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REVIEW 3 major objections 4 minor 38 references

Stability of slip channel flow revisited

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read With sufficiently large slip, both streamwise and spanwise wall slip switch the leading instability of channel flow to three-dimensional modes; spanwise slip can reduce the critical Reynolds number to about 336, while streamwise slip…

desk verdict The 3-D instability results are interesting and likely right, but the paper omits the derivation of the one boundary condition everything new depends on, so it needs a serious referee and an independent check. read the letter →

arxiv 1908.02027 v1 pith:GSJ754N3 submitted 2019-08-06 physics.flu-dyn

Chunshuo Chai , Baofang Song This is my paper · ORCID
classification physics.flu-dyn
keywords slipchannelflowlinearstabilityNavierboundaryconditionthree-dimensionalinstabilitycriticalReynoldsnumbernon-modaltransientgrowthanisotropicvelocity-vorticityformulation
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

This paper revisits the linear and non-modal stability of pressure-driven channel flow with tangential velocity slip at the walls, extending earlier analyses that focused on small slip and on spanwise-invariant two-dimensional modes. It claims that once the slip length is large enough, both streamwise slip ($\lambda_x$) and spanwise slip ($\lambda_z$) make three-dimensional oblique modes the leading instabilities, and the two directions act oppositely: streamwise slip only modestly raises the critical Reynolds number (about 5900 at $\lambda_x=0.05$ and 6280 at $\lambda_x=0.2$, with a small slip range where it falls below the no-slip value 5772), whereas spanwise slip lowers it dramatically (about 394 at $\lambda_z=0.2$ and 336 at $\lambda_z=0.25$). Streamwise slip suppresses non-modal transient growth, while spanwise slip enlarges it and shifts the optimal perturbation from streamwise rolls to tilted, long-wavelength structures. If correct, these results mean that anisotropic slip—rather than slip itself—can trigger earlier instability and stronger transient growth, which matters for flows over superhydrophobic surfaces where effective slip lengths can be large.

What carries the argument

The central object that carries the argument is the velocity–vorticity form of the linearized incompressible Navier–Stokes equations, in which the wall-normal velocity $u_y$ satisfies a fourth-order Orr–Sommerfeld-type equation and the wall-normal vorticity $\eta=\partial u_x/\partial z-\partial u_z/\partial x$ satisfies a second-order equation, coupled at the walls through the Navier slip condition. This formulation lets the authors build a spectral Fourier–Chebyshev eigenvalue problem and scan the four-parameter space $(Re,\alpha,\beta,\lambda)$ to locate the first unstable mode. For non-modal growth, they use an adjoint-based time-stepping method in primitive variables, iterated with a Krylov subspace solver, which gives the maximum transient growth $G(t)$ and the optimal perturbation. The key conceptual distinction is two-dimensional ($\beta=0$) versus three-dimensional modes, because the earlier conclusion that slip stabilizes the flow came from considering only the two-dimensional modes.

What would settle it

Run an independent primitive-variable linear stability calculation (solving the linearized Navier–Stokes equations directly with the Navier slip condition, without the velocity–vorticity $\eta$ boundary condition) for $\lambda_z=0.2$ at $Re=400$: if no mode with streamwise wavenumber $\alpha\simeq0.6$ and spanwise wavenumber $\beta\simeq1.27$ is unstable, the paper's $\eta$ boundary condition is incorrect; if the mode is reproduced, the claim is confirmed.

Watch

Extended reading notes

Core claim

The central discovery is that the linear stability of slip channel flow is controlled by the direction of slip. Using the velocity–vorticity formulation of the linearized Navier–Stokes equations with the Navier slip boundary condition at both walls, the paper computes eigenvalues over the streamwise and spanwise wavenumbers $(\alpha,\beta)$ for a wider range of slip lengths than earlier work. For streamwise slip ($\lambda_x$), two-dimensional modes ($\beta=0$) do become more stable as $\lambda_x$ grows, but above $\lambda_x\simeq 0.008$ three-dimensional modes are the most dangerous, so the critical Reynolds number stays near the no-slip value $Re_{cr}\simeq 5772$, reaching about 6280 at $\lambda_x=0.2$ and even dipping below 5772 for $0.07\lesssim\lambda_x\lesssim0.11$. For spanwise slip ($\lambda_z$), which does not alter the parabolic base flow, three-dimensional instabilities appear once $\lambda_z$ exceeds about 0.02 and the critical Reynolds number falls sharply, to about 394 at $\lambda_z=0.2$ and 336 at $\lambda_z=0.25$. Equal slip in both directions removes these three-dimensional instabilities and raises the critical Reynolds number to about $1.85\times10^5$ at $\lambda=0.05$. The paper also reports that spanwise slip enlarges the non-modal transient growth and moves the optimal perturbations to small finite streamwise wavenumbers, producing tilted streak structures instead of streamwise rolls.

Load-bearing premise

The three-dimensional results rest on the paper's unshown derivation of the boundary condition that couples the wall-normal vorticity $\eta$ to derivatives of the wall-normal velocity $u_y$ at the channel walls; if that boundary condition or its numerical implementation is wrong, the reported three-dimensional eigenvalues and critical Reynolds numbers would be invalid.

Editorial extensions

If this is right

  • Three-dimensional modes, not spanwise-invariant Tollmien–Schlichting waves, set the stability threshold for slip lengths above $\lambda_x\simeq0.008$ or $\lambda_z\simeq0.02$; any stability analysis that restricts to two-dimensional modes will overestimate the stabilizing effect of slip.
  • Spanwise slip can make a channel linearly unstable at Reynolds numbers an order of magnitude below the no-slip threshold—$Re_{cr}\simeq394$ at $\lambda_z=0.2$—so flows previously assumed linearly stable in this range can be unstable.
  • Streamwise slip does not strongly stabilize the flow: $Re_{cr}$ stays around 5900–6300 for $\lambda_x$ up to 0.2, with a small window of $\lambda_x$ where the flow is slightly less stable than no-slip flow.
  • Isotropic slip with equal streamwise and spanwise lengths suppresses the three-dimensional instabilities and greatly raises the critical Reynolds number, so the destabilizing effect is specifically due to anisotropy rather than to slip itself.
  • Spanwise slip changes the shape of the most amplified transient perturbations, replacing streamwise rolls with tilted long-wavelength structures, while streamwise slip reduces transient growth without changing the dominant structure.

Reading between the lines

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

  • If the critical Reynolds number for spanwise slip is as low as roughly 336, then in millimeter-scale channels with effective slip lengths on the order of 0.2 times the half-gap, linear instability should be observable at flow speeds far below the usual transition threshold; this is a testable experimental prediction with superhydrophobic surfaces that slip only in the spanwise direction.
  • The finding that spanwise slip destabilizes while leaving the base flow unchanged suggests the instability is produced by the modified wall condition acting on fluctuations rather than by reduced shear; an energy-budget or resolvent analysis of the three-dimensional eigenmodes could identify the production term responsible.
  • The same tilted three-dimensional modes likely connect the single-phase slip-channel results to oblique-wave instabilities reported in earlier two-fluid slippery channel studies; the paper's mode shapes could be compared directly with those earlier eigenfunctions to test whether the mechanism is shared.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper revisits the linear modal and non-modal stability of plane channel flow subject to Navier slip at the walls, treating streamwise and spanwise slip as two separate limiting cases of anisotropic slip and also considering isotropic slip. Using a velocity-vorticity formulation for modal stability and an adjoint-based time-stepper for transient growth, the authors report that sufficiently large streamwise slip (λx above about 0.008) or spanwise slip (λz above about 0.02) makes three-dimensional modes the leading unstable modes, in contrast to the two-dimensional leading modes of no-slip channel flow. They report that the critical Reynolds number is only mildly increased by streamwise slip (Recr about 5900 at λx=0.05 and 6280 at λx=0.2) but is drastically reduced by spanwise slip (Recr about 394 at λz=0.2 and 336 at λz=0.25), while equal isotropic slip strongly stabilizes the flow. They also report that streamwise slip suppresses transient growth, spanwise slip enhances it and changes the optimal perturbation structure, and isotropic slip yields transient growth dominated by the streamwise-slip effect.

Significance. If the reported three-dimensional instabilities are correct, the paper provides a substantive correction to the earlier conclusion, attributed to Lauga & Cossu (2005) and Min & Kim (2005), that velocity slip suppresses linear instability: that conclusion was based on two-dimensional modes, and this paper shows that sufficiently large anisotropic slip can make oblique/three-dimensional modes unstable at lower Reynolds numbers, even below the no-slip critical value for spanwise slip. The non-modal results, especially the shift of the optimal perturbation from streamwise-invariant rolls to finite-small-α oblique structures under spanwise slip, are also physically interesting and falsifiable. The paper's strengths include two explicit benchmark validations (no-slip transient growth against Reddy & Henningson (1993), and two-dimensional streamwise-slip critical Reynolds numbers against Ghosh et al. (2014)), a systematically explored parameter range, and internally consistent smooth trends in the reported eigenvalue data.

major comments (3)
  1. [§II.E, Eqs. (7)–(10)] The boundary condition for the wall-normal vorticity η is the load-bearing step for all three-dimensional modal results, but neither its derivation nor its final form is shown. The paper states that the η boundary condition 'can be derived using the slip boundary condition (2)' and that 'four boundary conditions coupling η and uy' result, yet the explicit conditions at y=±1 are absent. Since Eqs. (9)–(10) express ux and uz in terms of η and ∂uy/∂y, imposing Eq. (2) on ux and uz yields boundary relations involving η, ∂η/∂y, and second derivatives of uy; the reader cannot check the signs, the wall-normal orientation, or the handling of the (α²+β²)-¹ factor. If this derivation is incorrect, the eigenvalues in Figs. 4, 5, 8, and 9, and therefore every reported Recr value for three-dimensional modes, are not eigenvalues of the stated physical problem. The authors must present the full derivation and the final boundary conditions.
  2. [§III.A, Fig. 2; §III.B–C] No validation or convergence check exercises the three-dimensional coupled η–uy boundary condition. The two reported validations are a no-slip transient-growth calculation using the primitive-variable time-stepper and two-dimensional (β=0) streamwise-slip critical Reynolds numbers; neither involves the coupling of η with uy at nonzero β that is central to the new three-dimensional instability results. Moreover, no grid-convergence study is reported for any of the eigenvalue computations shown in Figs. 4, 5, 8, or 9. I request an explicit N-convergence test for a representative unstable three-dimensional mode, e.g., the λz=0.2, Re=394, (α,β)=(0.6,1.27) case, together with an independent check against direct temporal integration of the primitive-variable linearized equations (3) for the same parameters.
  3. [§III.D] The claim that three-dimensional instabilities 'disappear' under isotropic slip is supported too thinly. The text reports that for λx=λz=0.05 the first instability occurs at Re≃1.85×10⁵ and that the most unstable mode is still two-dimensional, but it does not state the range of (α,β) searched, the eigenvalue algorithm's convergence at such high Reynolds numbers, or how the absence of unstable three-dimensional modes was established. Because this negative result is used to contrast anisotropic and isotropic slip, it needs the same level of documentation as the positive instability searches.
minor comments (4)
  1. [§I] The sentence stating that subcritical transition 'can occur way below this Reynolds number at about Re=6601' is self-contradictory, since 6601 is above the no-slip critical value 5772; this appears to be a typo and should be corrected.
  2. [Throughout] There are numerous typographical errors that should be fixed in a revision, including 'Cartisian' (Cartesian), 'exerpiments' (experiments), 'boundanry' (boundary), 'stuctures' (structures), and 'steamwise' (streamwise).
  3. [Fig. 2(b) caption] The caption should identify clearly which curve is the reference data from Ghosh et al. and which points are the present results; the current text says the reference values are plotted as a solid line but does not state the symbol/line style for 'our results'.
  4. [§II.B] The claim that the adjoint system uses 'the same boundary condition (2)' would benefit from a one-line justification, since the adjoint Robin condition is not automatic for every non-self-adjoint formulation; a short derivation or citation would remove ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: slip lengths are specified model inputs and the stability results follow from solving the linearized equations, with validation against independent external benchmarks.

full rationale

The paper's central results are eigenvalues and transient-growth curves obtained by solving the linearized Navier-Stokes equations subject to the externally imposed Navier slip condition (2). The streamwise and spanwise slip lengths are prescribed model parameters, not fitted to the reported critical Reynolds numbers or optimal-growth curves, so there is no fitted-input-called-prediction pattern. The method is validated against two independent external references: the no-slip transient growth of Reddy & Henningson (1993) and the 2-D critical Reynolds numbers of Ghosh et al. (2014); these benchmarks do not depend on the paper's fitted values. The derivation of the wall-normal vorticity boundary condition in Section II.E is not shown, and the missing derivation is a legitimate correctness concern for the 3-D eigenvalues, but an omitted or unverified step is not circularity unless it reduces the output to an input by construction. No self-citation chain carries the load-bearing argument, no known result is merely renamed, and no prediction is equivalent to its inputs by definition. Accordingly, the honest finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

No new physical entities are introduced. Slip lengths are boundary-condition parameters with experimental motivation. The primary hidden load is the unshown derivation of the η boundary condition and the homogeneous-slip modeling assumption.

free parameters (2)
  • streamwise slip length λx = not fitted; parameter sweep (values include 0, 0.05, 0.1, 0.2, 0.5; up to 0.25 in Recr trends)
    Boundary condition parameter, not fitted. The central results depend on its value, with three-dimensional leading modes appearing above about λx = 0.008.
  • spanwise slip length λz = not fitted; parameter sweep (values include 0, 0.05, 0.1, 0.2, 0.5; up to 0.25 in Recr trends)
    Boundary condition parameter, not fitted. It controls the destabilization, with Recr dropping sharply for λz above about 0.02.
assumptions (3)
  • domain assumption The Navier slip boundary condition with homogeneous, constant, directionally independent slip lengths (Eq. 2) also applies to the perturbation velocity field and to the adjoint field.
    Used throughout the paper. The authors acknowledge in the Introduction that homogenization of the slip condition can be questionable for turbulent flows, citing Seo & Mani (2016), but adopt it for linear stability.
  • standard math The velocity-vorticity formulation (Eqs. 7-10) with six boundary conditions, including the slip-derived boundary condition for η, is equivalent to the linearized Navier-Stokes system with the Navier slip condition.
    Central to all modal stability results. The boundary condition for η is stated to be derivable but the derivation and expression are not shown (Section II.E).
  • domain assumption The base flow is a steady, parallel, parabolic profile with fixed bulk velocity; for streamwise slip the profile flattens with increasing λx, and for spanwise slip the profile is unchanged.
    Standard solution used in Section III. The no-slip parabolic profile and slip-modified parabolic profiles are exact solutions of the governing equations for the chosen normalization.

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Pith. "Pith review of Stability of slip channel flow revisited." pith.science (2026). https://pith.science/paper/GSJ754N3

@misc{pith2026190802027,
  author       = {Pith},
  title        = {Pith review of: Stability of slip channel flow revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GSJ754N3}},
  note         = {Machine review of arXiv:1908.02027}
}
read the original abstract

In this work, we revisit the temporal stability of slip channel flow. Lauga & Cossu (Phys. Fluids 17, 088106 (2005)) and Min & Kim (Phys. Fluids 17, 108106 (2005)) have investigated both modal stability and non-normality of slip channel flow and concluded that the velocity slip greatly suppresses linear instability and only modestly affects the non-normality. Here we study the stability of channel flow with streamwise and spanwise slip separately as two limiting cases of anisotropic slip and explore a broader range of slip length than previous studies did. We find that, with sufficiently large slip, both streamwise and spanwise slip trigger three-dimensional leading instabilities. Overall, the critical Reynolds number is only slightly increased by streamwise slip, whereas it can be greatly decreased by spanwise slip. Streamwise slip suppresses the non-modal transient growth, whereas spanwise slip enlarges the non-modal growth although it does not affect the base flow. Interestingly, as the spanwise slip length increases, the optimal perturbations exhibit flow structures different from the well-known streamwise rolls. However, in the presence of equal slip in both directions, the three-dimensional leading instabilities disappear and the flow is greatly stabilized. The results suggest that earlier instability and larger transient growth can be triggered by introducing anisotropy in the velocity slip.

Figures

Figures reproduced from arXiv: 1908.02027 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. (a) shows the comparison of our result with the reference values and obviously the two sets agree well. Therefore, our method can accurately calculate the transient growth of small perturbations. Note that for unstable modes, G(t) will exponentially grow at sufficiently large times. We also calculated the critical Reynolds numbers of β = 0 modes for a few streamwise slip lengths using the velocity-vorticity formulat… view at source ↗
Figure 3
Figure 3. FIG. 3. The velocity profiles of the base flow with [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The stability boundary for [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Contours in the wave number [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The stability boundary for [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Contours in the wave number [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The influence of spanwise slip on the most amplified pe [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The optimal perturbation of the mode ( [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]

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