13
Characteristic-value Problems in Hydrodynamic and Hydromagnetic Theory
S. CHANDRASEKHAR
MORTON D. HULL DISTINGUISHED SERVICE PROFESSOR
DEPARTMENTS OF PHYSICS AND ASTRONOMY, UNIVERSITY OF CHICAGO
13.1 Introduction
The general theme of both this book and its predecessor is modern mathematics for the engineer. “Modern mathematics” might be interpreted in this connection as new mathematical methods or techniques for solving problems in applied mathematics. Very often the discovery of new methods and techniques consists in reducing to an elementary level problems once considered difficult or complicated. Whether or not this point of view is justified under all circumstances, it is our purpose in this chapter to illustrate it by considering a very old and a very classical problem; it is a problem to which some of the great applied mathematicians have contributed. The problem concerns the stability of purely rotational flow between two concentric cylinders. The mathematical problems that arise are rather different when the fluid is considered nonviscous and when it is considered viscous. In the absence of viscosity, the subject was investigated by Lord Rayleigh in a famous paper;5 in the presence of viscosity, the subject was investigated in an equally famous paper by G. I. Taylor.6 Taylor’s investigation in fact provided the first example of a problem in hydrodynamical stability some aspects of which were successfully solved both analytically and experimentally. A complete solution of the basic mathematical problem was not available, however, until recently.1,2,3 We shall consider the two cases separately.
13.2 The Rayleigh Criterion for the Stability of Inviscid Rotational Flow
If we consider the fluid as incompressible and inviscid, then the equations of motion allow the rotational velocity uθ to be an arbitrary function V(r) of the distance r from the axis :
Since the equations of motion do not restrict V(r), the question arises: Are there any restrictions on V(r) that result from requirements of stability? Rayleigh has stated the following criterion:
In the absence of viscosity, a necessary and sufficient condition for a distribution of angular velocity Ω(r) to be stable is
everywhere in the interval; further, the distribution is unstable if Φ(r) should decrease anywhere inside the interval.
In establishing this criterion, Lord Rayleigh argued as follows:
If we restrict our attention to axisymmetric motions, then it is a direct consequence of the equations of motion that the angular momentum
L = r2Ω
of a fluid element, per unit mass, remains constant as we follow it with its motion. Suppose now that we interchange the fluid contained in two elementary rings, of equal heights and masses, at
r = r1andr = r2 > r1
If dr1 and dr2 are the radial extents of the rings, the equality of their masses requires
πr1 dr1 = πr2 dr2 = dS
(say). In view of the constancy of L with the motion, the fluid at r2, after the interchange, will have the same angular momentum (namely, L1) that it had at r1 before the interchange; similarly, the fluid at r1, after the interchange, will have the same angular momentum (namely, L2) that it had at r2. As a result, the change in the kinetic energy (or, what is the same thing, the change in the centrifugal potential energy) is proportional to
Remembering that r2 > r1, we observe that this is positive or negative according as L22 is greater than L12 or less than L12. Consequently, if L2 is a monotonic increasing function of r, no interchange of fluid rings such as we have imagined can occur without a source of energy, and this means stability. On the other hand, if L2 should decrease anywhere, then an interchange of fluid rings in this region will result in a liberation of energy, and this means instability.
13.3 Analytical Discussion of the Rayleigh Criterion
While the foregoing arguments of Rayleigh make his criterion a very likely one by drawing attention to its physical origin, one should like to establish it directly from the relevant perturbation equations. These equations are
and the equation of continuity,
In accordance with the general procedure of treating these problems, we analyze the disturbance into normal modes. In the present instance, it is natural to suppose that the various quantities describing the perturbation have a (t,θ,z) dependence given by
where p is a constant (which can be complex), m is an integer (which can be positive, zero, or negative), and k is the wave number of the disturbance in the z direction. Let ur(r), uθ(r), uz(r), and ω(r) now denote the amplitudes of the respective perturbations for which the (t,θ,z) dependence is given by the expression (13.8). Equations (13.4) to (13.7) then give
and
where
For our present discussion we shall restrict our attention to the axi-symmetric case, for which none of the quantities describing the perturbation depend on θ. Then
m = 0 σ = p
and the equations become
and
where
By eliminating ω from Eqs. (13.14) and (13.16), we find that
Now by making use of Eq. (13.15), we find, after some further reductions, that
We must seek solutions of this equation that satisfy the boundary condition
The problem presented by Eq. (13.18) and the boundary conditions (13.19) constitutes a characteristic-value problem. This problem is in fact of the classical Sturm-Liouville type,4 and by appealing to the standard theorems of the subject, we can reach the following conclusion:
The characteristic values of k2/p2 are all positive if Φ(r) is everywhere positive, and they are all negative if Φ(r) is everywhere negative. If Φ(r) should change sign anywhere in the interval (R1,R2), then there are two sets of real characteristic values that have the limit points + ∞ and − ∞.
Since a negative p2 means instability, it is clear that the foregoing statements regarding the sign of the characteristic values of k2/p2 are equivalent to a restatement of Rayleigh’s criterion.
An alternative way of establishing the same results is instructive. Multiply Eq. (13.18) by rur and integrate over the range of r. The integral on the left-hand side can be transformed by an integration by parts. We obtain
(say). From Eq. (13.21), it is immediately apparent that p2/k2 is positive if Φ(r) is everywhere positive, and is negative if Φ(r) is everywhere negative. The further result, that there exist unstable modes if Φ(r) is anywhere negative, can be deduced from the fact that we may regard Eq. (13.18) as the condition that
is a maximum, or a minimum, for a given
The ratio p2/k2 is then the value of I1/I2. If Φ(r) is negative anywhere, then I1 admits a negative value and therefore a negative minimum, so that one value at least of p2 is negative, and one mode of disturbance is unstable.
In deriving Rayleigh’s criterion for axisymmetric perturbations, we have not, of course, solved the mathematical problems completely: The complete solution will require a detailed consideration of the general nonaxisymmetric modes distinguished by m. It may be surprising, but it is true, that up to the present this more general problem has not been adequately solved. The reason is that in the general case the characteristic-value parameter occurs nonlinearly in the problem. It nevertheless appears possible that, by inverting the problem in the sense of regarding k2 as the characteristic-value parameter, some progress can be made. For example, one can show that, considered from this latter point of view, the general nonaxisymmetric problem allows a variational formulation exactly as in the axisymmetric case. In fact, if p is real then one finds that k2, given by the formula
together with the constraint
and the boundary condition
has an extremal character when the proper solutions of the problem are inserted in it. The problem of deriving Rayleigh’s criterion from Eq. (13.24) seems, however, to be a very delicate matter. But one conclusion is obvious: If Φ(r) is everywhere negative, then k2 cannot admit a positive characteristic value. For real p’s, the characteristic values of k are necessarily imaginary. Therefore for real k’s, p must necessarily be complex, and this means instability.
13.4 The Stability of Viscous Rotational Flow
We now turn to the question of how the inclusion of viscosity modifies the physical and the mathematical aspects of the problem. It is remarkable that, no matter how small viscosity may be, an arbitrary distribution of the rotational velocity (permitted when viscosity is exactly zero) is not consistent with the Navier-Stokes equations of hydrodynamics. Instead of Ω(r) being an arbitrary function, the allowed solution degenerates into a two-parameter family. The two parameters may, in fact, be taken as the angular velocities of the two cylinders between which the fluid is confined; that this should be so is physically understandable, since an experimenter can choose to rotate the two cylinders with different angular velocities at his discretion. It is, however, not under his discretion to distribute the angular velocities inside the fluid once he has decided to rotate the two cylinders at some assigned speeds; in other words, given the angular velocities of rotation of the two cylinders, the distribution of the angular velocity inside the fluid is uniquely determined. This is the content of the formula
which follows from the Navier-Stokes equations. In Eq. (13.26), A and B are two arbitrary constants; we may relate them to the angular velocities Ω1 and Ω2 with which the inner and the outer cylinders are rotated.
The problem of the stability of viscous flow between rotating cylinders reduces to the following: As the initial stationary flow, we consider one in which A and B have some preassigned values; we then ask whether, other things being equal, there are restrictions on A and B that follow from requirements of stability.
One might formulate the problem of stability in a somewhat different way. A distribution of angular velocity for arbitrarily assigned A and B will not, of course, satisfy Rayleigh’s criterion derived for nonviscous fluids, and one does not expect that in the presence of viscosity a violation of Rayleigh’s criterion would be followed instantly by instability. On general grounds, viscosity will have an inhibiting effect, and we expect to be able to transgress Rayleigh’s criterion to some finite extent before viscosity is unable to prevent the onset of instability. The question is: What is the extent of the transgression we are allowed?
When Ω has the form given by Eq. (13.26), Rayleigh’s discriminant becomes
On expressing A and B in terms of the angular velocities of the two cylinders and measuring r in units of the radius R2 of the outer cylinder, we find
where
Clearly,
so long as
Therefore, Rayleigh’s criterion applied to the distribution (13.26) requires that, for stability, the outer cylinder must rotate with an angular speed greater than η2 times that of the inner cylinder and in the same sense. In the (Ω2,Ω1) plane, the regions of stability are delimited by the positive Ω2 axis and the “Rayleigh line,”
Ω2 = Ω1η2
If the effects of viscosity are ignored, we must have instability to the left of the Rayleigh line; we should like to know how viscosity extends the domain of stability.
Formulated in an entirely general way, the mathematical problem that we must solve in order to establish a criterion for stability applicable under all circumstances is a difficult and complicated one, and it may be said that the complete solution has not yet been found. There is, however, one case that has been solved with relative completeness. This is the case when the gap
d = R2 − R1
between the two cylinders is small compared with the mean radius
½(R2 + R1)
With suitable approximations appropriate for this case, we eventually find that the characteristic-value problem we have to solve is the following:
and
where
and ν denotes the kinetic viscosity of the fluid.
Equations (13.32) and (13.33) must be considered together with the boundary conditions
The characteristic-value problem presented by Eqs. (13.32), (13.33), and (13.36) is typical of a large class of such problems that arise in the theory of hydrodynamic and hydromagnetic stability. For this reason it may be useful to describe how we might solve Eqs. (13.32) and (13.33) by a method that converges rapidly. The method of solution we shall adopt is the following: Since v is required to vanish at z = 0 and 1, we expand it in a sine series of the form
Having chosen v in this manner, we next solve the equation
obtained by inserting the series (13.37) in Eq. (13.32), and arrange that the solution satisfies the four remaining boundary conditions on u. With u determined in this fashion and v given by Eq. (13.37), Eq. (13.33) will lead to a secular equation for T.
When the details of the method described in the preceding paragraphs are carried out, we find that the process of solving the infinite-order characteristic equation for T, by setting the determinant formed by the first n rows and columns equal to zero and letting n take increasingly larger values, converges very rapidly indeed. Thus for α = − 2.5 and a = 5.00, 5.05, and 5.10, the values of T obtained in the third and the fourth approximations (the “order” of the approximation being the order of the determinant that is set equal to zero in the determination of T) are
It is seen that the values of T given in the third and the fourth approximations differ by only four parts in a thousand. The origin of this rapid convergence clearly lies in the splitting of the original equation of order six into a pair of order two and four, respectively, and satisfying the equation of order four exactly. This basic idea underlying the method is capable of extension and application to a wide class of problems.
13.5 On Methods of Solving Characteristic-value Problems in High-order Differential Equations
The methods described in the preceding section, while they are elementary, are adapted to the solution of a large class of problems that arise in problems of hydrodynamic and hydromagnetic stability. The basic idea is the following: When we wish to solve a characteristic-value problem in differential equations of high order, we try to separate the original differential equation and boundary conditions into two systems and seek to solve one of the systems exactly, and it is important that the order of the system we solve exactly is as high as can be managed. This manner of separation is useful even when a variational formulation is possible. In the latter connection, one carries out the variations with subsidiary constraints in the form of equations and boundary conditions, again of as high order as one can manage.
REFERENCES
1. Chandrasekhar, S., On Characteristic Value Problems in High Order Differential Equations Which Arise in Studies on Hydrodynamic and Hydromagnetic Stability, Amer. Math. Monthly, vol. 61 supplement, pp. 32–45, 1954.
2. ——, The Stability of Viscous Flow between Rotating Cylinders, Proc. Roy. Soc. London, Ser. A, vol. 246, pp. 301–311, 1958.
3. —— and W. H. Reid, On the Expansion of Functions Which Satisfy Four Boundary Conditions, Proc. Nat. Acad. Sci. U.S.A., vol. 43, pp. 521–527, 1957.
4. Ince, E. L., “Ordinary Differential Equations,” chap. 10, Longmans, Green & Co., Ltd., London, 1927.
5. Rayleigh, Lord, On the Dynamics of Revolving Fluids, Proc. Roy. Soc. London, Ser. A, vol. 96, pp. 148–154, 1917; also “Scientific Papers,” vol. 6, pp. 447–453, Cambridge University Press, Cambridge, England, 1920.
6. Taylor, G. I., Stability of a Viscous Liquid Contained between Two Rotating Cylinders, Phil. Trans. Roy. Soc. London, Ser. A, vol. 223, pp. 289–343, 1923.