ON PERIODIC MOTIONS IN MAGNETOHYDRODYNAMICS* Salvatore Rionero**

S O M M A R I O : Si dimostra ml teorema di uniciM ed un teorema di esistenza per i moti magnetoidrodinamid periodic; net tempo ed asintoticamente stabi/i in media, subordinatamente all'ipotesi the esista un morn asintoticamente stabile in media sufficientemente regolare, e si deducono, in particolare, analoghi teoremi per i moti stazionari. I1 procedimento adoperato costittdsce anche un semplice metodo per la effettiva costruzione dei moti periodici e dei moti stazionari.

(n) R~ + . V ' S M <

S U M M A R Y : A uniquoless theorem and an existence theorem for motions ao,mptotical ~, stable in the mean and periodically depending on time, are demonstrated in magnetohydrodynamics, on the hjpothesis that a motion sufficien@ regular and asymptotically stable in the mean exists. In particular, similar theorems for stead), motions ate deduced. The procedure used is also a simple method for constructing periodic and steady motions.

1.

Introduction.

Let S be a bounded domain, ±" its boundary, and let . f be the set of the regular solutions (i) (v, H , p) of the magnetohydrodynamics equations which verify the boundary conditions : v = v,_(P, t)

l

H = H,:(P, t)

( w >/0, P e x)

(I)

where vz and H e are assigned functions and where v is the velocity, H the magnetic field and p the pressure. In an earlier paper [1] we have established some conditions sufficient for the asymptotical stability in the mean of an element of o¢. Precisely we have proved that the condition sufficient for the asymptotical stability in the mean is that one of the following couples of relations is verified:

This work was done in the sphere of activity of the C.N.R. groups for mathematical research and, in Italian, has already been presented for publication in "Rend. Ace. So. Fis. Mat., Napoli".

I

o

!R:+ . - -1~ - P , . R d * < d 1 t R,,,* + -g-

R ,,,j"* <

where: Re, Rm, M, Pro, J* are respectively the Reynolds number, magnetic Reynolds number, Hartmann number, magnetic pressure number, current density number of the unpertubated motion; _R,*, Rm* are dimensionless numbers of the "kind of the Reynolds numbers (Sec. 2), and 6 ~_ 80 is the least positive root of tag %/x]2 = %/x[2. In this paper, assuming that S, vz and I-I~ depend periodical~ on the time t with the same period J ' , we intend to furnish some theorems on the possible periodic motions, with period 3", existing in J , which among other things, extend to magnetohydrodynamics the results established by J. Serrin in a work published in 1959 [2]. But in contrast to this work--which in effect is limited to characterizing the initial distribution of velocities of the possible periodic and asymptotically stable (in the mean) hydrodynamic motions--we in addition supply conditions sufficient for the e.~stence of periodic, asymptotically stable in the mean, magnetohydrodynamic motions, which in particular also ensure the existence of similar hydrodynamic motions. From the physical point of view, for the (I), this is equivalent to establishing sufficient conditions to ensure the existence of periodic motions in a domain having non-ferromagnetic rigid walls in prefixed motion (See. 2). However it is to be noted that, as in the Serrin work [2], sucl't existence depends upon the existence of a sufficiently regular asymptotically stable in the mean motion. After analytically setting the problem (Sec. 2) and recalling some formulas demonstrated in [1] (Sec. 3), four in number, we prove the two following uniqueness theorems which--in contrast to what happens in the theorems set in [3], [4], [5], [6], [7], ;8]--are unaffected by the initial distribution of velocity and magnetic field(U):

* * Istituto di Matematica, Universit~ di Napoli.

(1) We mean the solutions which satisfy the conditions (1) and (2) of Sec. 2. DECEMBER 1968

(3) Precisely, the initial distribution of velocity and magnetic field must verify only one of the global conditions (II). 207

~-A) I f there is in J[ a periodic motion on the time, with period , comp~ing with one of the conditions (II) of asymptotieal stabilio, , it is the only periodic motion on time with period if" existing in d .

2.

The magnetohydrodynamics system, are

B) I f there is in . f a (v, H ) motion, compO,ing with one of the conditions (II) of ao,mptotical stability in the mean, and if for each t >i 0 the sequences

l

IS = lim v(P, / + n.Y~) lim H(P, 1 + n,Y"-).

Giorgi's

O)

where v and H are the velocity and the magnetic field respectively, F the field force (typically, gravity), p the pressure, ~ the permeability, e the density and ~, and are the kinematical and magnetic viscosity respectively. Let S be a bounded domain of the three-space, Z" its boundary, and J the set of the solutions of (1) verifying the boundary conditions

or)

tl~oc

In number five we give subsequently conditions sufficient for the uniform convergence of sequences (III), and namely we prove that: C) Conditions sufficient for sequences (III) to be uniformO, converging in St, are that the functions of (III) be eqnicontinuous attd equibounded in St. In relation to the theorem B) it is to be noted that in addition to ensuring the uniqueness of the possible periodic motion, it also gives a construction procedure of its velocity and magnetic field. It is not a question of a real existence theorem, however, if further hypotheses are made on the asymptotically stable in the mean motion (v, H), it is possible to prove that the functions (IV) actually represent the velocity and the magnetic field of a stable periodic motion. Precisely, in number six, we prove the following existence theorem:

D) I f there is in ~¢ a (v, H, p) moron ver~,ing one of the conditions (II) of ao,mptotical stabiliO, in the mean, and if v, H attd all their fiirst arm second derivatives at'e equibounded Vt >1 O, Vp e S, furthermore, attd if the second pure space derivatives are equicontinuous, then there is in ~¢ one and ottO, one periodic motion with period .if" and that is the motion, the velocity and the magnetic field of which are given by (IV). Moreover this motion is also ao, mptoticalO, stable in the mean.

v = v..(P,

l

t)

(2)

H ---- H~,(P, t)

and the following regularity conditions: 1) that they are of class C1 in respect of all variables in each bounded time interval; 2) that they are of class Co in respect of the only space variables. Assuming that S, v r and H_, depend periodically on time with the same period if- and that there exists in J an asymptotically stable (in the mean) motion, we inted to determine conditions sufficient for ensuring the existence and uniqueness in o¢ of periodic asymptotically stable in the mean motions. From the physical point of view, for the (2), this is equivalent to set conditions sufficient for ensuring the existence of periodic stable magnetohydrodynamic flows in a rigid walls domain in a prescribed motion, in the case of the wall magnetic permeability's coinciding with the fluid permeability (this condition being sensibly verified if Z" is non-ferromagnetic). However, before establishing the above conditions, we shall report some formulas which will be useful on several future occasions.

3. In number seven we have considered the case that S, v r and H_~ are independent of time and, by making use of the theorem D), we have obtained a theorem concerning the existence and uniqueness of steady motions. Finally in number eight, by making use of results obtained in [9], we have established that theorems similar to the theorems A), B), C), D), also exist for non-isothermic magnetohydrodynamics, at least when consideration is given to the heat conduction equation in the Boussinesq approximation. 208

in

I

(III)

converge uniform O, in S,, then the ono, possible motion of J depending periodical o, on time with period J ' , is that of velociO, and magnetic field given by:

equations,

- ~ - + rot ( H A v) = q, loH

{ u . ( P , t)} = {v(P, t + , J ) }

{h,,(P, t)} = {H(P, t + ,,.~')}

Analytical formulation of the problem.

Some recalls.

Let d be the maximum diameter of S and let (v, H) be a motion of .f. We shall set: V" ----1.u.b.[v(& 0[, H * = I.u.b.IH(P, t), J = l.u.b.lrot a I

(Vt ~ o, P ~ s) R, = V d v

R,, '

=

Vd q"

j, = d.j, ~

l'JI. *

p.,

_

tl o

H*O V 2

MECCANICA

M =

p

p

... H * d

*

tads ,

Re

*

R,,,

The (6) is precisely the relation we intended recalling. We remark now that, as demonstrated in [1] and as, on the other hand, it follows from (6), if at least one k(v, H) (i -----1, 2, 3, 4) is negative, then the (v, I t ) magnetohydrodynamic flow is asymptotically stable in the mean.

m*d~

) . l = ~ . max [Re+2~--~; ~(R,,t+M--~)]I,

4.

).~ = .V/~. max

[Re, +

M__~/];

L~/,~

ha = max

Proof of the theorems A) and B).

M__~¢/~1

" tC6

P,d*\

"

,~ (R_~m+

/J

"3

2V"~

We shall start to demonstrate the uniqueness theorem A). The proof follows immediately from (5) when one observes that if one of the ~,(v, H) is negative, K + IVis a decreasing function and therefore it cannot be periodic. Therefore if the (v, H ) motion has period ~'-, it is the only periodic motion of period ~ - existing in a . Let us now proceed to demonstrate theorem B) which is itself a uniqueness theorem, but it further supplies a construction procedure of the possible periodic motion. To this end we set:

* 1 R, + ~ P,,,RJ* - - 6;

(,

,

'_L,. R,. + -2- R,,I*--.

)]

6

u.(P, t) = v(P, t + . J - )

(3)

where Re, Rm, M, Pm are Reynolds', Reynolds magnetic, Hartmann and magnetic pressure of motion (v, H) numbers(3). In the dimensionless numbers Re* and Rm*, - - m and m* are the lower and upper bound of the characteristic values of the defornaation matrix of the (v, H ) flow. The dimensionless number ~ is the least positive root of tag %/x12 =%/x12 and results ~ "- 80. We shall further indicate as:

I K(v, - - v) =

.['s ~(vl - - v)~ dS .l's /,(H~ - - H)~ i S

(i = 1, 2, 3, 4)

(7) P~S~.

L, l[

'q'I f,, l Ev,(;.

+

/_L_'[HI(P, t ) - - H(P, t)] '~I ~S,- e~p 2n,,.Y )a(v, H). (8) o

But, by hypothesis, one at least of the h(v, H ) is negative, therefore, setting

the "kinetic and magnetic energy of the difference motion of two motions (vl, H1) and (v, H) of J . This set, we note that, as shown in [1], there subsists the relation (4) : d 2,, -c/t (K + iV) ~< -'fi-o 2,. (K + ll::'),

nJ-)

Then if the (vl, H1) ~ o¢ motion has period ~--, according to the periodicity assumption on S, vz and H z and setting tl = t, & = t + n~- in (6), it follows:

+ (4)

'(Hi - - H) :

h.(P,/) = H ( P , t +

n = 0 , 1,2 ....

(5)

u(P, t) : lira u,,P, t) : lira v(P, t + n J°'-)

(9) lh(P, t) = lim h,,(P, t) : lim H(P, t = n~-')

for each fixed t >t 0 it follows that:

vl(P, t) = u(P, t)

l

and consequently

(10)

Ht(P, t) = h(P, t)

(K + IV),2 ~< (K + IV)q • exp

2 , , ( & - &) do 2s(v, H),

(t., >/ t,/> 0).

(6)

(3) To point this out we shall then also write R,(v), Rm(v), M(H), etc., and we shall do likewise for the quantities 2, writing ~,(v, H). (4) The (5) is immediately obtained from. (9x), (13), (14), (16), (17), (19) of [1]. DECEMBER 1~8

which proves the theorem. In particular it follows from the demonstration that it is sufficient for the sequences {v(P,~Y')}, {H(P,n.~)} to be uniformly converging in So, provided there is V l ( P , o> =

t

v(P,

P ~ So

(11)

(Hi(P, 0) = lim H ( P , n~d-). n - * ~o

209

But, for the uniqueness theorems [31, [4]. [5], [6], [7], [8] initial data determine a unique motion, and consequently there we also obtain the following theorem which in the hydrodynamic case is included among the results set by Serrin in his work [2]:

I) I f there is a (v, H) e dr motion compO,ing with one of the (II) and if the sequences {v(P, nJ-)}, {H(P,ng--)} converge ,miformly in So, then the only possible motion of . f dependitg periodicall), on time with period ~'-, is that which complies to the initial data ( 11 ).

with V and H * finite by hypothesis, it comes out that: [ K ( v ' - - v ) + W ( H ' - - H ) ] , t ~< ~< - ~1- J . "sq [ ( 2 ' / ) -° + V/t( 2 I - I * ) 2 ] dSq <.

,

~< ~ d ' ~

~< T

From the procedure adopted for the B) theorem demonstration, it appears that the theorem remains valid, even if the two sequences {u,}, {hn} do not converge uniformly in St, only under the condition that each of them contains a subsequence which converges uniformly in St. For this purpose however it is convenient to note that the following theorem exists.

II) IJ" there is a (v, H) e ~¢ motion verifying one of the conditions (II) of ao,mptotica/ stabiliO, in the mean arm if v arm H are equibounded Vt >1 O, V P ~ S, then the ttniformly convelgi~ snbsequences contained it, {u,,} and {h.} converge to the same

fnnetion. Let {ur.}. {u.,,,} and {h..}, {h..,} be uniformly converging subsequences contained in {u.} and {h,,} respectively. We must prove l lira u~ = lim ur.,

(12)

Qi'..n h%-~ li?:_ h%," Setting:

,)= vl,,,, + (,,,--n),q

(,,,> ,,, ,> q

H ' ( P , t) = H I P , t + (1--q):~-]

k/>/ 0, P ~ S , /

(13)

with m, n, I and q positive integers, then obviously, according to the periodicity assumption, (v', H ' ) is an J motion. Therefore from (6) follows [ K ( v ' - - v)

q-

W(H'

--

H)]t,, ~<

~< [ K ( v ' - - v) q- W ( H ' - - H)] exp 2,.(&d.--o

h) 2*(v, H)

(14)

where 2" is that of the & wkich, for assumption, is negative. On the other hand, from (13) and (31) it follows:

0

, VII/> 0

(16)

and therefore from (14) we get

[K(v'-5. Proof of the theorem C).

( I'/'~-F - -,,H ' ° * )

v) + 1 U ( H ' - -

8

'rda

H)],,,

( I.'"'-'+ /',, H 2.)/ exp 2,,(&-d~

t,)

2*(v, H),

Putting t.o = t + n.Y~ and letting n-+cro, we get lim

Likewise putting t2 =

.l't (u,, - - u,)" dS, = 0.

t+

lira l , ¢ 1 ~ ,*~

qJ - and tt = t it follows that:

J's, (hi •

--

h q ) "2 d,~'t =

O.

.

Letting m, n, /, and q tend to infinity through the sequences {,;,,}, {,',} {s,,} and {s,} we get the (12). Theorem II) is thus proved. For its applications it is useful to note: III) The theorem is valid for a fixed t >>. 0 even if v and H are not equibounded (VI >t O, V P E S), provided that the sequences {v(P,t + n.9-)}, {H(P.t + n3")} are equibounded(S). IV) The v and I t fnnctions are equibounded (Vt >1 0, VP ~ S) both if the motion (v, H) verifies the (IIz), namelj, 2z < 0, and if J # o and if(v, H) verifies the (II~) or (IIs), namely & < 0 0r 24 < 0. V) The sequences (v(P.t + nJ')}, {H(P, t + noq')} are equicontinuous in St, Vt >i O, if theY, H first derivatives with respect to the space coordinates are equibounded Vt >. O, V P E S. We are now in the position to prove the theorem C). The sequence {v(P, t + nJ-)} is bounded and equicontinuous in St by hypothesis, hence by theorem of AscoliArzel~t, each of its subsequences contains a subsequence which converges uniformly to a vector u(P, t) which, for the theorem II), is independent of the subsequence. Therefore it follows that the whole sequence {v(P, t + + n.7)} converges uniformly towards u(P, t) in SeQ). (3) In fact, in this case [ U ( v ' - - v ) + ~ ( H ' - - H ) ] f - a is still an independent constant of m, n, /and q. (~) In fact, if it were not so, given an e > 0, infinite elements could be found such as: u=~, urn2.... , u....... for which the inequation titan> I t +

!vV,,t)l

lIH'(P, t)] 210

.< v

H*

(15)

is valid for each m., at least in one point of St. But this is impossible because {11,.,}, by theorem of Ascoli-ArzelA and by theorem II), contains a subsequence which converges uniformly towards u(P, t) in Sv MECCANICA

In a similar way, all we have stated is applicable to { H ( P , t + ,,J')}. In the application o f theorem C), it is useful to note

that for the observations IV) and V) it results that:

cgh.

I c'~-- + rot (hn A u,,) ~-- ,/A2hn

i

div h, = div u. ----0

dun

' gradp . . . . .

VI) The sequences {v(P, t + n~-')}, {H(P, t + n3-)} are equicontinuous and eqvtibouttded in St, Vt >1 O, if (v, H) verifies (IIx) and if the v, H first derivatives with respect to the space coordinates, are equibomMed Vt >i O, V P ~ S.

6.

Q~

(19) + /, rot hn A ha + 0vA.ou,,, + ~F .

Letting n - + co and noting that for the (17) and (18), we may invert the symbols which appear in (19a), (1%) and the right-hand side of (193), with the sign of limit, it follows that

P r o o f o f the t h e o r e m D).

Let us start by stating that by the hypothesis made on v, H and their first derivatives, for theorem C) and observation V), the sequences {v(P, t + n.Y)}, {H(P, t + n~--)} converge uniformly in St, VI >10. Their limit vectors u(P, t), h(P, t) are equicontinuous in St for each fixed l >/ 0 and, by Ascoli-Arzel~t theorem, they will also be continuous in each closed and bounded subset of the fourspace set D~ described by St on varying of t in [0, ~]. We now prove that u hand h have continuous first derivatives in the interior of D ~ and furthermore the following relations are valid

GU c):,q

-- l i m n-. ~

i div h = div u -~ 0 lira gradpn = - -

(20) du o~+

proth Ah+

ov. l z u + oF.

Therefore the sequence {gradp.}also converges uniformly, and let A(P. t) be its limit. But, as we may easily verify(7), rot A = 0, so that it is possible to find a function p* so that gradp* -----lira grad p,. Then u, h , p * verify (1) and: n~O0

for their procedure construction, also the boundary conditions, namely (u, h, p*)E J . Moreover, according to the periodicity assumptions on S and boundary data, it follows that:

dUn c).\'i

?)h t c--~- + rot (h A u) = ,7,'12h

"

(17) Oh r)x i

-- l i r a n -, ~

c)h,,

u(P, / + m ~ ) = lim v(P, t + mJ - + nJ ) ---rl -,

C)X i

= lira vii) , t + (m + n)~--] = u(P, l) The sequence {Ou,,/Ox~} is bounded and equicontinuous in D ~ by hypothesis on the first and second derivatives of v hence the Ascoli-Arzel~t theorem is applicable to each closed and bounded set D* ~ D.~. Consequently each subsequence of {Oun/Oxt) contains a subsequence uniformly converging in D*. This demonstrates, by means of a well known theorem, that u has first derivatives in respect to xl(i = 1, 2, 3, 4) and that Ou/Oxt is precisely the limit vector of each uniformly converging subsequence. Moreover it proves that the whole sequence {Ou,,/Oxt} converges uniformly towards OutOx¢ and also the relation (171). In the same way we may proceed to demonstrate the formula (17~) and also to prove the existence and continuity of the pure second derivatives of u and h and the relations C)2U

- - =

- - =

= lim H I P , t + (m + n).~] = h(P, t) It -"

that is, (u, h, p*) is periodic and has period 9"-. To complete the proof of the theorem it is enough to show that 2i(u, h) <~ i,(v, H)

t~)2Un

lim-

6"h,~

(2t)

u(P, t) = lira v(P, t + nJ ) //~,c

it follows that

(i = 1, 2, 3 ) .

u~< v + O Setting p,t = p(P, t + nJ-), where p is the (v, H) motion pressure, we note that, by their construction procedure and according to the periodicity assumptions, u,,, hn, pn verify (1), hence it follows that: DECEMBER 1968

(i = 1, 2, 3, 4)

for then, by (II) and theorem A), (u, h, p*) is unique and asymptotically stable in the mean. We limit ourselves to proving that 2z(u, h) ~<"2z(v, H ) ; in the same way, it is possible to prove (21) with i = 2, 3, 4. Let us note that from

lira - - -

08) fi"h

h(P, ! + #z.f) = lira H(P, ! + m.7 + n.~-) =

(1)

(7) It is enough to observe that for each fixed t/> 0 and each closed curve F ~ St, we get.d'r gradp. • dP = 0; and also that we must take the {gradpn} uniform converging into account. 211

But u is periodic, hence u~< sup lv(P,t)[~< V"

VI>I O, 19~3"

follows. In the same way we get: h ~ suplH(P, t)l = H*, therefore it follows that 2x(u, h)~< 2x(v, FI).

the equation of non-isothermic magnetohydrodynamics in the approximation of Boussinesq [10], the regularity conditions (1) and (2) of Sec. 2 and the boundary conditions : v(P, t) = vl(P, t) = vz(P, t) H(P, t) = I-I,(P, t) = FIr(P, t)

V/t> 0, P E L"

(22)

T(P, t) = T,(P, t) = Tz(P, t) 7.

Steady case.

then, as shown in [9], one gets: Let us examine briefly the case that S, vr and H r are time-independent. It is clear that, in this case also, the proved theorems are valid, but now °o°J can be any one positive number. Concerning the steady flows, the following theorem subsists: VII) Under the hypotheses of theorem D), the necessary and sufficient condition for the existence in . f of a stead), motion, is that the limits:

v / K + W/"+ Pr~'~ V [h. ~ <<"

<~(VKO ~- WIO~- Pr}'t V~t ' ,...00) !exp [ ~

(3 ~2~,,_ x/R---~)t]

Pry~<~ l

lim v(P, t) = u*(P) t~oo

!exp [ ~ ( 3 ~ % , , - - ~ / R - 0 ) t ]

Pr~,~>~l

lim H(P, t) = h*(P)

exist. Such lim#s give respectivePi the velocity and the magnetic field of the steady motiou. The condition is necessary for the theorem B). On the other hand, it is an obvious consequence of the asymptotical stability of (v, H). The sufficience follows from theorem D), taking into account that now in (20) we must make the substitution: h--~ h*, u--+ u*. Furthermore it is to be considered that the (u*, h*) motion is not only asymptotically stable in the mean, but it is the only steady flow belonging to ~¢ and no periodic flows belong to.

where a, g, ~. are respectively, the coefficient of thermal expansion, the field force (typically, gravity)vector and the thermometric coefficient, and where:

= max I grad,

IR~ = Pr --

E x t e n s i o n o f the p r e c e d i n g results to the noni s o t h e r m i c magnetohydrodynamics.

We show briefly now that theorems analogous to those of the preceding numbers subsist also for the non-isothermic magnetohydrodynamics, at least when consideration, is given to the equation of heat conduction in the Boussinesq approximation. For this end we observe that if (v, H, 2") and (vl, H1, 7"i) are respectively the velocity, the magnetic field and temperature of two flows satisfying

212

=

1

o(7

~g~4

(Rayleigh number)

"

(Prandtl number)

- T) dS

X

)'~ 8.

-I

1 2i(v, H). 3 ~2

Let S, Vr, H r and Tr depend periodically on time with the same period J " and let o¢' be the set of motions verifying respectively the equations of non-isothermic magnetohydrodynamics in the Boussinesq approximation [10], the regularity conditions (1) and (2) of See. 2 and the boundary conditions (22). It is cleat" tbeu that the theorems of the preceding numbers are valid, when we take into account the temperature T attd substitute ).~, with Ra/3n2 + 2, (i = 1, 2, 3, 4). Received 21 February 1968.

MECCANICA

REFERENCES [1 ] S. RIONERO, Sulla stabilit~ asintolica in media in magnetoidrodinamica, Ann. Mat. Pura ed Appl., IV, Vol. 76, pp. 75-92. [2] J. SERalN, A note on the existenca of periodic solutions of the Navier-Stokes equations, Arch. Rat. Mech. Analysis, 3, pp. 120-122, 19S9. [3] R. NARDINX, Due teoremi di unicit;z ndla teoria delk onde magnetoidrodinamkhe, Rend, Sem. Mat., Univ. Padova, 20, 1952. [4] R. NARDmI, Due teoremi di unicita nella magnaoidrodinamica dei fluidi compressibili, Boll. U.M.I., 7, 1952. [5] R. P. KANWALL, Oniqainess of magnetohydrodinamic flow,

DECEMBER 1~8

Arch. Rat. Mech. Anal., 4, 1960. [6] I. FEttR^v.i, Su tin teorema di unidth per le equazioni ¢kll'idromagnetismo, Atti Sem. Mat. Fis., Modena, 9, 1960. [7] I. FERRARI, Sul teoreraa di unicit;~ etc. di un fluido compressibile in un dominio illimitato, Atti Sem. Mat. Fis., Modena, 10, 1961. [8] R . H . DYER and D. E. EDMUNDUS,A lady, chess theorem in magnetohydrodynamks, Arch. Rat. Mee. Anal., 8, pp. 254262, 1961. [9] S. RIONERO, Sulla stabilit~ asintotica in media in magnetoidrodinamica non isoterma, Ricerche di Matematica, V, XVI, pp. 250-263, 1967.

[10] S, CHANDRASEKH&R,Hydrodynamic and hydromagnetic stability, Clarendon Press, Oxford, 1961, or cfr. [9].

213