0.0 0.0189 0.0607 0.1110 0.1718 0.2431 0.3246 0.4158 0.5158 0.6239 0.7395 0.8620 0.9909 1.1255 1.2656 1.4108 1.5607 1.7150 1.8735 2.0360 2.2023 2.3722 2.7224 3.1783 3.6532 4.1461 4.6561 5.1826 5.7252 6.2832 6.8565 7.4445 8.0469 8.6635
0.0 0.0354 0.1085 0.1866 0.2713 0.3633 0.4629 0.5700 0.6845 0.8058 0.9337 1.0677 1.2074 1.3525 1.5026 1.6575 1.8167 1.9802 2.1477 2.3189 2.4938 2.6721 3.0387 3.5145 4.0085 4.5199 5.0479 5.5919 6.1514 6.7260 7.3153 7.9190 8.5367 9.1682
0.0 0.0935 0.2815 0.4715 0.6641 0.8599 1.0591 1.2619 1.4683 1.6783 1.8919 2.1089 2.3292 2.5528 2.7794 3.0091 3.2416 3.4769 3.7148 3.9553 4.1984 4.4438 4.9418 5.5769 6.2254 6.8869 7.5610 8.2474 8.9458 9.6560 10.3779 11.1112 11.8559 12.6117
0.0 0.2475 0.7428 1.2388 1.7359 2.2342 2.7339 3.2349 3.7374 4.2413 4.7467 5.2535 5.7617 6.2713 6.7821 7.2943 7.8077 8.3223 8.8381 9.3550 9.8731 10.3923 11.4339 12.7417 14.0559 15.3762 16.7027 18.0352 19.3737 20.7181 22.0684 23.4245 24.7866 26.1544
= 0.72
H,, /H,
= 0.8
H,/H,
ties. The magnetic field has no contribution in the equilibrium restriction (11) because we have taken the intensity of the magnetic field in the two fluids to be originally the same.
3. Perturbation analysis If we allow a small departure from the equilibrium state, every physical quantity Q( r, cp, z; t) can be expressed as its equilibrium part, plus a
= 1.0
H, /H,
= 2.0
H,/H,
= 5.0
fluctuation part
Q(r, cp, z; t> = Q,(r)
+ c(t)Q,(r,
cp, z>,
04
where E is the amplitude of the perturbation at time t and where Q(r, q,, z) stands for each of P', P, II', II,u',H’, u, H and the radial distance of each fluid jet. The latter are, if we consider a single Fourier term of a sinusoidal wave, given by r, = a + eOa exp[ at + i( kz +
mq)] ,
r, = b + e,,b exp[ at + i(kz + m(p)],
(13) 04
303
A. E. Radwan / Magnetohydrodynamic stability
Table4 p’/p = 2.0 Ho/H,=0
Ho/H,
= 0.1
x
o/w
x
o/ 7XT/pa
0.0 0.05
0.0
0.0
0.0
0.035168 0.069463 0.102347 0.133432 0.162414 0.189036 0.213075 0.234315 0.252541 0.267519 0.278980 0.286607 0.289990 0.290114 0.288591 0.281665 0.268096 0.246070 0.212172 0.157927 0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.652 0.70 0.75 0.80 0.85 0.90 0.95 0.985
0.034812 0.068748 0.101267 0.131977 0.160568 01.86782 0.210388 0.231164 0.248887 0.263311 0.274155 0.281075 0.283636 0.283641 0.281254 0.273102 0.257921 0.233591 0.195871 0.132635 0.012494
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.661 0.70 0.75 0.80 0.85 0.90 0.95 1.00
H,,/H,
= 0.3
HI = [ -ikH,/(a2
0.0
0.0
0.0
0.0
0.0
0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.577 0.60 0.65 0.70 0.75 0.80 0.85 0.855 0.860 0.866
0.031822 0.062736 0.092170 0.119694 0.144961 0.167663 0.187512 0.2044216 0.217454 0.226954 0.231950 0.232702 0.232121 0.226485 0.213671 0.191261 0.153968 0.081717 0.075619 0.057596 0.017184
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.418 0.45 0.50 0.55 0.60 0.601 0.603 0.605
0.024783 0.048529 0.070540 0.090247 0.107129 0.120654 0.130217 0.135050 0.135436 0.134052 0.125392 0.105263 0.060280 0.058766 0.055579 0.052149 0.048423 0.044326 0.034485 0.028170 0.013785
0.05 0.051 0.053 0.055 0.057 0.059 0.061 0.064 0.065 0.067 0.069 0.071 0.073 0.075 0.077 0.079 0.081 0.083 0.085 0.087 0.091
0.004070 0.004113 0.004191 0.004258 0.004314 0.004358 0.004388 0.004407 0.004406 0.004390 0.004356 0.004302 0.004226 0.004124 0.003993 0.003828 0.003623 0.003369 0.003050 0.002641 0.001190
+ S;ri)] grad II,,
(16)
mcp)],
0.609 0.613 0.615 0.618
II; = [ -u/( u2 +
clip)] grad
~~ = [ -ikH,/(
a2 + ~?a)]
II; = [Bl,(kr)
II;. grad III;,
(19) (20)
- CK,(kr)]
x exp[ at + i( kz + mcp)]
(21)
and Hr==
v{[Km(kr)] exp[ut+i(kz+mg,)]}. (22)
Here A, B, C and E are constants to be determined, I,( kr) and K,( kr) are the modified Bessel functions of the first and second kind of the order m; s2, and aa are the Alfven wave frequencies of the gas and liquid jets 0, = ( pH;k2/p)1’2,
(23)
ii!; = ( p’H;k2/p’)1’2,
(24)
(17)
P,,=(-T/a)(l-m2-k2a2) xexp[at+i(kz+mcp)],
o/ 7 T/pa
o/ TXT/pa
(15)
II, = AI,,,( kr) exp[ at + i( kz +
= 0.1
x
grad II,,
+ Q:)]
H,,/H,
= 0.5
44%
where cc, is the initial amplitude at time t = 0, u is the temporal amplification, k is (real) the longitudinal wavenumber and m is (integer) the azimuthal wavenumber. By inserting the expansions (12)-(14) into the basic equations (l)-(9), the relevant perturbation equations are identified. Based on the linear perturbation technique and the (cp, z)-dependence, every relevant perturbed quantity can be written as a function of r times exp[ot + i(kz + mcp)]. Consequently by following the same procedure as in refs. [2,10], the linearized system of the basic equations are solved and the non-singular solutions are given by ur = [-o/(02
H, /H,
(18)
where CL’is the coefficient of the magnetic permeability of the liquid of the outer jet.
304
A.E. Radwan / Magnetohydrodynamic stability
Table 5 p’/p = 2.0 Ho/H,=0
H,/H,
x
a/ J----x T/pa
1.00 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0 0.1735 0.3272 0.4564 0.5798 0.7019 0.8247 0.9489 1.0750 1.2032 1.3337 1.4665 1.6017 1.7393 1.8792 2.0214 2.1660 2.3129 2.4622 2.6136 2.7674 2.9234 3.2420 3.5693 3.9051 4.2494 4.6018 4.9623 5.3308 5.7070 6.0909 6.3838
= 0.1
H,/H,
= 0.3
a/ 7xT/pa 0.986 1.00 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0194 0.0898 0.1973 0.3428 0.4696 0.5918 0.7133 0.8356 0.9596 1.0855 1.2137 1.3442 1.4770 1.6122 1.7498 1.8897 2.0321 2.1768 2.2328 2.4731 2.6247 2.7785 3.2535 3.5810 3.9171 4.2615 4.6142 4.9749 5.3436 5.7201 6.1042 6.3973
0.861 0.90 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
= 0.5
H&H,
o/JT/;;;Is
x
a/ FxT/pa
0.0109 0.1218 0.2029 0.3311 0.4485 0.5640 0.6803 0.7983 0.9186 1.0413 1.1666 1.2945 1.4250 1.5581 1.6937 1.8318 1.9724 2.1154 2.2608 2.5588 2.8660 3.0230 3.3437 3.6731 4.0111 4.3575 4.7121 5.0748 5.4454 5.8238 6.2098 6.5043
0.619 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.35 2.55 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0037 0.0837 0.1985 0.2984 0.3984 0.5010 0.6070 0.7164 0.8294 0.9457 1.0653 1.1881 1.3139 1.4427 1.5743 1.7088 1.8459 2.1280 2.4202 2.7221 3.0333 3.1924 3.5172 3.8508 4.1930 4.5435 4.9021 5.2689 5.6435 6.0258 6.4157 6.7131
4.Boundary conditions The solution given by eqs. (15)-(22) must satisfy the following boundary conditions at the interfaces r = a and r = b. (9 The continuity of the normal components of the velocities of the fluids across the interface, eq. (13), at r = a and simultaneously they must be compatible with the velocity of the deformed interface (13) at r = a. The normal component of the velocity (n u’) ’ of the (outer) liquid jet must be compatible with the velocity of the deformed interface, eq. (14), at r = b. l
Ho/H,
= 0.7 a/ F T/pa
0.092 0.10 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.15 1.35 1.55 1.75 1.95 2.15 2.35 2.55 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0004 0.0040 0.0174 0.0531 0.1017 0.1612 0.2302 0.3073 0.3917 0.4824 0.5789 0.7870 1.0126 1.2536 1.5081 1.7749 2.0532 2.3421 2.6413 2.9501 3.2683 3.4308 3.7625 4.1029 4.4518 4.8089 5.1741 5.5473 5.9282 6.3168 6.7128 7.0147
(iii) The normal component of the magnetic fields H and H ’ must be continuous across the interface, eq. (13), at r = a. (iv) The normal component of the magnetic fields H’ and HVaC must be continuous across the interface, eq. (14), at r = b. (v) The normal component of the total stress tensor must be continuous across the (gas-liquid) interface, eq. (13), at r = a. The foregoing boundary conditions, (i)-(iv), are satisfied such that A = --c()+2+ B = [a’yK;((y)
sz~)/xI;(x), - b’&(x)]
(2% G(x,
Y),
(26)
A. E. Radwan / Magnetohydrodynamic stability
305
Table6 p’/p = 2.0 H,/H,
x
0.0 0.05
0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
= 0.72
Ho/H,
= 0.8
H,/H,=I.O
H,/H,
o/w
a/ 7 T/pa
a/ 7 T/pa
a/ 7 T/pa
0.0
0.0
0.0
0.0073 0.0304 0.0672 0.1165 0.1766 0.2461 0.3237 0.4085 0.4997 0.5966 0.6987 0.9169 1.1513 1.4001 1.6618 1.9354 2.2199 2.5150 2.8199 3.1343 3.4580 3.7905 4.1317 4.4814 4.8393 5.2054 5.5793 5.9610 6.3503 6.7471 7.0495
0.0188 0.0600 0.1084 0.1655 0.2312 0.3050 0.3861 0.4740 0.5679 0.6673 0.7718 0.9945 1.2332 1.4862 1.7521 2.2097 2.3183 2.6173 2.9261 3.2444 3.5718 3.9081 4.2529 4.6052 4.9676 5.3371 5.7144 6.0994 6.4920 6.8920 7.1968
0.0354 0.1073 0.1822 0.2614 0.3456 0.4349 0.5294 0.6290 0.7335 0.8425 0.9560 1.1950 1.4489 1.7161 1.9955 2.2863 2.5877 2.8991 3.2201 3.5503 3.8893 4.2370 4.5930 4.9571 5.3292 5.7092 6.0968 6.4919 6.8944 7.3041 7.6161
(27)
= 2.0
0.0
Ho/H,
a/ 7 T/pa 0.0
0.0934 0.2784 0.4604 0.6400 0.8180 0.9950 1.1720 1.3493 1.5276 1.7072 1.8882 2.2555 2.6303 3.0131 3.4037 3.8022 4.2083 4.6218 5.0426 5.4703 5.9050 6.3463 6.7942 7.2486 7.7092 8.1762 8.6492 9.1284 9.6135 10.1045 10.4766
0.2470 0.7344 1.2098 1.6730 2.1253 2.5685 3.0044 3.4346 3.8605 4.2833 4.7038 5.5410 6.3762 7.2122 8.0504 8.8918 9.7369 10.5857 11.4383 12.2950 13.1553 14.0194 14.8871 15.7583 16.6329 17.5110 18.3924 19.2771 20.1650 21.0562 21.7266
normal stress tensor (condition (vi) reads)
3%
II, + q,a exp[ at + i( kz + m(p)] ar
(28) (2% t p’
qy
=
chw;(x)
- MawY)~
(30)
where x( = ku) and y( = kb) are the longitudinal dimensionless wavenumbers. The continuity of the
= 5.0
=p,,.
II,l +
~,,a
1 an;,
exp[ at + i( kz + m(p)] ar
) (31)
On utilizing eqs. (lo), (17), (18), (21), (23)-(27), (29) and (30) for the condition (31), the following
A. E. Radwan / Magnetohydro&namic
306
stability
Table7 p’/p = 5.0 H,,/H,
= 0
Ho/H,
H,/H,
= 0.1
= 0.3
Q/ F*T/pa
H, /H, = 0.5
a/ 7xT/pa
Ho/H,
x
o/ 7.xT/pa
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.629 -0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.034965 0.068245 0.099074 0.127102 0.151187 0.174291 0.193415 0.209567 0.222739 0.232888 0.239928 0.243712 0.244329 0.244011 0.240480 0.232598 0.219547 0.199951 0.171169 0.126557 0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.621 -0.65 0.670 0.75 0.80 0.85 0.90 0.95 0.985
0.034611 0.067542 0.098028 0.125715 0.150458 0.172213 0.190976 0.206749 0.219516 0.229225 0.235777 0.239008 0.239318 0.238665 0.234366 0.225527 0.211214 0.189811 0.158018 0.106289 0.009968
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.551 -0.60 0.65 0.70 0.75 0.80 0.85 0.855 0.865 0.866
0.031639 0.061636 0.089222 0.114016 0.135834 0.154585 0.170211 0.182647 0.191793 0.197487 0.199480 0.199483 0.197380 0.190576 0.178050 0.157943 0.126086 0.066402 0.055862 0.021571 0.013930
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.402 0.45 0.50 0.55 0.60 0.601 0.603 0.605 0.607 0.611 0.615 0.617 0.618
0.024640 0.047678 0.068284 0.085966 0.100384 0.111243 0.118203 0.120787 0.120790 0.118232 0.109160 0.090527 0.051258 0.049960 0.47230 0.044296 0.041113 0.033716 0.023876 0.016785 0.011676
0.050 0.051 0.053 0.055 0.057 0.059 0.061 0.064 -0.065 0.067 0.069 0.071 0.073 0.075 0.077 0.079 0.081 0.083 0.087 0.089 0.091
0.004046 0.004088 0.004164 0.004229 0.004283 0.004325 0.004353 0.004369 0.004367 0.004349 0.004314 0.004258 0.004180 0.004078 0.003946 0.003781 0.003577 0.003324 0.002603 0.002055 0.001172
eigenvalue relation is derived a2 = (T/pa3)(1
- m2 - ~‘>W;:,+,“,G,(x~
-(~~H,Z/pa~)x~yl,(x)L,“,G,(x,
Y) b2 2 -YL,”
-(y’H,Z/p’u2)(p’/p)x21~(x) [
xG,&
Y>,
Y>
1
(32)
with G,_‘k
Y)
5. General discussions
Eq. (32) is the stability criterion of a liquid jet having a gas jet (of different density) acting as a
o/ 7.xT/pa
= 0.7 o/ J--T/pa
mantle upon the inertia, capillarity and electromagnetic forces. It is a quadratic equation in the temporal amplification (I. The eigenvalue relation (32) relates (I to the fundamental quantities ~;~:~;$1’2 as well as ( ~‘u~/~_L’H,)~‘~ and as a unit of time, the two kinds of modified Bessel functions and their derivatives, different combinations (L,mY,Lr and G,(x, y)) of the Bessel functions, the densities and radii ratio (p’/p) and (b/u) and with the wavenumbers x, y and m. By means of the dispersion relation (32) the characteristics and the (in-)stability states can be identified. The marginal stability can be obtained by setting u = 0. It is worthwhile to mention here that the relation (32) is somewhat complicated, to some extent, because we are dealing with double interfaces and consequently two kinds of perturbation. That kind of disturbance is totally different from those which are considered in refs. [1,2,8-lo], where only one single interface is considered.
A.E. Radwan / Magnetohydrodynamic
This relation, for small axisymmetric disturbances m = 0, coincides with our recent result [12], if we neglect the basic flow velocity there. For the stability discussions of the relation (35) we may refer to ref. [12] or/and [9] in neglecting the streaming effect or/and viscosity influence. If p’ = 0, Ho = 0 and at the same time y tends to infinity; the model is a full liquid jet (submerged in a vacuum) subject to inertia and capillary forces. The corresponding dispersion relation can be determined from (32) as
6. Stability discussions Since the problem under consideration is somewhat more general, several eigenvalues of different hydrodynamic and hydromagnetic problems can be recovered as limiting cases from the general eigenvalue relation (32). If p = 0, H,, = 0 and simultaneously y tends to infinity; the model is a hollow cylinder endowed with surface tension and is acted upon by the liquid inertia force. The dispersion relation of this case can be obtained from (32) as cJ2= (T/pV)[
307
stability
a*=
(T/pa3)[xI;((x)/lm(x)](l
-rn*-X2).
- xKx-4/Kn(41
(36)
X(1-d-x*).
(35)
This is the classical
Ho/H,= 0.3
H,, /H,
dispersion
relation of a cir-
Table8 p'/p= 5.0 Ho/H,=0
Ho/H,= 0.1
x
w/JT/pn3
x
m'/7x T/pa
1.00
0.0
1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.1374 0.2564 0.3545 0.4467 0.5371 0.6273 0.7179 0.8095 0.9022 0.9964 1.0919 1.1890 1.2876 1.3878 1.4896 1.5930 1.6979 1.8044 1.9125 2.0221 2.1334 2.3604 2.5935 2.8326 3.0776 3.3283 3.5868 3.8468 4.1143 4.3872 4.5954
0.986 1.00 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0155 0.0715 0.1562 0.2686 0.3647 0.4560 0.5458 0.6356 0.7260 0.8174 0.9101 1.0042 1.0997 1.1968 1.2954 1.3957 1.4975 1.6009 1.7058 1.8124 2.0303 2.1416 2.3687 2.6020 2.8412 3.0864 3.3373 3.5939 3.8561 4.1238 4.3968 4.6051
a/
0.867 0.90 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
7x T/pa
0.0088 0.0982 0.1626 0.2622 0.3515 0.4380 0.5242 0.6109 0.6987 0.7878 0.8785 0.9707 1.0646 1.1601 1.2573 1.3561 1.4567 1.5588 1.6627 1.8752 2.0942 2.2060 2.4344 2.6689 2.9095 3.1559 3.4081 3.6661 3.9295 4.1985 4.4729 4.6821
= 0.5 a/
0.619 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.35 2.55 2.75 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
H,,/H,= 0.7
47xT/pa
0.0031 0.0704 0.1639 0.2425 0.3193 0.3967 0.4756 0.5564 0.6390 0.7237 0.8103 0.8989 0.9894 1.0818 1.1761 1.2723 1.3703 1.5716 1.7799 1.9949 2.2165 2.3297 2.5608 2.7981 3.0414 3.2906 3.5356 3.8063 4.0725 4.3442 4.6212 4.8325
u/
0.091 0.10 0.15 0.25 0.35 0.45 0.55 0.75 0.75 0.85 0.95 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
7 T/pa
0.0004 0.0039 0.0168 0.0498 0.0923 0.1422 0.1979 0.3234 0.3234 0.3920 0.4639 0.5389 0.6972 0.8657 1.0435 1.2298 1.4241 1.6260 1.8352 2.0514 2.2743 2.5037 2.7394 2.9812 3.2291 3.4828 3.7423 4.0074 4.2779 4.5539 4.8352 5.0496
A.E.Radwan / Magnetohydrodynamicstability
308 Table9 p'/p= 5.0 x
0.0 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
H,/H, = 0.12
Ho/H,= 0.8
H&H,
o/ F T/pa
w/ 7 T/pa 0.0 0.0187 0.0581 0.1016 0.1503 0.2039 0.2623 0.3249 0.3914 0.4614 0.5348 0.6111 0.7723 0.9437 1.1244 1.3138 1.5115 1.7164 1.9287 2.1480 2.3740 2.6065 2.8453 3.0902 3.3411 3.5978 3.8601 4.1281 4.4015 4.6802 4.9642 5.1806
a/ F T/pa 0.0 0.0325 0.1039 0.1708 0.2373 0.3048 0.3740 0.4455 0.5194 0.5960 0.6752 0.7569 0.9281 1.1087 1.2983 1.4964 1.7023 1.9158 2.1364 2.3638 2.5979 2.8383 3.0848 3.3373 3.5957 3.8597 4.1293 4.4043 4.6847 4.9703 5.2611 5.4825
0.0 0.0073 0.0294 0.0630 0.1057 0.1558 0.2116 0.2724 0.3373 0.4060 0.4781 0.5533 0.6313 0.8810 0.1059 0.1246 0.1441 0.1643 0.1853 0.2070 0.2293 0.2523 0.2759 0.3002 0.3250 0.3504 0.3764 0.4030 0.4301 0.4578 0.4859 0.5074
cular liquid column. Regarding its (in-)&ability we may refer to Chandrasekhar (ref. [2], p. 537). If p # 0, p’ # 0, ZZ, = 0 and y tends to infinity, then the model is a gas-core liquid jet. The driving forces in such a case, are capillarity and the inertia of the nonmiscible fluids. The corresponding dispersion relation is established from (32) in the form
=l.O
Ho/H,= 2.0
Ho/H,= 5.0
a/ F T/pa 0.0 0.0928 0.2695 0.4314 0.5810 0.7214 0.8557 0.9861 1.1143 1.2413 1.3681 1.4951 1.7516 2.0129 2.2796 2.5524 2.8311 3.1156 3.4059 3.7017 4.0029 4.3092 4.6205 4.9368 5.2578 5.5834 5.9136 6.2482 6.5873 6.9306 7.2782 7.5416
w/ F T/pa 0.0 0.2455 0.7109 1.1336 1.5186 1.8745 2.2089 2.5280 2.8363 3.1370 3.4325 3.7246 4.3032 4.8794 5.4566 6.0368 6.6207 7.2087 7.8008 8.3968 8.9967 9.6002 10.2071 10.8172 11.4303 12.0464 12.6652 13.2868 13.9109 14.5375 15.1666 15.6400
By resort to the recurrence relations [13] of the modified Bessel functions and the each non-zero real value of x that Z,,,(x) is positive and monotonic increasing while K,,,(x) is monotonic decreasing but never negative. One can show that Z;(x) is positive definite while K:(x) is negative for each non-zero real value of x. By the aid of these arguments we can prove, for each non-zero real value of x, that
Using this inequality
for the criterion (37) we
A. E. Radwan
/ Magnetohydrodynamic
stability
309
Table10 p’/p = 10.0 H,,/H,
=0
H,,/H,
= 0.1
H,, /H,
= 0.3
Ho/H,
= 0.5
H,/H,
= 0.7
x
a/FxT/pa
o/m
x
o/m
x
o/Jvz
x
o/ lf-- T/pa
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.550 0.598 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.034635 0.066350 0.094253 0.118298 0.138738 0.155899 0.170071 0.181482 0.190288 0.196575 0.200361 0.201592 0.201591 0.200180 0.195740 0.188038 0.176403 0.159772 0.136091 0.100167 0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.598 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.985
0.034284 0.065667 0.093259 0.117007 0.137162 0.154040 0.167926 0.179041 0.187534 0.193483 0.196894 0.197752 0.197700 0.195745 0.190764 0.182321 0.169708 0.151669 0.125635 0.084125 0.007866
0.05 0.10 0.15 0.20 0.25 0.30 0.35' 0.40 0.45 0.50 0.523 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.855 0.865 0.866
0.031340 0.059925 0.084881 0.106118 0.123830 0.138272 0.149668 0.158169 0.163850 0.166694 0.167022 0.166583 0.163267 0.156304 0.144925 0.127685 0.101308 0.053059 0.044614 0.017210 0.011112
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.384 0.40 0.45 0.50 0.60 0.601 0.603 0.605 0.607 0.609 0.611 0.613 0.615 0.618
0.024408 0.046354 0.064962 0.080011 0.095134 0.099504 0.103936 0.104819 0.104599 0.101007 0.092139 0.042399 0.041317 0.039046 0.036607 0.033965 0.031067 0.027834 0.024132 0.019697 0.009628
0.050 0.051 0.053 0.055 0.059 0.063 0.064 0.065 0.067 0.069 0.071 0.073 0.075 0.077 0.079 0.081 0.083 0.085 0.087 0.089 0.091
0.004008 0.004048 0.004121 0.004183 0.004271 0.004308 0.004309 0.004304 0.004284 0.004246 0.004188 0.004108 0.004004 0.003872 0.003708 0.003505 0.003254 0.002943 0.002545 0.002007 0.001143
conclude, through its analytical discussions, that the model of a gas-core liquid jet is unstable only in the axisymmetric disturbances m = 0 for certain wavelengths which are longer than the circumference of the gas jet. If p = 0, p’ # 0, H,, f 0 and y tends to infinity, then the model is a hollow jet which acts upon the capillarity liquid inertia and electromagnetic forces. The eigenvalue relation (32) in such a case degenerates to a2 = (T/p’a3)[
-xKk(x)/K,(x)](l
-m*
X(~‘H,2/p’a2)[-x/(I~(x)K,(x))l
- x2)
9 (39)
hollow jet in the domain m = 0, 0 G x -C 1 and increases the stability of the surface tension in the domains (m = 0, x > l), and (m a 1 for all x values). Moreover, one can show that the capillary instability of the hollow cylinder is completely suppressed if EC, > (fiH,/2) (where H,( = (7’/ pR,,)“*) has a unit of magnetic field) and then stability sets in. Now returning to the general case of compound MHD liquid jets where the stability criterion is given by eq. (32) with eqs. (33) and (34). If the surface tension influence is paramount over that of the magnetic field, the relation (32) degenerates to
where use has been made of the Wronskian w,(Ll(X)~
u*= (T/pa’)
L(x))
= I,(x)KA(x)
-IA(x)&(x)
= -x-l.
(40)
As H,, = 0, eq. (39) reduces to (35) which is the capillary stability criterion of a hollow cylinder. The magnetic field has a stabilizing influence and therefore reduces the capillary instability of the
VqJA (x >
X
YLc4qJ + (d/P) X(1-m*-x2).
[
5 -.Yq
1
4x4 (41)
310
A.E. Radwan / Magnetohydrodynamic
stability
Table11 p’/p = 10.0 Ho/H,=0
Ho/H,
x
w/ 7xT/pa
1.00
0.0
1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.15 2.85 2.95 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.1079 0.2000 0.2750 0.3449 0.4131 0.4807 0.5486 0.6169 0.6861 0.7561 0.8272 0.8993 0.9725 1.0468 1.1223 1.1989 1.2167 1.3557 1.4358 1.5170 1.5994 1.7676 1.9402 2.1173 2.2987 2.4844 2.6142 2.8682 3.0662 3.2683 2.4223
= 0.1
H,/H,
= 0.3
a/ 7xT/pa
0.986 1.00 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0122 0.0564 0.1227 0.2096 0.2829 0.3521 0.4198 0.4871 0.5547 0.6230 0.6920 0.7620 0.8331 0.9052 0.9784 1.0527 1.1282 1.2049 1.2827 1.3617 1.4418 1.6056 1.7738 1.9466 2.1238 2.3053 2.4911 2.6811 2.8751 3.0733 3.2754 3.4296
= 0.5
w/ 7xT/pa
0.867 0.90 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0070 0.0781 0.1287 0.2059 0.2742 0.3398 0.4047 0.4698 0.5355 0.6020 0.6695 0.7381 0.8079 0.8788 0.9509 1.0242 1.0987 1.1745 1.2514 1.3296 1.4894 1.6539 1.8230 1.9967 2.1748 2.3572 2.5440 2.7349 2.9299 3.1290 3.3321 2.4869
Utilizing the recurrence relations [13] of the modified Bessel functions together with their properties, we can prove for every non-zero real value of x, Y and (p’/p) that the following combinations of Bessel functions satisfy the inequalities. L,“>O,
H,/H,
Lx”y< 0,
{ XYL,~,M~NL(X~ Y)} ‘0 and never change sign for all axisymmetric (m = 0) and non-axisymmetric (m f 0) modes provided
H, /H,
a/ J-‘x T/pa
0.619 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 2.05 2.15 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0025 0.0577 0.1325 0.1938 0.2527 0.3115 0.3710 0.4316 0.4934 0.5565 0.6210 0.6868 0.7540 0.8226 0.8925 0.9638 1.0364 1.1103 1.2619 1.4186 1.5802 1.7466 1.9177 2.0933 2.2134 2.4578 2.6465 2.8395 3.0365 3.2315 3.4426 3.5989
= 0.1 a/ 7 T/pa
0.092 0.10 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 4.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
0.0003 0.0038 0.0160 0.0454 0.0812 0.1215 0.1653 0.2121 0.2615 0.3132 0.3671 0.4231 0.5409 0.6658 0.7973 0.9351 1.0788 1.2281 1.3827 1.5426 1.7074 2.5014 2.0514 2.2303 2.4137 2.6014 2.1934 2.9895 3.1897 3.3939 3.6020 3.7606
that YLY”’ (b/a
)’
(43)
is fulfilled. From the viewpoint of the restriction (43) and the inequalities (42), the stability discussions of the relation (41) predict the following results. The concentric liquid jets are only unstable in the axisymmetric mode m = 0 if the perturbed wavelength X is longer than the circumference of the gas-mantle jet 2~~2. While the model is stable in the axisymmetric disturbance m = 0 as long as
A.E. Radwan / Magnetohydrodynamic
X G (2~) and in all non-axisymmetric modes m 2 1 for all (short and long) wavelengths. In order to examine the magnetic field influence and its effect on the capillary (in-)stability it is found convenient to write the general dispersion relation (32) in the dimensionless form
Wn(x~ Y>,
(40
311
stability
where and
H, = (WV$‘*
H,’ = ( T/p’p’ )1’2.
The last two terms (of (44)) which include Ho represent the effects of the magnetic fields interior and exterior to the gas jet. For one single interface it is proved (cf., refs. [2,10]) that such a homogeneous magnetic field always has a stabilizing influence, whatever the kind of models considered. From the analytical discussions of (44) it is also found that the magnetic fields inside and outside the compound jet have stabilizing effects for all values of (p//p), x (f 0) and y ( f 0) provided that condition (43) is satisfied. The stabilized influence of the magnetic field increases with in-
Table 12 p’/p = 10.0 x
0.0 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.65 3.85 4.05 4.25 4.45 4.65 4.85 5.00
Ho/H,
= 0.12
H,, /H,
= 0.8
H,,/H,
=l.O
Ho/H,
= 2.0
H,,/H,
= 5.0
w/ 7 T/pa
a/ 7 T/pa
w/ 7 T/pa
w/ 7 T/pa
w/ 7 T/pa
0.0 0.0072 0.0280 0.0574 0.0930 0.1331 0.1767 0.2234 0.2727 0.3244 0.3784 0.4345 0.5524 0.6776 0.8094 0.9475 1.0916 1.2413 1.3963 1.5566 1.7218 1.8919 2.0667 2.2460 2.4298 2.6179 2.8102 3.0067 3.2073 3.4119 3.6204 3.7792
0.0 0.0186 0.0552 0.0926 0.1321 0.1742 0.2190 0.2665 0.3164 0.3687 0.4232 0.4799 0.5991 0.7258 0.8592 0.9990 1.1448 1.2963 1.4531 1.6152 1.7823 1.9542 2.1207 2.3119 2.4974 2.6873 2.8813 3.0795 3.2818 3.4880 3.6981 3.8582
0.0 0.0348 0.0988 0.1557 0.2087 0.2604 0.3123 0.3653 0.4199 0.4762 0.5344 0.5944 0.7200 0.8527 0.9921 1.1378 1.2895 1.4469 1.6096 1.7775 1.9503 2.1279 2.3101 2.4967 2.6877 2.8829 3.0822 3.2856 3.4930 3.7042 3.9192 4.0830
0.0 0.0919 0.2563 0.3933 0.5108 0.6163 0.7146 0.8088 0.9008 0.9919 1.0828 1.1741 1.3589 1.5480 1.7419 1.9408 2.1446 2.3531 2.6933 2.9301 3.0051 3.2307 3.4601 3.6933 3.9301 4.1704 4.4141 4.6612 4.9116 6.1651 5.4219 5.6165
0.0 0.2432 0.6763 1.0335 1.3353 1.6014 1.8446 2.0734 2.2929 2.5066 2.7167 2.9248 3.3384 3.7526 4.1696 4.5904 5.0153 5.4444 5.8774 6.3141 6.7542 7.1974 7.6437 8.0926 8.5440 8.9978 9.4538 9.9119 10.3721 10.8343 11.2984 11.6476
A.E. Radwan
312
/ Magnetohydrodynamic
creasing magnetic field intensity. Therefore as is expected under the restriction (43) the capillary instability will be omitted and completely suppressed above a certain value of ZZ,,relative to H, and H,’ and then the stability sets in. This means that, in the unstable domain 0 d x < 1 for m = 0, either the growth rate u/( T/p a 3)1’2 or the critical wavenumber x, or both are decreasing with increasing intensity of the magnetic field until it completely shrinks and the model will be stable for all problem parameters in all modes of perturbations m z 0 under the restriction (43). Finally we intend to investigate the MHD stability of a gas core liquid jet which is a special case arising from our present general case, as the cylindrical surface of the outer jet is cited at infinity. In such a case the stability criterion is given from (32) in the dimensionless form a2
= { (1 - m2 - x2)
(VW3) +(H~/H,)2[Z~(x)K~(x)l
-‘} (4%
w?(x), where Z%(x) = xZ~(x)K~(x)[Z,(x)K~,(x) -
Z:(x) K;(x)
= :(Z,&) = -f@,-,(x)
+ Z?n+*(x)), + L+,(x)),
(46) relations
>O
111Lord Rayleigh,
(cf., PI (47) (48)
and to each non-zero real value of x that Zm(x), Z:(x) and K,(x) are always positive while KA(x,) is negative one can show that El(x)
axisymmetric modes m > 0 for all x, (p’/p) and (Ho/H,) values. If one considers the combined effect of the capilarity and electromagnetic forces simultaneously, the capillary instability is decreasing by increasing the magnetic field intensity. Moreover, it is found that above the critical value ,/m of Ho, the capillary instability is completely suppressed and stability sets in. Noting that in identifying that critical value of H,, the inequality Z,(x)K,(x) G : for x > 0. These analytical results are confirmed numerically for different values of (p’/p) = 1.0, 2.0, 5.0 and 10.0, then for each value of (p’/p) the capillary instability of the gas-core liquid jet is examined by considering several values of (Ho/H,) = 0, 0.1, 0.3 . . . 5.0. It is found that the critical values of (Ho/H,) above which the capillary instability is completely suppressed is H,,/H, = 0.702. . . for all values of (p//p), see tables 1-12. It is also remarkable that however much larger the density of the outer liquid is than that of the gas jet, there are always instability domains. This shows that the ratio (p’/p) plays a secondary role in the stability of a compound liquid jet.
References
b’/d~~~x)~m(x)l -l.
By an appeal to the recurrence ref. [13]):
stability
(49)
for all m and (p//p) values. In view of the inequality (49) the gas-core liquid jet at Ho = 0 is capillary unstable for the axisymmetric mode m = 0 if the wavenumber x is less than unity, while it is stable for m = 0 if x > 1 and for m 2 1 for all x values. In neglecting the surface tension influence, eq. (45) (by the aid of the relations (47) and (48)) indicates that the magnetic field has a strong stabilizing effect for all axisymmetric and non-
[31 [41 PI [61 171 PI [91 PO1
1111 WI [I31
The Theory of Sound (Dover, New York, 1945); Scientific papers Vols. I, II and III (Cambridge, 1902). S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981). C.H. Hertz and B. Hermanrud, J. Fluid Mech. 131 (1983) 271. V.A. Surin, V.N. Evchenko and V.M. Rubin, J. Eng. Phys. (USA) 45 (1983) 1091. P.D. Esser and S.I. Abdel-Khahk, J. Fluids Eng. (USA) 106 (1984) 45. L.Y. Cheng, Phys. Fluids 28 (1985) 2614. J.M. Kendall, Phys. Fluids 29 (1986) 2086. A.S. Monin, Sov. Phys. USP (USA) 29 (1986) 843; Phys. Abst., Sci. Abst., series A (1987) 1104. A.E. Radwan and S.S. Elazab, Simon Stevin 61 (1987) 293. D.K. Cahebaut and A.E. Radwan, Europ. Phys. Sot., 10D (1986) 11; A.E. Radwan and CaIlebaut, Proc. Int. Conf. on Computing in Plasma Physics, Garmisch, Fed. Rep. of Germany (1986) p. 6. A.E. Radwan and D.K. CaIlebaut, Proc. Belg. Phys. Sot. P.P. (1985) 21. A.E. Radwan, J. Phys. Sot. Japan 58 (1989) 1225. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).