Chandrasekhar's 1953 theory of the linear stability of electrically conducting viscous flow between rotating coaxial cylinders in a uniform axial magnetic field derived its marginal-stability equations under an unproven assumption of the principle of exchange of stabilities (PES) — that instability sets in through a stationary mode of zero temporal frequency — and acknowledged three items for future work: extension to counter-rotating cylinders, inclusion of a pressure gradient, and a rigorous proof of the PES. Later literature closed the first two items (Chandrasekhar and Elbert 1961; Chandrasekhar, Elbert and Lebovitz 1961; Kurzweg 1964; Chang and Sartory 1967) and settled the PES in one regime: Herron (2004) proved it at zero magnetic Prandtl number for insulating walls, co-rotation, and outward-decreasing circulation. What remains open is the rigorous onset-character determination for the eigenproblem the 1953 theory was actually built on — perfectly conducting walls, co-rotating cylinders with circulation decreasing outwards, and magnetic Prandtl number effectively zero: prove the PES where it holds, and rigorously establish whether the oscillatory (overstable) onset that numerical evidence indicates at high Hartmann number for conducting walls actually occurs.

Background

Between two coaxial cylinders rotating at different angular speeds, a viscous fluid can sustain a purely azimuthal flow V(r)=Ar+B/rV(r)=Ar+B/r (Couette flow, the constants AA and BB being fixed by the no-slip conditions at the walls). G. I. Taylor showed in 1923 that when the inner cylinder rotates fast enough this flow is centrifugally unstable: above a critical value of the Taylor number TT (the ratio of centrifugal to viscous forces) axisymmetric toroidal vortices appear. If the fluid is electrically conducting and a uniform magnetic field HH is imposed along the cylinder axis, the Lorentz force acting on the currents induced by any radial motion opposes that motion, so the field inhibits the instability and raises the critical Taylor number. Chandrasekhar (1953) formulated and solved this magnetohydrodynamic (MHD) stability problem for two cylinders rotating in the same direction with a narrow gap d≪R0d\ll R_0 (R0R_0 the mean radius), walls modelled as perfect electrical conductors, and magnetic Prandtl number Pm=ν/ηPm=\nu/\eta (kinematic viscosity ν\nu over magnetic diffusivity η\eta) effectively zero, as in liquid metals where η≫ν\eta\gg\nu. Restricting to axisymmetric disturbances, and replacing the velocity-profile coefficients by their values at the mean radius (the averaged approximation), he reduced the marginal state to an eighth-order characteristic-value problem, which in the narrow-gap limit takes the form [(D2−a2)2+Qa2]2ψ=−Ta2(D2−a2)ψ[(D^2-a^2)^2+Qa^2]^2\psi=-Ta^2(D^2-a^2)\psi for the azimuthal magnetic perturbation amplitude ψ\psi, where aa is the axial wavenumber and Q=μ2H2d2σ/(ρν)Q=\mu^2H^2d^2\sigma/(\rho\nu) is the Chandrasekhar number (the square of the Hartmann number, the ratio of magnetic to viscous forces). The eight boundary conditions are ψ=0\psi=0, Dψ=0D\psi=0, (D2−a2)ψ=0(D^2-a^2)\psi=0 and D[(D2−a2)2+Qa2]ψ=0D[(D^2-a^2)^2+Qa^2]\psi=0 at each wall, ζ=(r−R0)/d=±1/2\zeta=(r-R_0)/d=\pm 1/2. A variational procedure gave the critical Taylor number as a function of QQ, and mercury experiments (Donnelly and Ozima 1960) confirmed the predicted magnetic inhibition. To set up the marginal equations, however, Chandrasekhar assumed the principle of exchange of stabilities (PES): the temporal eigenvalue ω\omega of the linearized disturbance problem is real, so that at marginal stability ω=0\omega=0 and instability sets in as a stationary pattern rather than an oscillatory, overstable wave. For the non-magnetic Taylor problem this question was taken up in Chandrasekhar's (1954) companion paper and in Gandhi (1964); for the MHD problem a proof was lacking, and the 1953 paper acknowledged this, together with the restrictions to co-rotation and to the absence of a pressure gradient, as matters for future work. Subsequent developments closed the geometric extensions: Niblett (1958) treated insulating walls and tested the boundary-condition effect experimentally; Kurzweg (1963) allowed arbitrary magnetic Prandtl number; Chandrasekhar and Elbert (1961, Part II) extended the theory to counter-rotating cylinders, with explicit results for equal and opposite rotation, and Kurzweg (1964) computed critical values for all rotation ratios; Chandrasekhar, Elbert and Lebovitz (1961) solved the pressure-driven counterpart (flow between coaxial cylinders maintained by a pressure gradient acting round the cylinders, i.e. the curved-channel or Dean configuration, with an axial field); Chang and Sartory (1967) found that for perfectly conducting walls at high Hartmann number the onset can occur through oscillatory rather than stationary axisymmetric modes, although the narrow-gap averaged formulation used in 1953 eliminates these modes. The PES itself was proved only in 2004, and only in part: Herron established it at Pm=0Pm=0 for all Chandrasekhar numbers when the cylinders co-rotate with circulation decreasing outwards (the Rayleigh centrifugal-instability condition: the specific angular momentum rV(r)rV(r) decreases with increasing radius) and the walls are insulating, for both finite and narrow gaps. For the conducting-wall eigenproblem of the original 1953 theory, no rigorous onset-character result exists; modern reviews (Rüdiger et al. 2018) treat the hydromagnetic Taylor–Couette spectrum largely as an object of direct numerical computation.

Problem Statement

Determine, by rigorous argument, the character of the marginal mode in the linear stability problem of hydromagnetic Couette flow — an electrically conducting, incompressible viscous fluid between rotating coaxial cylinders in a uniform axial magnetic field, for axisymmetric disturbances — in the regime of this eigenvalue problem that no existing theorem covers. Specifically: (1) The source problem. For perfectly conducting cylinder walls, magnetic Prandtl number tending to zero, and co-rotating cylinders with circulation decreasing outwards, prove or refute that the temporal eigenvalue ω\omega is real at marginality (the principle of exchange of stabilities), so that the marginal equations obtained by setting the time derivatives to zero — the basis of Chandrasekhar's (1953) critical Taylor numbers — are valid; settle this both in the narrow-gap averaged formulation and in the full finite-gap problem. (2) Conducting walls, oscillatory onset. Chang and Sartory (1967) found numerically that at high Hartmann number oscillatory axisymmetric modes can become more critical than the stationary ones: rigorously confirm or refute the existence of oscillatory marginal modes for the perfectly conducting wall problem; a rigorous demonstration of an oscillatory marginal mode more critical than the stationary ones at certified parameter values refutes the principle of exchange of stabilities. The oscillatory-mode question is the refutation direction of item (1), not an independent question: an oscillatory marginal mode more critical than the stationary ones refutes the principle of exchange of stabilities for the full finite-gap problem, whereas a proof of the principle of exchange of stabilities for the full finite-gap problem confirms it; item (1) requires a determination in both of its settings, the narrow-gap averaged formulation and the full finite-gap problem, and neither setting implies the other. Both items concern the configuration of Chandrasekhar's (1953) original theory and carry one common objective: completing the rigorous foundation of the 1953 marginal-stability theory by establishing, for this eigenvalue problem, whether onset is stationary or oscillatory. The already-settled regime — insulating walls, co-rotation, Pm=0Pm=0 (Herron 2004) — is excluded. Extensions to counter-rotating cylinders, to finite magnetic Prandtl number, and to a global delineation of a stationary/oscillatory boundary in multi-parameter space are not part of the source question; the literature on these extensions (for example Chandrasekhar and Elbert 1961; Kurzweg 1963, 1964) remains historical background. The determination of item (1) is to be settled by proof where the PES holds and by a rigorous construction at certified parameter values where it fails; uncontrolled numerical evidence does not settle the question.

References

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