Couette-Taylor instabilities in the small gap regime: the very counter-rotating case

Dongfen Bian; G\'erard Iooss

REVIEW 4 major objections 5 minor 19 references

Couette-Taylor instabilities in the small gap regime: the very counter-rotating case

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For strongly counter-rotating cylinders, the first Taylor-Couette instability becomes non-axisymmetric and is governed by two coupled Ginzburg-Landau equations.

desk verdict A serious first amplitude-equation treatment of the strongly counter-rotating small-gap Couette-Taylor regime, with a real gap: the center-manifold step is delegated, not verified, and the GL coefficients lack convergence data. read the letter →

arxiv 2608.10951 v1 pith:ADCRH3J4 submitted 2026-08-11 math.AP physics.flu-dyn

Dongfen Bian , Gérard Iooss This is my paper · ORCID

classification math.APphysics.flu-dyn MSC 35B3276E0776D05

keywords Couette-Taylorinstabilitysmall-gaplimitcounter-rotatingcylindersnon-axisymmetriccoupledGinzburg-Landauequationshelicoidalwavesribboncenter-manifoldreduction

topics Navier-Stokes Regularity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

For a viscous flow between two cylinders that counter-rotate sufficiently strongly (rotation-rate ratio $\mu$ below $\mu_c \approx -0.8$), the first Couette-Taylor instability is no longer axisymmetric but takes the form of azimuthal travelling waves. The paper shows that in the small-gap, slow-rescaled-rate, high-Reynolds limit, the slow modulation of these left- and right-travelling waves is governed by a system of two coupled complex Ginzburg-Landau equations. It computes every coefficient of this amplitude system, including the cubic nonlinearities, numerically from the linearised eigenfunctions and the adjoint eigenvector. It then determines when helicoidal waves (travelling in both axial and azimuthal directions) and ribbon waves (standing axially, travelling azimuthally) exist and are linearly stable, and identifies a third-order ODE system for more exotic spatially modulated solutions whose classification is left open.

What carries the argument

The central object is the limit Navier-Stokes system (7), obtained in the SGSRRHR limit, together with its two-dimensional critical eigenspace spanned by the eigenfunctions $\zeta_1 = e^{i(\alpha_c z + \beta_c y)} U_1(x)$ and its reflected counterpart $\zeta_2 = S\zeta_1$. The mechanism carrying the argument is the center-manifold reduction of this limit system to the two coupled complex Ginzburg-Landau equations, whose symmetry group forces the normal form (5)-(6). The coefficients $b$ and $c$ are the load-bearing numbers: they are computed from the quadratic interaction operator $B$ and the adjoint eigenvector via the Fredholm-type formulas (40)-(41), and their real parts determine the existence and stability of helicoidal waves (one amplitude nonzero) and ribbon waves (equal amplitudes, standing axially).

What would settle it

Compute the coefficients $b$ and $c$ (or their real parts) for a fixed $\mu$ in $(-0.85, -0.8)$ using an independent numerical method, or run direct Navier-Stokes simulations in the SGSRRHR limit to measure the critical Taylor number and the stability of helicoidal versus ribbon waves; if the signs of $b_r$ and $c_r$ (or the predicted thresholds $\mu_1$, $\mu_2$, $\mu_3$) do not match the paper's predictions, the central claim fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that, in the SGSRRHR regime with $\mu < \mu_c$, small-amplitude solutions of the limit Navier-Stokes system near the critical Taylor number $T_c(\mu)$ are described at leading order by the amplitude equations (5)-(6), namely two coupled complex Ginzburg-Landau equations for the amplitudes $A$ and $B$ of the two non-axisymmetric critical modes. All coefficients of this system—the linear dispersion terms and the cubic coupling coefficients $b$ and $c$—are obtained from the eigenfunctions $\zeta_1, \zeta_2$ and the adjoint eigenfunction via explicit formulas (40)-(41), evaluated by Chebyshev collocation. The reduced system admits helicoidal-wave and ribbon-wave solutions, and their existence and stability are governed by the signs of the real parts $b_r, c_r$, $b_r + c_r$, and $b_r - c_r$, with thresholds at $\mu_1 \approx -0.814$, $\mu_2 \approx -0.848$, and $\mu_3 \approx -0.8$. The paper also reduces the search for stationary spatial modulations to a third-order ODE system (18). If these claims hold, they give a systematic weakly nonlinear description of the non-axisymmetric onset in the small-gap limit.

Load-bearing premise

The load-bearing premise is that the center-manifold reduction to the two coupled Ginzburg-Landau equations faithfully represents the full Navier-Stokes system near onset in the SGSRRHR limit, so that the numerically computed coefficients are the true reduction coefficients and the reduced solutions lift to genuine solutions of the original equations.

Editorial extensions

If this is right

For $\mu < \mu_c \approx -0.8$, the first linear instability is non-axisymmetric, with critical axial wavenumber $\alpha_c(\mu)$ and azimuthal wavenumber $B_c(\mu) > 0$, so the onset branch differs fundamentally from the axisymmetric Taylor-vortex branch.
The coupled Ginzburg-Landau system predicts that helicoidal waves exist and are stable for $\mu_2 < \mu < \mu_1$ (with $\mu_1 \approx -0.814$ and $\mu_2 \approx -0.848$), while ribbon waves are stable for $\mu < \mu_2$, with the intervals determined by the signs of $b_r$, $c_r$, $b_r + c_r$, and $b_r - c_r$.
If the coefficients are correct, the weakly nonlinear behaviour near onset for any fixed $\mu$ in the strongly counter-rotating regime is fully captured by the two-equation amplitude system, so the competition between helicoidal and ribbon states is decided by the ratio $c_r/b_r$.
Provided the lifting to Navier-Stokes is justified via the cited reduction theory, each periodic solution of the reduced system corresponds to a genuine periodic flow in the small-gap limit, giving explicit travelling-wave and standing-wave patterns.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the reduction holds, the same two-mode Ginzburg-Landau structure should appear in other hydrodynamic systems with a reflection symmetry coupling two counter-propagating waves, suggesting the stability criteria derived here are generic rather than specific to the Taylor-Couette geometry.
The threshold values $\mu_1 \approx -0.814$, $\mu_2 \approx -0.848$, and $\mu_3 \approx -0.8$ could be compared with finite-gap experiments or direct numerical simulations at moderate radius ratios; agreement would support the small-gap limit as a quantitative model beyond its formal derivation.
The third-order ODE system (18) is likely to admit spatially periodic or homoclinic orbits that correspond to modulated wave packets; proving or disproving their existence would settle the open classification problem raised in the paper.
A direct check of the paper's coefficients using a different discretization (for instance finite elements or shooting methods) would test the reliability of the Chebyshev-collocation results, since no convergence study in the number of collocation points is reported.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The genuinely new thing here is the first amplitude-equation treatment of the strongly counter-rotating small-gap Couette-Taylor regime, where the primary instability is non-axisymmetric. The authors derive a two-coupled Ginzburg-Landau system, compute its coefficients from the linear eigenfunctions and adjoint problem via Chebyshev collocation, and use the reduced system to map out helicoidal-wave and ribbon-wave stability, plus some exotic solutions governed by a third-order ODE. Appendix 4.4 is a serious piece of coefficient bookkeeping; the symmetries (zeta2 = S zeta1, the quadratic interactions) are handled cleanly, and the sample values at mu = -1 are concrete. The paper is also honest that the exotic solutions are not fully classified.

The soft spots are real but not fatal. The load-bearing issue is the omitted Step 3. The paper says the lift from solutions of (5)-(6) to genuine solutions of the limit Navier-Stokes system can be obtained by repeating [9] 'up to minor changes.' That might be true, but it is not demonstrated. The original result in [9] does not obviously cover two coupled GL equations with both axial and azimuthal modes and y-modulation; the spectral-gap and nonlinear-estimate hypotheses are neither checked nor cited as a ready-made theorem covering this exact case. Until that step is supplied, (5)-(6) is a formally derived envelope system, not a proven reduction. This is exactly what a referee should ask for.

Second, the stability regions in Sections 3.2-3.3 depend on numerically computed coefficients b_r and c_r, and on the crossing values mu1, mu2, mu3. There is no convergence study in N and no error estimate, only a claim of extended precision. That is a moderate issue, not a fatal one; the formulas are explicit and reproducible, but the paper should ship data or a convergence table.

The citation pattern is fine. [1,2] are the same group's prior related results and are cited as such; [9] is a sensible source for the reduction strategy; the numerical work of Deguchi and Nagata is acknowledged.

Who should read it: anyone working on Taylor-Couette amplitude equations or on formal center-manifold reductions in hydrodynamic stability. It deserves a serious referee, but the referee should press on the reduction step and the numerics. My verdict would be conditional acceptance: require the authors to verify that [9] applies, or state precisely which theorem does, and add a convergence study for the coefficients.

Referee Report

4 major / 5 minor

Summary. The paper studies the Couette-Taylor instability in the small-gap, slow-rotation, high-Reynolds limit for strongly counter-rotating cylinders (mu < mu_c ~ -0.8), where the primary instability is non-axisymmetric. From the Navier-Stokes equations the authors recall the limit system (7), perform its linear stability analysis, and derive a system of two coupled complex Ginzburg-Landau equations, (5)-(6), for the amplitudes of the left- and right-travelling azimuthal waves. All linear and cubic coefficients are expressed through eigenfunctions and the adjoint problem, and the cubic coefficients b and c are evaluated numerically by Chebyshev collocation. The reduced equations are then used to discuss existence and stability of helicoidal and ribbon waves and to derive a third-order ODE system for spatially modulated solutions. The main new content is the two-amplitude Ginzburg-Landau description, the explicit coefficient formulas in Appendix 4.4, and the resulting stability diagram based on the signs of b_r, c_r and b_r+c_r.

Significance. If the reduction is valid, the paper provides a systematic weakly nonlinear description in a regime that previous small-gap analyses did not cover, and it gives explicit, checkable formulas for the amplitude-equation coefficients in terms of the linearized eigenfunctions and adjoint problem. The stability criteria for helicoidal and ribbon waves are presented transparently, and the reduction to a third-order ODE for spatially modulated states is a useful starting point. The paper is nevertheless conditional in an essential way: the step that lifts solutions of the Ginzburg-Landau system to genuine solutions of the limit Navier-Stokes system is delegated to reference [9] and not proved, and the numerical coefficients that determine the stability thresholds are reported without a convergence study. These two issues are load-bearing for the paper's central claims.

major comments (4)

[Section 1, Step 3] The central claim that the Ginzburg-Landau system (5)-(6) describes the limit Navier-Stokes system (7) rests on the omitted Step 3. The sentence 'This last step can be done by repeating the proofs in [9], up to minor changes' is not sufficient as it stands: the manuscript does not verify that the hypotheses of the reduction theory in [9] hold for the present problem, which is time-dependent, has two coupled complex amplitudes, and is posed on the unbounded y-axis where the linearized spectrum is not gapped when modulations are allowed. Please either provide a reduction theorem for (7) with the spectral-gap and nonlinearity hypotheses stated and verified, or explicitly restrict the claims to the formally derived amplitude system. As written, the existence and stability statements for helicoidal and ribbon waves in Sections 3.2 and 3.3 are theorems only about the reduced system unless Step 3 is supplied.
[Section 4.4.3 and Figure 1] The stability thresholds mu_1, mu_2, mu_3 and the signs of b_r, c_r and b_r+c_r are computed numerically, but the paper contains no convergence study in the number N of Chebyshev collocation points, no error estimates, and only one reported numerical value (b and c at mu=-1). The statement that extended precision guarantees well-converged results is not a substitute. Please include a table of b and c for increasing N at representative values of mu, and preferably an independent check of the crossing points, so that the stability diagram in Figure 1 is reproducible and the thresholds are credible.
[Section 4.3, expansion near (29)] The perturbation expansion of U lists U_0, U_100, U_010, U_001, U_110, U_011 and U_020, but the coefficient U_200 is never introduced; nevertheless U_200 appears in the displayed equation for a_4. This makes the formula for a_4 unverifiable as written. Please include the (B-B_c)^2 term in the expansion and define U_200, or restructure the calculation so that every coefficient used is defined.
[Section 4.4.1, computation of Phi_0011] The text states Phi_0011 = S Phi_1100 and then concludes Phi_0011 = (0, phi_1100_y, -phi_1100_z)^t = Phi_1100. Since phi_1100_y is real and phi_1100_z is pure imaginary, the vector (0, phi_1100_y, -phi_1100_z) is the complex conjugate of Phi_1100, not Phi_1100 itself. If the intended identity is Phi_0011 = conjugate of Phi_1100, this should be stated explicitly, because the sign convention enters the c-coefficient formula (41).

minor comments (5)

[Sections 2.2, 3.1, 3.2 and Theorem 1] The notation for the azimuthal wavenumber is overloaded: B is the rescaled variable B=beta R, B_c is the critical value of B, while zeta_1 contains e^{i beta_c y}; later beta is used for a modulation wavenumber in either y or tilde y. Please make the relation between beta, B, R, y and tilde y consistent, especially in Theorem 1 and equation (14), where y and tilde y appear to be used interchangeably.
[Throughout] The value of mu_c is given as approximately -0.8 in the abstract and introduction and -0.785 in Section 3.2; please unify the values and state the numerical precision.
[Figure 1] The caption describes crossings at mu_3, mu_1 and mu_2, but the figure does not show markers or a table of the computed b_r(mu), c_r(mu) values. Adding a small table or markers would substantially improve verifiability.
[Section 4.3 and Appendix 4.4] The scalar product (22) involves only the x and y components; the paper should state explicitly that the z-component is eliminated through the divergence-free condition, and that the nonzero normalization <zeta_1, zeta_1^*> is checked numerically.
[Section 3.3] There are typos such as 'exist on the super-critical size' which should read 'side'; similar small typographical errors occur elsewhere and should be corrected.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: coefficients are obtained by spectral projection; the only flagged issue is the delegated Step 3, a proof gap rather than a circular reduction.

full rationale

The load-bearing coefficient computation is self-contained: Section 4.3 gives the linear dispersion coefficients from the eigenproblem (24), the adjoint (30) and the solvability conditions, and Section 4.4 computes the cubic coefficients b and c by solving forced linear systems for the harmonics Φ2000,...,Φ1001 and projecting the quadratic interactions onto the adjoint eigenvector via formulas (38)-(41). These are direct spectral projections, not fits: no stability conclusion is used as an input, and the linear part of the GL system (9) is intentionally constrained to reproduce λ0, which is a construction condition rather than a prediction. The helicoidal/ribbon wave existence and stability then follow from the ODE system (10)-(13) using the independently computed signs of b_r and c_r. The one caveat is the explicit omission of Step 3: 'This last step can be done by repeating the proofs in [9], up to minor changes. We will omit this step here and refer to [9] for more details.' I weigh this as a proof gap and a self-citation, not as circularity: [9] is a general spatial-dynamics theorem with independent content, and no equation in the paper is equivalent to its own input by construction. The numerical convergence caveat for the Chebyshev evaluation is likewise a rigor issue, not circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities, forces, or fields. Its axioms are the standard assumptions of asymptotic reduction and numerical discretization. The free parameters are numerical crossover values and the discretization parameter. The central claim rests on the validity of the limit system from prior work and on the applicability of the Ginzburg-Landau approximation, both of which are reasonable domain assumptions but not proven here.

free parameters (3)

µc (critical rotation ratio) = ≈ -0.8 (numerically observed)
The value µ_c ≈ -0.8 marks the transition between axisymmetric and non-axisymmetric primary instability. It is determined numerically from the linear instability analysis and is an input for the paper's analysis, not derived analytically.
µ1, µ2, µ3 (crossing values of br, cr) = ≈ -0.814, ≈ -0.848, ≈ -0.8 respectively
These values are inferred from the numerical evaluation of the cubic coefficients as functions of µ. They determine the stability and existence regimes of helicoidal and ribbon waves.
Number of Chebyshev collocation points N = not specified
The numerical spectral method relies on a discretization parameter N; the paper does not state the value used, nor provide a convergence study. This is an implicit free parameter of the numerical procedure.

assumptions (5)

domain assumption The limit Navier-Stokes system (7) is the correct leading-order approximation of the full Navier-Stokes equations in the SGSRRHR limit.
The derivation is only recalled from previous work [1,2] and its main steps are stated in Appendix 4.1. The validity of this asymptotic reduction is assumed, and it is the foundation on which the rest of the analysis rests.
domain assumption The center-manifold reduction and the Ginzburg-Landau equation approximation are applicable, including the existence of a spectral gap and normal form properties as in [9].
The paper relies on the theory in [9] (Iooss, Mielke, Demay) and states that the step of constructing genuine Navier-Stokes solutions from reduced equation solutions can be done by repeating proofs in [9]. This is a substantive assumption about the validity of the reduction in this specific non-axisymmetric setting.
domain assumption All eigenvalues of the linearized problem except the critical pair have negative real parts bounded away from zero uniformly in the distinguished limit.
This spectral gap condition is needed for the center-manifold reduction. The paper does not prove this, it only numerically observes that there exists a critical Taylor number and a single eigenmode that becomes unstable.
ad hoc to paper The numerical spectral method (Chebyshev collocation) converges and produces accurate values of the coefficients b and c.
The paper says 'All computations are performed with extended precision to guarantee well-converged results' but does not give explicit error bounds or convergence checks. The validity of the stability conclusions depends on the accuracy of the computed signs of br, cr, br+cr, br-cr.
domain assumption The symmetries of the rotating frame and the z-reflection S are preserved in the reduced system and are the only relevant symmetries for the bifurcation.
The Ginzburg-Landau system is restricted to forms commuting with phase rotations and the swap symmetry. It is assumed that no other modes or symmetries participate in the bifurcation.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Couette-Taylor instabilities in the small gap regime: the very counter-rotating case." pith.science (2026). https://pith.science/paper/ADCRH3J4

@misc{pith2026260810951,
  author       = {Pith},
  title        = {Pith review of: Couette-Taylor instabilities in the small gap regime: the very counter-rotating case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ADCRH3J4}},
  note         = {Machine review of arXiv:2608.10951}
}

abstract

In this paper, we study the Couette-Taylor instability of a viscous fluid between two rotating cylinders in the small-gap, slow rescaled rotation rate, high Reynolds number regime, focusing on the very counter-rotating case $\mu < \mu_c \approx -0.8$ where the primary instability is non-axisymmetric. Starting from the Navier-Stokes equations, we derive a limit system that captures the leading-order dynamics and compute the critical Taylor number $T_c(\mu)$ together with the critical axial and azimuthal wavenumbers. Near criticality, the weakly nonlinear behaviour is governed by a system of two coupled complex Ginzburg-Landau equations. All coefficients of this amplitude system including the cubic nonlinear terms are evaluated numerically from the linearised eigenfunctions and the associated adjoint problem. The reduced equations admit helicoidal waves (travelling in both the axial and azimuthal directions) and ribbon waves (standing axially, travelling azimuthally), and their existence and stability criteria are discussed. We also examine more exotic spatially modulated solutions that satisfy a third-order dynamical system, whose complete classification remains an open challenge.

Figures

Figures reproduced from arXiv: 2608.10951 by the authors.

**Figure 1.** Evolution of the point (br, cr) as a function of µ from µ = µc to µ = −1 (from right to left) together with the diagonals br = cr and br = −cr. The blue curve starts on the upper right part of the figure at µ = µc and successively crosses cr = −br at µ = µ3, br = 0 at µ = µ1 and cr = br at µ = µ2. It ends on the left part of the figure at µ = −1. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

**Figure 2.** Numerical study of the critical point as a function of [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[9]

Iooss, A

G. Iooss, A. Mielke, Y. Demay, Theory of steady Ginzburg-Landau equation in hydrodynamic stability problems, Europ.J. Mech. B/Fluids8 , 3, 229-268, 1989

work page 1989
[1]

D. Bian, E. Grenier, G. Iooss, Z. Yang, Couette-Taylor instabilities in the small gap regime, arXiv:2601.14806, 2026. 27

work page arXiv 2026
[2]

D. Bian, E. Grenier, G. Iooss, Z. Yang, Couette-Taylor instabilities for counter-rotating cylinders in the small gap regime, arXiv:2603.19704, 2026

work page arXiv 2026
[3]

Chossat, G

P. Chossat, G. Iooss, The Couette-Taylor problem,Applied Mathematical Sciences, Vol. 102, Springer, 1994

work page 1994
[4]

Davey, The growth of Taylor vortices in flow between rotating cylinders, J

A. Davey, The growth of Taylor vortices in flow between rotating cylinders, J. Fluid Mech., 14, 336–368, 1962

work page 1962
[5]

Deguchi, M

K. Deguchi, M. Nagata, Bifurcations and instabilities of Taylor-Couette flow in the counter-rotating regime,J. Fluid Mech., 678, 156–183, 2011

work page 2011
[6]

DiPrima, P.M

R.C. DiPrima, P.M. Eagles, Amplification rates and torques for Taylor- vortex flows between rotating cylinders,Phys. Fluids, 20, 171–175, 1977

work page 1977
[7]

Drazin, W.H

P.G. Drazin, W.H. Reid, Hydrodynamic stability,Cambridge University Press, 1981

work page 1981

Show all 19 references

[8]

Haragus, G

M. Haragus, G. Iooss, Local bifurcations, centre manifolds, and normal forms in infinite dimensional dynamical systems,Springer, 2011

2011
[10]

Meseguer, F

A. Meseguer, F. Marques, On the competition between axisymmetric and non-axisymmetric instability modes in Taylor–Couette flow,J. Fluid Mech., 452, 63–108, 2002

2002
[11]

Meseguer, F

A. Meseguer, F. Mellibovsky, M. Avila, F. Marques, Instability mechanisms and transition scenarios in the Taylor-Couette system,Eur. Phys. J. Special Topics, 146, 249–268, 2007

2007
[12]

Nagata, Bifurcations in Couette flow between almost co-rotating cylin- ders,J

M. Nagata, Bifurcations in Couette flow between almost co-rotating cylin- ders,J. Fluid Mech.169, 229-250, 1986

1986
[13]

Nagata, Taylor-Couette flow in the narrow-gap limit,Phil

M. Nagata, Taylor-Couette flow in the narrow-gap limit,Phil. Trans. R. Soc. A., 381, 2023

2023
[14]

Nagata, Taylor-Couette system in the narrow-gap limit, revisited, with a corrigendum to the paper by Nagata (2023, Philosophical Transaction A.), Phil

M. Nagata, Taylor-Couette system in the narrow-gap limit, revisited, with a corrigendum to the paper by Nagata (2023, Philosophical Transaction A.), Phil. Trans. R. Soc. A.480, 2024

2023
[15]

Pellew, R.V

A. Pellew, R.V. Southwell, On maintained convection motion in a fluid heated from below,Proc. Roy. Soc. A, 176, 312-343, 1940

1940
[16]

Schneider, Global existence results for pattern forming processes in in- finite cylindrical domains - Application to 3D Navier-Stokes problems,J

G. Schneider, Global existence results for pattern forming processes in in- finite cylindrical domains - Application to 3D Navier-Stokes problems,J. Math. Pure Appl.78, 265-312, 1999. 28

1999
[17]

Taylor, Stability of a viscous liquid contained between two rotating cylinders,Phil

G.I. Taylor, Stability of a viscous liquid contained between two rotating cylinders,Phil. Trans. R. Soc. Lond. A, 223, 289–343, 1923

1923
[18]

Yudovich, Secondary flow and fluid instability between rotating cylin- ders.J.Appl

V.I. Yudovich, Secondary flow and fluid instability between rotating cylin- ders.J.Appl. Math. Mech, 30, 4, 822-833, 1966

1966
[19]

Yudovich

V.I. Yudovich. Free convection and bifurcation.Prikl. Mat. Meh.31, 101- 111 (Russian); translated asJ. Appl. Math. Mech.31, 103-114, 1967 29

1967

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [9]

Iooss, A

G. Iooss, A. Mielke, Y. Demay, Theory of steady Ginzburg-Landau equation in hydrodynamic stability problems, Europ.J. Mech. B/Fluids8 , 3, 229-268, 1989

work page 1989

[2] [1]

D. Bian, E. Grenier, G. Iooss, Z. Yang, Couette-Taylor instabilities in the small gap regime, arXiv:2601.14806, 2026. 27

work page arXiv 2026

[3] [2]

D. Bian, E. Grenier, G. Iooss, Z. Yang, Couette-Taylor instabilities for counter-rotating cylinders in the small gap regime, arXiv:2603.19704, 2026

work page arXiv 2026

[4] [3]

Chossat, G

P. Chossat, G. Iooss, The Couette-Taylor problem,Applied Mathematical Sciences, Vol. 102, Springer, 1994

work page 1994

[5] [4]

Davey, The growth of Taylor vortices in flow between rotating cylinders, J

A. Davey, The growth of Taylor vortices in flow between rotating cylinders, J. Fluid Mech., 14, 336–368, 1962

work page 1962

[6] [5]

Deguchi, M

K. Deguchi, M. Nagata, Bifurcations and instabilities of Taylor-Couette flow in the counter-rotating regime,J. Fluid Mech., 678, 156–183, 2011

work page 2011

[7] [6]

DiPrima, P.M

R.C. DiPrima, P.M. Eagles, Amplification rates and torques for Taylor- vortex flows between rotating cylinders,Phys. Fluids, 20, 171–175, 1977

work page 1977

[8] [7]

Drazin, W.H

P.G. Drazin, W.H. Reid, Hydrodynamic stability,Cambridge University Press, 1981

work page 1981