ELSEZVIER

1 March 1999

Physics Letters A 252 ( 1999) 222-232

PHYSICS LETTERS A

Linear and nonlinear magnetohydrodynamic waves in twisted magnetic flux tubes

Y,D. Zhugzhda a*b*l, V.M. ~~~akov c a Kiepenheuer-institut fiir Sonnenphysik, Schiineckstr. 6, O-79104 Freiburg, Germuny

h IZMIRAN, Troitsk, Moscow Region 142092, Russia c School of Mathematical and Computational Sciences, Vniversiry of St Andrew.% St Andrews, Fife KY16 9SS, Scotland, UK

Received 25 September 1997; revised manuscript received 6 August 1998; accepted for publication 18 December 1998 Communic~~ by M. Porkolab

Abstract

Dynamics of axisymmetric Alfven and slow magnetoacoustic waves in weakly twisted magnetic flux tubes is considered. Linear dispersion relations for the waves are derived and analyzed in the presence of the twisting. The weakly nonlinear dynamics is shown to be governed by the Korteweg-de Vries equation. Nonlinear slow body waves appear as a narrowing of tube in a low /3 plasma and widening of tube, when ,i3 > 1. Nonlinear AlfvCn torsional waves appear as a widening (p > 1) and narrowing (/3 < 1) of tube, accompanied by the local increase of the tube twisting. @ 1999 Published by Elsevier Science B.V.

PACS: 52.35.-g; 52.35.M~; 52.35.Sb; 52.35.T~ Keywords: Twisted magnetic flux tubes; Nonlinear waves; KdV equation

1. Int~ducti~n

Magnetized plasmas in astrophysicai, geophysical and laboratory conditions are not uniform, but usually occur as filament structures stretched along the magnetic field, with properties of the plasma being inhomogeneous across the field. Such structures are commonly called magneticJlux tubes [ 11. The magnetohydrodynamic waves supported by the flux tubes are different from the waves in homogeneous media [ 21. An analytical treatment of these tube waves is di~cult and approximate approaches are of crucial importance. In particular, the thin- flux-tube approximation [3-51 is extensively used, allowing a reduction of the 3D probIem to ID. The linear and nonlinear waves in current-free flux tubes were explored by many authors [2,6-13,151. The derivation of the thin-flux-tube approximation for twisted flux tubes [5] offers new possibilities for the treatment of linear and nonlinear waves.

’ E-mail: yu~f~kis.uni-~iburg.de.

0375-9601/99/$ - see front matter @ 1999 Published by Elsevier Science B.V. All rights reserved. PII SO375-9601(99)00014-6

ED. zhrrgzhda, KM. ~a~ria~~/~~ys~~s Letters A 252 (1999) 222-232 223

The aim of this work is to investigate linear and weakly non~ine~ dyn~ics of axisymme~ic magnetohy- drodynamic waves in weakly twisted magnetic flux tubes. Only the case of weakly twisted flux tubes, which are stable with respect to the kink instability, is considered. Tbe paper is organised in the following way. The derivation of a fin-flux-Tut approximation is outlined in the second section. Using the approximation, linear dynamics of the waves in a twisted flux tube is considered in the third section. The linear results are applied to a derivation of evolutionary equations for weakly nonlinear slow body and Alfven waves (Section 4). Then the properties of weakly nonlinear slow (Section 5) and Alfven (Section 6) waves in the twisted flux tubes are discussed.

2. Governing equations

A symme~ic magnetic flux tube with a straight axis of symmet~ is considered. ~ylindricai coordinates are used (I”, 4, z), and only axisymmetric motions (a/~?# E 0) are considered. The thin flux tube approximation is based on the assumption that radial variations of all physical quantities are described adequately by their low order radial derivatives, evaluated on the axis of the flux tube. This means that variables are expanded in a Taylor series. The condition of the tube symmetry imposes certain obvious restrictions on physical variables. The r- and ~-com~nents of vector quantities have to vanish on the axis (r = 0). The magnetohydrodynamic (MHD) equations for an ideally conducting, inviscid, compressible fluid are taken as

( C3V p -8;t(v.V).v >

=-vp+4* I-(VxB)xB,

~+v.pv=o, (2)

dB -=Vx(vxB), dt (3)

(4)

V*B=O, (5)

where v - (u,, vb, u,) is the flow velocity, B = (B,, B+ B2) is the magnetic field, p is the fluid density, p is the pressure and T is the temperature. Additional terms, like radiative exchange, viscous or Ohmic dissipation can also be consistently included in the consideration, but we restrict ourseives to adiabatic processes only. To simplify the problem, the thin-flux-tube approximation derived by Zhugzhda IS] is used. To obtain the thin~flux-tube approximation, ali physical quantities are expanded in a Taylor series in the tube radius r with coefficients depending on .z and t,

p=J5fp&-k..*, p=~-kp&-... , T=rii+T2r2+.., ,

ur = wr-t-u,3r3+... , “4 = a?- -I- u&r3 -b 1. r , u, = u + ff,*r* + f . . ,

BP = i&r + z&r3 -I- . . . , 234 = Jr + Bg;g3 -I- . . . t & = B, 4” l&p2 -I-. . L ,

(V x VI* =2J2+4u$Q-... , (VxB),=2J+413~y-2+... , (6)

where coefficients are function of z and t, f2 is the zeroth order value of the angular velocity, 2J is the zeroth order value of the current density. Substituting the expansions (6) into the set of equations ( I>-( 5) and collecting terms of the same power, we replace each of the equations by the infinite set of equations for coefficients of expansion (6), because the equations have to be satisfied for all values of the radial coordinate

224 LD. Zhugzha!a. VM. Nakariakov/Physics Letters A 252 (1999) 222-232

r. Thus, the treatment of Eqs. (l)-(5) can replaced by the infinite set of linked equations for coefficients of (6). The thin-flux-tube approximations are obtained by truncating of the infinite set of equations. Zhugzhda [ 51 made a truncation for the case, when only the first terms in the expansions (6) are taken into account.

Thus, the set of equations ( 1) -( 5) is reduced to the set of approximate equations for a thin tube [ 51, which read

C?A --+u: -2Av=O,

aB dt +uE+2Bv=O,

= Pext 9

(7)

(8)

(9)

(11)

(13)

(14)

where the tilde is dropped, A = rR2 is the tube cross section, and R is the tube radius. Eq. (14) is a condition of the total pressure balance at the tube boundary. The external pressure, pext, is assumed to be constant. The set of equations is valid for thin flux tube of a finite diameter.

In the limit of the infinitely thin flux tube, the variables B+ and r+ tend to zero and Eqs. (9) and ( 10) for them are dropped out of the set, as well as the terms in the boundary conditions ( 14), which are proportional to the cross section of the tube. If Eqs. ( 12) and ( 13) are written as a flux conservation condition BA = const, the set of equations (7)-( 14) reduces to the well-known thin-flux-tube approximation of Roberts and Webb [ 61. The treatment of linear and nonlinear waves in the twisted flux tubes is not possible in the Roberts and Webb approximation, because it is valid only for untwisted tubes.

3. Linear waves in force-free flux tubes

The thin-flux-tube approximation derived by Zhugzhda [ 51 allows us, for the first time, to obtain a dispersion equation for waves in a twisted magnetic flux tube. Linearizing Eqs. (7)-( 14) and introducing harmonica1 dependence of perturbations on time and coordinate, B N exp(iot - ikz), we obtain the dispersion equation

(W2 Ao Co2 - c;‘kq (to2 - Cik2) - z - C;k=)=(W= - Cs’k2)

c; + c; J;AoC; (o= - C;k2) (o= - C;k=) + 2u2k2C; _ o -- 2rB;: (Cj +c$

- . (15)

LD. Zhugdda. KM. ~~~~/Ph~si~s Letters A 252 [I9991 222-232 225

The first term in the dispersion equation corresponds to wave propagation in the infinitely thin untwisted tube (JO = 0, A0 = 0). The second term shows the influence of the finite cross section. The third term shows the effect of the twisting. It is clear that dispersion appears only due to the second term. Introducing the fast and slow speeds, Ck:, dispersion equation ( 15) can be rewritten as

(C:+C,2-Kc~)(w*-C~k*)(J-C~k*)+ co= - C;k2)*(co2 - Cs’k*) = 0,

where

(16)

(17)

5 = CA” + 2K (3C,2C,2 + 4C,4 - Cj) + K”(Cs” - 6C$,2 + C;) .

The parameter K is

(18)

J;Ao Aoa* 2R2 0 K=-=-=_, 27rB; 8~ 8

(19)

where ff = Jo/B0 and RO is the radius of the flux tube. If the parameter K tends to zero, the fast speed C.+ tends to the Alfvtn speed CA, and the slow speed C_ tends to the tube speed Cr. In the case of the untwisted tube, the dispersion relation ( 16) splits into the dispersion relations for the slow and torsional Alfven waves,

c-u* = C,k2 + Ao

47r( cj + Ci) (o* - Cs’k2) (w* - C:k=) , (20)

o2 = C*k* A ’ (21)

In the untwisted case, the torsional mode is not dispersive. AlfvCn torsional and slow magnetosonic sausage waves are modified by the twisting of the tube. The speed C+ and C_ may be considered as the ~~~~e~ Affvh speed and modi$ed tube speed, respectively.

In the case of weak dispersion (small k*Ao), when w % C&k, the dispersion equation ( 16) can be approxi- mated by

For the slow body waves in untwisted tube, (20) approximately gives

io2 xC,k2 If A& k2 477TT(c; + c;) *

(22)

(23)

This dispersion equation coincides with the dispersion equation for the slow body sausage waves obtained by Roberts ( 1981) without using the thin-flux-tub approximation, except the factor N 1 before Ao. As shown by Zhugzhda [ 5 1, this is due to the truncation of the infinite set of equations, which replaced magnetohydr~yn~ic equations for the case of expansion of dependent variables over the tube radius. The truncation of this set of equations is, at the same time, the truncation of the radial eigenfunction of the problem as well. The value of the first root of the truncated radial eigenfunction slightly differs from the root of the Bessel function, which is an exact eigenfunction of the problem. Zhugzhda [S] proposed to the in~~uction of an “effective” cross section and “effective” radius of the tube,

A = 4A&;*, R=ZR[,‘, (24)

226 KD. Zhugzhda, VM. Nakariakov/Physics Letters A 252 (1999) 222-232

where & = 2.4 is the first root of the zeroth order Bessel function. The “effective” cross section makes the

approximation not only an exact one but is valid for non-thin j7u.x tubes as well. We replace A by A in the following analysis. In the case of weakly twisted flux tube, it is assumed that the same effective cross section

can be used in (23). The case of weak dispersion (22) corresponds to the long wavelength limit, when the wavelength is much

larger than the tube diameter, k*Ao < 1. This limit is akin to the shallow water approximation. However, the weak dispersion limit for torsional AlfvCn waves appears also in the case of a weakly twisted flux tube, when K < 1 and the phase velocity is approximately

c+c: I,? . ( > In this case, the approximate dispersion relation (22) reads

w2 x c2 k2

+ 1 + AoK2p2( ’ - fl)

167r k*

which then is valid for any values of k*Ao providing the twisting is small enough. Finally, it is important to mention that resonant absorption and phase mixing of MHD waves do not appear

in the problem, because a uniform tube is considered. This is the same approach as for untwisted flux tube, where resonance absorption and phase mixing occur only if tube is transversally non-uniform [ 151.

4. Nonlinear wave equation

Consider small perturbations of the variables from the stationary state of the force-free flux tube, which is

defined by values pe, pa, Bo, JO and do,

ll=ii, p=Po+P, P=po+P, B=Bo+B,

J=Jo$J, O=JZ, d=Jto+A, u=ij.

The tilde will be omitted in the following consideration. Substituting these expressions into the set of equations (7)-( 14), we obtain the following equations,

au ap PO;+~=NL

JP f3B du Bo at - PO at + POBO z = N2 ,

a.0 Bo &l Jo LJ’B -_ &

--+ 47rpo dz

- - = N3 , 47rpo az

8.7 LIB au 2 a-J Boclr-Jo;l;+JoBo--Bo-=N4, dZ 8Z

(27)

(28)

(29)

%,*!!f=N at Sdr 5’ ad --2&v=N,, aI

f3B at -I- 2Bov = NI ,

(34)

On the right-hand sides, the nonlinear terms are gathered. The nonlinear terms appear due to (Y . V) -17 and ( V x B) x B terms in momentum equation ( 1 >, the V - pv term in mass conservation equation (2), V x (v x B) term in induction equation (3), and u - VT and pV - u terms in energy equation (4). Retaining only quadratic nonlinear terms, we write down the right-hand sides of the above equations,

+g pog+g+gg. ( > (35)

Eqs. (27)-f 34) may be combined into the quadratically nonlinear, quasi-hyperbolic equation for B,

where we use the D’Alembert’s operators

The left-hand side of Eq. (36) can be rewritten as

(C,~+Cr?...~C~)~+D_B+~~~~n2B=C~RHS.

where RHS is the right-hand side of Eq. (36) and

(36)

(37)

(38)

(39)

228 KD. Zhugzhda, KM. N~~kuriako~~/Ph_vsics Lerters A 252 (1999) 222-232

For the considered case of quadratically nonlinear waves, all dependent variables may be expressed in terms of B, using the linear parts of Eqs. (27 ,-( 34). After this procedure, Eq. (38) becomes a self-consistent nonlinear wave equation for the weakly nonlinear axisymme~ic slow and Alfven torsional waves in a twisted, straight

magnetic flux tube.

5. KdV equations for slow body and torsional Alfvch modes

There are three small parameters in the problem, connected with smallness of the tube equilibrium cross-

section .&, twisting I( and the wave amplitude. In the following, we assume that effects of these parameters are of the same order and proportional to a dimensionless parameter 0 < ,U < 1, In addition, we consider an evolution of a sirtggle mode, neglecting nonlinear coupling of the mode with backwardly propagating waves and other modes. The presence of the small parameter /.L allows us to apply the method of slowly varying

amplitudes. We pass to the frame of reference, moving with the linear speed of the mode, that is,

7 = yt , l=:-C&r,

where r is a slow time, showing the wave evolution due to the weak nonlinearity and dispersion. All physical variables in the right-hand side of Eq. (36) are expressed in terms of B as

(40)

Ci dB

C=sg)iq‘ c:c;pa B

‘= &(I$C;) ’

Jo (c; - cj,cc; -c;, - cjc.; B

J=Bo (Ci -cq,(Ci -Cj) ’ GPO B

‘= Bo(C$ 4;) ’

G cf l1 = B&z; - cs’t

B-&z, Jo a=--- C,C,z

4?rpe cc: - C,2)fC$ - Ci) B. (41)

Substituting expressions (4 1) into (36) and keeping terms of the first order of /.L only, we rewrite the nonlinear

wave equation (36) as

a4B -- ar@

E =

(42)

(43)

(44)

LD. Zhugzhda, VIM ~~r~~v/Fhysics kiters A 2.52 (1999) 222-232 229

(45)

Integrating (42) three times with respect to 6 and choosing the constants of integration as zero, we obtain the KdV equation

(46)

which is valid for long wavelength, weakly nonlinear, slow body and torsional Aifven waves in twisted flux tubes. Moreover, the KdV equation is valid for torsional waves of arbitrary wavelength in a weakly twisted tubes. The KdV equation (46) is valid for a tube with free boundaries (peXt = const on the tube boundary}. Dynamics of a tube with rigid boundaries (u, = 0 on the boundary) is different. For example, Merzljakov and Ruderman [ 161 showed that dynamics of a thin layer (a planar analogy of the tube} with the rigid boundaries is not governed by the KdV equation.

In the following consideration, our analysis is restricted to soliton soIutions of KdV, which are a zero frequency limit of a more general cnoidal wave solution,

(47)

where

L = 2 ( 3S/‘BUe) 1’2 , V = cB,/3 (48)

is the wavelength and speed of the soliton in the moving frame of reference. An analysis of waves described by this partial solution can give us an important insight into a more general case.

Since the expressions (43) and (44) for the coefficients S and E are rather cumbersome, we consider specific cases when the expressions arc simplified.

6. Nonlinear slow body waves

With C- substituted into the expressions for S and E, KdV equation (46) describes nonlinear slow body waves. In the limiting case of an untwisted flux tube, iy = 0, Eq. (46) reduces to the equation derived by Zhugzhda and Nakariakov [ 13,141. If K < 1 and ,f3 < 1, the coefficients of KdV equation (46) are

&E (Yf I)(1 -3K)CA

2Bo/3’/2 . (49)

According to (47) and (49), the slow soliton is a retarded soliton, because its speed in the laboratory frame is less than the speed of linear waves C-. An analysis of (47) and (41) gives that the soliton is a narrowing of the tube, which is accompanied by increasing of the magnetic field, decreasing of the gas pressure and density, and an acceleration of the plasma in the soliton throat.

In the case of strong magnetic field p < 1, the soliton speed and length are not affected pronouncedly by the weak twisting iy << 1. But, when the twisting reaches some certain value, the nonlinear coefficient E changes the sign. In the case B > 1, the sign is changed by weaker twisting. For example, the change of the sign of E

230 ED. Zhugzhda, VM. Nukuriukov/?‘hysics Letters A 252 (1999) 222-232

occurs for K = 0.1 (cv72+~ = OX), if ,B = 15, and for K = 0.001 (a%$ = 0.09) t if B = 50. When the coefficient of nonlinearity is negative, the solution (47) is rewritten as

(50)

and the absolute value of E is to be used in expressions (48) for soiiton speed and length. In this case, the

nonlinear slow body waves present a widening of the tube, which are running in the laboratory frame with velocity larger than modified tube velocity C-. The plasma is compressed and heated in the tube widenings.

7. Nonlinear torsional Alfvkn solitons

With C = C+, Eq. (46) describes Alfven torsional waves in twisted magnetic flux tube. In the case of a

weakly twisted tube (K 6~ 1) filled by a low-p plasma, the coefficients S and E reduce to

s ~ sZoCAP”K’ 273.

I1 -B+K(3-4P

CA &M

CP- 1)Bo

According to (51 ), the dispersion for the Alfven waves comes from the finite twisting K. In the limit K --+ 0, Eq. (46) does not describe the AlfvCn waves appropriately, because the value B tends to 0 itself in this case. Indeed, the substitution of R instead of B in (46) shows that both 6 and E vanish in this limit, giving the expected result that the non-dispersive Alfven waves are not affected by the quadratic nonlinearity (see e.g.

Ref. [ 171). The torsional solitons are described by expression (50). with the wavelength and speed given by the formulas

(53)

Thus, the AlfvCn soliton is a propagating widening of the tube. The speed of the soliton exceeds the modified Alfven velocity C+. The Alfvtn soliton could produce strong twisting in a weakly twisted flux tube, when the full $-component of the magnetic field on the surface of flux tube, as it follows from (41), is approximately

Bd, M B;” - 4BBo

(1 - ,f3)B;)’ (54)

where BF’ is the $-component of the field on the surface of the undisturbed tube and B is defined by solution (50). The soliton produces the strongest twisting of the tube, when the equilibrium twisting of the

tube is very small, Bi”’ < Bo. The effect increases with B + 1. In the soliton, the twisting is accompanied by a fast rotation of the plasma. The azimu~al component of the plasma velocity on the boundary of a weakly twisted tube is approximately

(55)

The temperature, density and gas pressure drop in the soliton. When j3 > I, both parameters 6 and E change the sign. In this case the soliton solution of the KdV equation

is (47), where the absolute values of parameters (5 I ) and (52) have to be used. In this case, the Alfven

ED. Zhugzhda, KM. Nakariakr,v/Physics Letters A 252 (1999) 222-232 231

soliton appears as a narrowing of the tube, where twisting and rotation increase according to (54) and (55),

while both the pressure and density drop. However, the changes of signs of 6 and E do not occur exactly at j3 = 1 and, moreover, do not occur at the same values of j3. It means, that there is a rather small range of

parameters /3 and K, where the coefficients S and E have different signs. Obviously, in the case of nonlinear wave propagation along the tube with longitudinally varying parameters /3 and K, the crossing of the region

of p = 1 has to be accompanied by some special effects. This is the region where twisting by the soliton could increase strongly and the wave corresponding to a narrowing transfers to the widening wave or vice versa. This effect is not described by the quadratically nonlinear theory and higher order nonlinear theory has to be applied. Note that the same analytical difficulty occurs in the vicinity p M 1 for plane linearly polarized

nonlinear AlfvCn waves [ 171.

8. Discussion

The analysis of nonlinear waves in twisted tubes is restricted by the case of weakly nonlinear and weakly

dispersive waves, when nonlinear effects are balanced by dispersion. The theory of weakly nonlinear dispersive waves is a necessary step on the way to the understanding of nonlinear dissipation, which has to occur when the dispersion can no longer balance the nonlinear effects. The dissipation can be included in consideration,

leading to the Korteweg-de Vries-Burgers equation as the governing evolutionary equation, but this is out of the scope of the paper. However, the dissipationless case provides us with clues for the cons~ction of a nonlinear dissipation scenario. Considering an untwisted tube, Zhugzhda and Nakariakov [ 131 speculated that supersonic flows in the throat of the tube narrowing, created by a nonlinear slow body wave, can result in a shock. The same scenario can be valid for slow waves in a twisted flux tube. We should also mention that, in the model considered, effects of resonance absorption and phase mixing, which can be responsible for advanced dissipation, do not appear, because the tube is taken to be uniform.

In addition, for weakly nonlinear torsional waves, the breakdown of the balance of nonlinear effects and dispersion can result in a violation of the validity of ideal magnetohydr~ynami~s in the regions of strong twisting and low density, which appear due to nonlinear Alfvdn waves in a twisted flux tube. If this is true, kinetic effects will have to be taken into account. But this is only a starting idea for a future investigation.

The appearance of the dispersion of torsional waves in a twisted flux tube and, as a result, the existence of weakly nonlinear dispersive torsional AlfvPn waves is very important for astrophysical and geophysical plasma problems, because magnetic fields are far from being potential in almost all cases. The dispersion of AlfvCn torsional waves is not connected with the long wavelength approximation. Consequently, the validity of the KdV equation for these waves is connected only with validity of thin flux tube approximation of Zhugzhda. This

approximation is a first order approximation and it is valid when the squared ratio of the length of the wave considered to the tube radius is much less than unit, (A/Ra)* < 1. This is a weaker restriction in comparison with A/R0 << 1, which has to be fulfilled for thin flux tube approximations [6,7] of zeroth order.

Acknowledgement

One of the authors (YZ) thanks the Alexander von Humboldt Foundation and Russian Science Foundation (grant 96-02-16491) for support of this research. Authors are grateful to B. Roberts, MS. Ruderman and A.W. Hood for a number of stimulating discussions.

References

( 11 C.T. RusselI. E.R. Priest, L.C. Lee, eds., Physics of Magnetic Flux Ropes (American Geophysical Union. Washington, DC, 1990).

232 ED. Zhugzhda, Wf. Nakariakov/Physics Letters A 252 (1999) 222-232

f 2 I E3. Roberts, in: Advances in Solar ~~netohydr~ynamics, E.R. Priest, A.W. Hood, eda (Camb~dge Univ. Press, Cambridge, 1992) p. 105.

[ 31 8. Roberts, A.R. Webb, Solar Phys. 56 ( I578) 5.

141 H.C. Spruit, Astron. Astrophys. 98 ( 1981) 155, ISI Y.D. Wugzhda, Phys. Plasmas 3 (19%) IO.

/ 61 B. Roberts, Phys. Fluids 28 ( 1985) 3280.

171 G. Herbold, P. Ulmschneider, H.C. Spruit, R. Rosner, Astmn. Astrophys. 145 (1985) 157.

181 A. Ferriz-Mas, M. Schiissler, V. Anton, Astron. Astrophys. 210 ( 1989) 425. [PI P.M. Edwin, B. Roberts, Wave Motion 8 (1986) 151.

[ IO] Y.D. Zhugzhda, V. Bromm, P. Ulmschneider, Astron. Astrophys. 300 (1995) 302.

1 I I 1 V.M. Nakariakov, Y.D. Zhugzhda, P. Ulmschneider, Astron. Astrophys. 312 ( 1996) 691.

1121 B. Roberts, A. Mangeney, MNRAS 198 (1982) 7.

[ 13) Y.D. Zhugzhda, V.M. Nakariakov. Phys. Lett. A 233 (1997) 413.

[ 14 J Y.D. Zhugzhda, V.M. Nakariakov, Solar Phys. 175 ( 1997) 107.

I 15 1 M. Goossens, in: Adv~ces in Solar ~aguetohydr~ynamics, E.R. Priest, A.W. Hood, e&s. (~amb~dge Univ. Press, Camb~dge, 1992) p. 105.

1 ISJ E.G. Merzljakov, M.S. Ruderman. Solar Phys. 103 (1986) 259.

I 1’71 R.H. Cohen, R.M. Kulsrud, Phys. Fluids 17 ( 1974) 2215.