433
NORMAL MODE EXPANSION AND STABILITY OF COUETTE FLOW*
J. EISENFELD
Rensselaer Polytechnic Institute
1. Introduction. Chandrasekhar [1] has considered the effect of an axial magnetic
field of constant intensity on inviscid Couette flow. He found that the axisymmetric
normal mode perturbations satisfying the linearized equations are stable when the angular
velocity is increasing in the radial direction. We now consider an arbitrary axisym-
metric perturbation and we show that under the above stability condition the perturba-
tion energy is bounded in time. Moreover, the bound can be made arbitrarily small by
choosing the initial perturbation values sufficiently small. Hence, the Couette flow is
"stable" with respect to axisymmetric perturbations.
The conclusion readily follows once it has been demonstrated that an arbitrary
perturbation is a sum of normal modes. The completeness of the normal modes, which
is of interest in itself, will be derived from the fact that they are the eigenfunctions of a
one to one, linear transformation which is symmetric and completely continuous. This
is true even in the absence of the above stability condition.
2. The linearized equations. The linearized equations governing the motion of a
perturbation are (cf. [1]):
(ur), - 2Que - (Ar), = -(Si), , (2.1)
W, + (re + 2n)ur - ^ (he), = 0, (2.2)
(«.). = "(*). - (2-3)
(hr)t - H(ur)z = 0, (2.4)
(he), — H(ue)z — rQ'hr = 0, (2.5)
(K)t - H(u.). = 0, (2.6)
and the equations of continuity
(Or + Ur/r + (ut), = 0 (2.7)
and
(hr) + hr/r + (hl)l = 0. (2.8)
Here the vectors u = (ur , u9 , w2) and h = (hr , h6 , hz) denote the perturbations in
the radial, transverse, and axial directions of the perturbed velocity and magnetic
intensity. Q(r), P(r), p, n and H denote the equilibrium angular velocity, pressure,
density, magnetic permeability and magnetic intensity; the notation indicates which
'Received February 23, 1967; revised manuscript received August 16, 1967.
434 J. EISENFELD [Vol. XXVI, No. 3
equilibrium quantities are functions of r and which are constant. In addition, a> denotes
the perturbation of the quantity P/p + ixH2/(8irp) and primes denote differentiation
with respect to the radial variable r, which varies on a finite interval 0 < a < r < b.
The end points correspond to the cylinder walls at which the radial velocity vanishes, i.e.,
uT(a) = ur(b) = 0 (2.9)
The axial variable z varies from minus infinity to infinity. The perturbation has finite
energy, i.e., it is square integrable.
We consider the equations obtained by taking the Fourier transforms of the perturba-
tion variables. This amounts to replacing the operation of partial differentiation with
respect to z by multiplication by ik in (2.1)-(2.8) where fc is the wave number. Eliminat-
ing a> between (2.1) and (2.3) we obtain
ik{(ur)t - 2Q&, - ^ hr) = ((*.), - ^ (k))r (2.10)
where - denotes Fourier transformation. We see from (2.7) that there is a function
X such that
u, = ikx, u, = — -j; (rx)r • (2.11)
Observe from (10.4) that (2.11) is valid at k = 0. Setting
wi = x. w2 = h, , (2.12)
(2.10) and (1.2) may be written as
((DD* - k2)w1)„ + 0\{DD* - k2)Wl - Atyw, + 2QQAikv>, = 0, (2.13)
and
(tt>2) t ( + U\w2 + 2 UikilAWi — 0, (2.14)
where
<f> = 4Q2 -f- r(02)' and (2.15)
4xp
are Rayleigh's discriminant and the "Alfven frequency" respectively. Also
D = (•), and D* = £ (»■(•))r
are differential operators. Equations (2.4)-(2.8) and (2.11) serve to represent the per-
turbation variables in terms of and w2 . We now have a 2 X 2 system of equations
which we can write in the form
(.Pw)„ + tfAPw + Sw = 0 (2.16)
where
P =
(DD* - k2) 0
0 1J
S =
k2(j> —20 2Aik
2QQ Aik 0 j
(2.17)
1968] NORMAL MODE EXPANSION AND STABILITY OF COUETTE FLOW 435
and the vector w = (w>, , w2) satisfies the boundary condition
Wi(a) = Wi(b) = 0. (2.18)
3. The normal modes. The normal modes are those solutions of (2.16) of the form
w,(r)e'". The functions w, , also referred to as the normal modes when no confusion
arises, are the eigenfunctions of
\Pw = Sw, Wi(a) = w,(b) = 0. (3.1)
Here we made the substitution
X = a2 - 0; . (3.2)
It will be seen from (4.5) that
DD*y - k2y = 0, y(a) = y(b) = 0
has no nontrivial solution; hence, its Green's function, G(r, /), exists. It follows that,
P has the inverse
P'1 =
where
J 0
0 1
(3.3)
•/(/) = ~ f G(r, r'W) dr' (3.4)
J a
is "Green's operator." Eq. (3.1) may now be written as
Tw = \w (3.5)
where T = P'lS.
4. The inner product. It will now be shown that there is a symmetric, positive
definite, bilinear form (•, •)„ for which
(77, g)» = (/, Tg)v . (4.1)
Let L2 denote the space of square integrable functions on (a, b). For / and g in L2
let
(/. g)r = [ f gr dr, (4.2)
J a
where bars denote complex conjugation. Upon integrating by parts one sees that DD*
is symmetric with respect to (4.2) for functions vanishing at the end-points. It follows
that J is also symmetric. Let
DP = {/ : D2U J2zL2, A (a) = A (6) = 0[. (4.3)
Then on DP , (Pf , g) where
(/, g) = (/i, + (U , g^r • (4.4)
Since, after integrating by parts,
(~(DD* - k2)f, f), = f (|D*/|2 + k2 |/|2)r dr > 0, (4.5)
J a
436 J. EISENFELD [Vol. XXVI, No. 3
we see that the inner product
(/, 9)p = (Pf, g)r (4.6)
is symmetric and positive definite on DP . Moreover, S is symmetric with respect to
(4.4); hence,
(:Tj, g)P = (PP-'Sj, g)r = (SI, g), = (/, Sg)T = (/, PP^Sg), = (/, Tg)P on DP .
5. Complete continuity. With respect to the norm,
IMIp = (w, w)V2 = (IIZ>*w?i1+ k2 ll^illr + ||w2||r),/2 (5.1)
where ||-||r = (•, •)DP is a pre-Hilbert space. Since
a INI* < ||-|[? < 6 11-112
we see that || • ||r is equivalent to the L2 norm, || • ||2 . Moreover, D*j is in L2 if and only
if Dj is in L2. Therefore, the completion of DP , II, is precisely the set of functions in the
product $2 X L2 where S2 is the Sobolev space of functions which are continuous on
(a, b), vanish at the end points and have "strong" first derivatives in L2 (cf. [6]). To
see that T is completely continuous on H we compute
\\Tw\l = [(Jv, v)T + ||fiWl||r2]1/2 (5.2)
where v = k2<t>w! — 2W,Aikw2 and J is given by (3.4). We wish to show that if w(n) is a
bounded sequence in II then there is a subsequence of integers for which \\T(w(ri) —
w(m)) ||„ —> 0 as n, m —> <». Since J is an integral operator with a bounded kernel, it is
completely continuous on L2 . Since the sequence v(n) is bounded in L2 , Jv(ri) has a
Cauchy subsequence in L2 . Therefore \J(v(m) — u(m)), v(n) — v(m)\ < \\Jv(m) —
Jv(n) 1111v{m) — v(n) \ \ —»0 as n, m —> co on a subsequence of integers. We may now assume
that the subsequence is the original sequence. In view of (5.2) it remains to show that
||u>i(n)||r has a Cauchy subsequence, but this is a special case of Rellich's Theorem (cf.
[5, p. 30]). It also may be deduced from the fact that w^in) is the image of the completely
continuous operator, (D*)_1 = r (-)s-1 ds and that D*wl(n) is bounded in L2 .
6. Differentiability of the eigenfunctions. It will be shown that in the process
of "completing" DP no new eigenfunctions are added. In fact, it will be seen that the
eigenfunctions are as differeDtiable as the equilibrium angular velocity, and that they
vanish at the end points. Suppose that w is an eigenfunction. This means that
J(k2<j)Wi — 2Q£lAikw2) = Xwj (6.1)
and
2W,AikWi = \w2 . (6.2)
Since wl is continuous and vanishes at the end points the same is true of iv2 by (6.2).
In view of (6.1) wi has two continuous derivates since the kernel of J is the Green's
function of a second order differential equation. Now if £2 is twice differentiable, the same
is true of w2 . Repeating this argument as many times as necessary we see that w is as
differentiable as 0.
We see also from (6.1) and (6.2) that if X == 0 then vh = 0 and J(0,w2) = 0. Applying
(DD* — k2) to the latter gives w2 — 0. Here we are assuming that £2 does not vanish
identically in any subinterval. It follows that zero is not an eigenvalue, i.e., T is one
to one.
1968] NORMAL MODE EXPANSION AND STABILITY OF COUETTE FLOW 437
7. The completeness theorem. It has been shown in the preceding sections that
the normal modes are the eigenfunctions of a symmetric, completely continuous, one
to one, linear transformation of the complete Hilbert space of vectors, w = (wx , w2)
such that Wi , its derivative, DiOi , and w2 are square integrable on (a, b). Convergence
in the space is taken in the sense of (5.1). If follows (cf. [4]) that for any vector / in the
space
/ = £ (/> wn)Pwn (7.1)
where the normal modes wn are normalized by ||w„||p = 1. Moreover, the normal modes
are orthonormal, i.e.,
(wn , wm)P = 5„m . (7.2)
8. Distribution of the characteristic frequencies. We now assume the stability
condition (cf. Sec. 1):
((fi2(r)))' > 0. (8.1)
It will be shown that the characteristic frequencies a„ , wrhich are related to the eigen-
values X„ by (cf. (3.2))
K = "l — , (8.2)
are real, nonzero, 5^ , but tend to as n —> co . The last two properties are true
in the absence of (8.1) and follow immediately from the fact that X„ —> 0 as n —» °° and
that X„ ^ 0, since they are the eigenvalues of an operator with properties stated in Sec.
7. The first two properties follow from the relation:
a"n/SfA > J r(Sf)' \wx\2 r dr j 4> \wi\2 r dr (8.3)
where w, is the first component of the nth normal mode. To verify (8.3) observe that
by substituting for w2 from (6.2) into (6.1) and then applying DD* — k2 to both sides
W\ satisfies
DD*Wl + k\4n2tfA\~2 + cfrK1 - l)W\ = 0. (8.4)
Multiplying both sides of (8.4) by rwx , integrating from a to b and then integrating by
parts using (3.1), we obtain
4X„"2 + BX-1 = C
where
A = 4/b2^
B = /c2
f £22 |u>! |2 r dr > 0,
"a
f 4>{r) |wi|2 r dr > 0,
J a
and
C = k2 J |wj |2 r dr + J \D*wt |2
r dr > 0.