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arxiv: 2512.00404 · v2 · submitted 2025-11-29 · ⚛️ physics.flu-dyn · cond-mat.soft· math-ph· math.MP

Linearized instability of Couette flow in stress-power law fluids

Krishna Kaushik Yanamundra (1) , Lorenzo Fusi (2) ((1) Department of Mechanical Engineering , Texas A&M University , College Station , TX , USA , (2) Dipartimento di Matematica e Informatica "U. Dini'' , Universit\`a degli Studi di Firenze
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50134 Firenze Italy)
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Pith reviewed 2026-05-17 03:48 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.softmath-phmath.MP
keywords Couette flowstress-power law fluidslinearized stabilitynon-monotonic constitutive curveascending and descending branchesboundary conditionsfluid instability
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The pith

In stress-power law fluids, plane Couette flow is stable on ascending branches of the constitutive curve and unconditionally unstable on the descending branch.

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

The paper examines linearized stability for plane Couette flow in fluids whose stress-strain rate relation is non-monotonic. These fluids arise from a thermodynamic model with a non-convex dissipation rate that permits three steady states when both plates move at fixed speeds. Linear analysis shows that the two base flows on the ascending parts of the curve remain stable to small disturbances, while the base flow on the descending part grows unstable. When one boundary is traction instead of velocity the base state becomes unique and its stability follows the same ascending-stable or descending-unstable rule. The result ties the choice of boundary conditions directly to whether the flow stays steady or breaks down.

Core claim

Under velocity boundary conditions the system admits three steady-state solutions. Linearized stability analysis reveals that the two solutions on ascending constitutive branches are unconditionally stable, while the solution on the descending branch is unconditionally unstable. For mixed traction-velocity boundary conditions, the base state is unique. Stability depends solely on whether the prescribed traction lies on an ascending (stable) or descending (unstable) branch of the constitutive curve.

What carries the argument

The non-convex rate of dissipation potential that produces the non-monotonic stress-power law constitutive relation and its ascending and descending branches.

If this is right

  • Under fixed-velocity boundaries three steady Couette states exist, with only the descending-branch state unstable.
  • Mixed traction-velocity boundaries produce a single base state whose stability is fixed by its location on the constitutive curve.
  • Flow stability is controlled by the interaction between the imposed boundary conditions and the non-monotonic shape of the material response.
  • Perturbations decay on ascending branches and grow on the descending branch.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same branch-dependent stability rule may apply to pressure-driven channel flows of the same fluids.
  • Selecting wall traction on an ascending branch could be used to maintain steady flow in processing applications.
  • Direct measurement of perturbation growth rates on the descending branch would provide a quantitative test of the linear analysis.
  • Similar stability switches may appear in other materials whose constitutive curves are non-monotonic, such as certain concentrated suspensions.

Load-bearing premise

The model permits three distinct steady flows for the same boundary speeds and treats small disturbances as evolving according to linear equations.

What would settle it

An experiment that drives a stress-power law fluid at a wall speed corresponding to the descending branch and records whether small velocity perturbations grow in time or decay.

Figures

Figures reproduced from arXiv: 2512.00404 by (2) Dipartimento di Matematica e Informatica "U. Dini'', 50134, College Station, Firenze, Italy), Krishna Kaushik Yanamundra (1), Lorenzo Fusi (2) ((1) Department of Mechanical Engineering, Texas A&M University, TX, Universit\`a degli Studi di Firenze, USA.

Figure 1
Figure 1. Figure 1: Constitutive response of the generalized stress p [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the function F(c1) defined in (4.12) with a = 0.032, b = 1, Γ = 10−3 , n = −1.2 and for various Re. The range in which 3 solutions occur is (Rem, ReM), with Rem = 23.2 and ReM = 42.2. From Figs. 2 and 3, it is evident that the solution corresponding to the first ascending branch can exist only for (Rel , Reu) belonging to the region (a) ∪ (b). On the other hand, the solution corresponding to the th… view at source ↗
Figure 3
Figure 3. Figure 3: Regions of existence of the three solutions. [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: mi(Rel , Reu) corresponding to the three base states v (b,i) x (y), i = 1, 2, 3. States (i) and (iii), lying on the ascending branches of the constitutive curve, are unconditionally stable, whereas state (ii), associated with the descending branch, is unconditionally unstable. not infinite). This implies that the demarcation lines between regions (a), (b), and (c) do not represent “true” marginal stability… view at source ↗
Figure 5
Figure 5. Figure 5: Plot of the functions m1, m3 versus (i) Re with k = 1 and (ii) k with Re = 32. more stable than the first one. Finally, in Figs. 6(i), 6(ii) we show the streamlines R(ψ˜) = const of the perturbation defined in (5.30) k = 1, 6, Re = 80 (third solution). We observe that the increase of k results in a stronger shearing of the fluid. (i) k = 1, Re = 80 (third solution). (ii) k = 6, Re = 80 (third solution) [P… view at source ↗
Figure 6
Figure 6. Figure 6: Streamlines R(ψ˜) = const for two representative wavenumbers, computed for the third (stable) base state with parameters a = 0.032, b = 1, Γ = 10−3 , n = −1.2. 7.2. Case 2: Mixed Boundary Conditions Let us now consider the case in which a stress is prescribed on the upper wall, and velocity is imposed on the lower. The stress in the fluid layer is constant and equal to the one imposed on the upper wall. Th… view at source ↗
Figure 7
Figure 7. Figure 7: v (b) x,y vs s (b) xy , eq. (4.4). The material parameters are n = −1.2, a = 0.032, b = 1, Γ = 10−3 . Rel = 0, Reu = 10, 35, 180. The coordinates of the dot represent the values of the shear stress and the corresponding velocity gradient. In [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Stability diagram. Stress Reu/Re applied on the top surface. stability/instability regions are symmetric with respect to the line Reu = 0, which allows us to limit our analysis to the portion Reu > 0. From the figure it can be seen that, regardless of the velocity imposed on the lower wall, there exist two critical values Re1 u , Re2 u (Re1 u = 22, Re2 u = 101) such that, for Reu > 0 outside the interval [… view at source ↗
Figure 9
Figure 9. Figure 9: Function m(Rel , Reu; k; v (b) x ) as a function of Rel and k with Reu = 15, 125, n = −1.2, a = 0.032, b = 1, Γ = 10−3 . (i) Streamlines for k = 1. (ii) Streamlines for k = 6 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Streamlines R(ψ˜) = const for two choices of the wavenumber k (Re = 15, Rel = 0, a = 0.032, b = 1, Γ = 10−3 , n = −1.2). 8. Conclusion In this work we derived the generalized stress power law model from the thermodynamic framework of Rajagopal and Srinivasa by constructing a rate of dissipation potential that is non-convex for a range of parameters. This approach provides a natural thermodynamic origin fo… view at source ↗
read the original abstract

This paper examines the linearized stability of plane Couette flow for stress-power law fluids, which exhibit non-monotonic stress-strain rate behavior. The constitutive model is derived from a thermodynamic framework using a non-convex rate of dissipation potential. Under velocity boundary conditions, the system may admit three steady-state solutions. Linearized stability analysis reveals that the two solutions on ascending constitutive branches are unconditionally stable, while the solution on the descending branch is unconditionally unstable. For mixed traction-velocity boundary conditions, the base state is unique. Stability depends solely on whether the prescribed traction lies on an ascending (stable) or descending (unstable) branch of the constitutive curve. The results demonstrate that flow stability in these complex fluids is fundamentally governed by both boundary conditions and constitutive non-monotonicity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a linearized stability analysis of plane Couette flow for stress-power law fluids with non-monotonic constitutive behavior arising from a non-convex dissipation potential. Under velocity boundary conditions, three steady states are possible, with stability depending on the branch: stable on ascending, unstable on descending. For mixed boundary conditions, stability is determined by the branch of the prescribed traction.

Significance. This work contributes to understanding stability in complex fluids by linking it directly to constitutive non-monotonicity and boundary conditions. The thermodynamic derivation of the model is a positive aspect, providing a principled basis for the multi-valued response. If the linearization is accurate, the results offer falsifiable predictions for flow behavior in such materials.

major comments (2)
  1. [Constitutive linearization] The step where the constitutive relation is linearized around each base state using the derivative from the non-convex potential is critical. The manuscript should provide the explicit form of the linearized operator and confirm that the effective viscosity is negative on the descending branch to support the unconditional instability claim. Without this, the central stability conclusions rest on an unverified assumption.
  2. [Eigenvalue problem] In the stability analysis section, the linearized equations lead to an eigenvalue problem; the paper should specify how the spectral properties are determined (analytically or numerically) to establish unconditional stability or instability.
minor comments (2)
  1. [Abstract] Consider adding a sentence on the key mathematical approach, such as the form of the perturbation equations.
  2. [Notation] Ensure consistent use of symbols for stress, strain rate, and branches throughout the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive suggestions. We address each major comment point by point below. The requested clarifications strengthen the presentation of our analytical results without altering the core findings.

read point-by-point responses

  1. Referee: [Constitutive linearization] The step where the constitutive relation is linearized around each base state using the derivative from the non-convex potential is critical. The manuscript should provide the explicit form of the linearized operator and confirm that the effective viscosity is negative on the descending branch to support the unconditional instability claim. Without this, the central stability conclusions rest on an unverified assumption.

    Authors: We agree that explicit presentation of the linearization step improves clarity. In the revised manuscript we have added the explicit form of the linearized constitutive operator in the stability analysis section: the perturbation stress is related to the perturbation strain rate by the local slope of the constitutive curve evaluated at the base state. This slope (effective viscosity) is positive on ascending branches and negative on the descending branch, as required by the non-convex dissipation potential. The negative sign directly produces a positive growth rate for all wavenumbers, establishing unconditional instability. A short derivation confirming the sign and its consequences has been inserted. revision: yes

  2. Referee: [Eigenvalue problem] In the stability analysis section, the linearized equations lead to an eigenvalue problem; the paper should specify how the spectral properties are determined (analytically or numerically) to establish unconditional stability or instability.

    Authors: The spectral properties are obtained analytically. After linearization, the system reduces to a simple algebraic eigenvalue problem for the growth rate whose real part has the same sign as the effective viscosity (negative for instability on the descending branch, positive for stability on ascending branches). No numerical discretization or solution of a differential eigenvalue problem is required. We have revised the relevant section to state this reduction explicitly and to display the closed-form eigenvalue expression for both velocity and mixed boundary conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; stability follows from direct linearization of the given constitutive model

full rationale

The derivation begins with a constitutive relation obtained from a non-convex dissipation potential, identifies the three uniform base states under velocity BCs, and then applies standard linearized perturbation equations whose spectral properties are computed from the branch-specific derivatives. No step reduces a claimed prediction to a fitted parameter or self-referential definition; the stability classification (ascending branches stable, descending unstable) is an output of the eigenvalue analysis rather than an input. No load-bearing self-citation, uniqueness theorem imported from the same authors, or ansatz smuggled via prior work is present in the provided chain. The analysis is self-contained against the model equations and does not rename a known empirical pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on a thermodynamically motivated constitutive model whose non-convex dissipation potential is taken as given; no explicit free parameters or new entities are named in the abstract.

axioms (1)
  • domain assumption The rate of dissipation potential is non-convex, producing non-monotonic stress-strain rate response.
    This assumption is invoked to generate the three possible steady states and the ascending/descending branches.

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