Pith. sign in

REVIEW 3 major objections 7 minor 44 references

Lattice Boltzmann simulation reveals supercritical bifurcation in flow mode transitions of power-law fluids in the four-roll mill

T0 review · 3 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read Four-roll mill, long assumed to produce simple extensional flow, is shown to host two previously unreported vortex modes whose disappearance at a critical Reynolds number is a supercritical bifurcation strongly controlled by roller radius…

desk verdict Solid numerical observations of new vortex modes in the four-roll mill, but the 'supercritical bifurcation' label is asserted, not demonstrated, and should be either backed by stability analysis or softened. read the letter →

arxiv 2501.04655 v1 pith:GRUUB42E submitted 2025-01-08 physics.flu-dyn

Yuan Yu , Xiao Jiang , Qingqing Gu , Chuandong Lin , Qingyong Zhu , Hai-zhuan Yuan This is my paper · ORCID
classification physics.flu-dyn MSC 76A0576M28
keywords four-rollmillpower-lawfluidlatticeBoltzmannmethodsupercriticalbifurcationquadrifoliatevortexmodedumbbell-shapedquad-vortexextensionalflowstagnationpoint
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

At the center of a four-roll mill, the flow is usually described as pure hyperbolic extension: fluid is pulled in along one axis and pushed out along the other, with zero velocity at the stagnation point. This paper argues that this picture is incomplete at low roller speeds. Using lattice Boltzmann simulations of Newtonian and power-law fluids, the authors find a quadrifoliate vortex mode — four symmetric counter-rotating vortices clustered around the stagnation point — and, for shear-thickening fluids, an intermediate dumbbell-shaped quad-vortex state in which the vortices detach from the center and align along one axis. As the Reynolds number rises, the vortices shrink monotonically and disappear at a critical value, which the paper identifies as a supercritical bifurcation to simple extensional flow. The result matters because four-roll mills are standard tools for stretching droplets, cells, and polymers, and any experiment run below the critical speed is actually probing a different base flow than intended.

What carries the argument

The analysis is carried by a two-relaxation-time regularized lattice Boltzmann (TRT-RLB) model augmented with a one-point second-order curved boundary scheme, which the paper presents as the first application of this TRT-RLB variant to power-law fluids in curved geometries. The quantitative backbone is a pair of equivalent bifurcation criteria: the stagnation-point gradients $\partial_x u_x$ and $\partial_y u_y$ computed by central differences, and the vortex lengths $l_x$ and $l_y$ measured along the symmetry axes from the point where the axial velocity first vanishes to the center $O$. Agreement between the two criteria marks a direct transition; persistence of $l_x$ after $l_y$ has vanished marks the dumbbell mode. The phase diagram in the $(n, Re)$ plane is built with a step of $\Delta Re = 0.1$ near the critical lines, and the roller geometry is fixed by the geometric condition $(r+I)^2 = 2(r+h/2)^2$ to minimize container-wall influence.

What would settle it

A high-resolution experiment or simulation that tracks tracer particles in a Newtonian four-roll mill with $r=9$, $h=1$, $\delta\approx2.56$ should see the four central vortices shrink to zero and the stagnation-point velocity gradients reverse sign around $Re=36$; if the vortices persist at $Re=50$ or disappear below $Re=30$, the claimed bifurcation point is wrong. For $n=1.3$, a grid-converged simulation at $Re=36$ should show the dumbbell-shaped mode with $l_y=0$ but $l_x>0$; absence of that state under resolution refinement would refute Transition II.

Watch

Extended reading notes

Core claim

The central claim is that the four-roll mill's stagnation flow undergoes a supercritical bifurcation: as $Re$ increases, the vortex lengths $l_x$ and $l_y$ decrease continuously to zero at a critical Reynolds number $Re_c$, and the stagnation-point velocity gradients $\partial_x u_x$ and $\partial_y u_y$ reverse sign at the same point. For Newtonian fluids under the reference geometry ($h=1$, $r=9$, $\delta\approx 2.56$) that critical value is $Re_c = 36$. The paper further claims that the transition path is controlled by the power-law index $n$: for $0.7 \le n < 1.0$ the quadrifoliate mode passes directly to simple extensional flow (Transition I), while for $1.0 \le n \le 1.3$ it passes through the dumbbell-shaped quad-vortex mode (Transition II), in which $l_y$ vanishes before $l_x$. The authors state that the roller radius $r$ is the dominant geometric parameter — larger $r$ enlarges the vortices, can shift $Re_c$ beyond the studied range, and for $r=4$ suppresses the vortex mode entirely — whereas the roller-container gap $\delta$ has only a weak effect.

Load-bearing premise

The load-bearing premise is that the TRT-RLB solver and curved-boundary scheme resolve the near-stagnation velocity gradients accurately enough that the sign of $\partial_x u_x$ and $\partial_y u_y$ at the center is trustworthy, since the authors show grid-dependent nonphysical states near the critical Reynolds number for $n = 0.7$–$0.9$ that disappear only with refinement to $1600\times1600$ or $3200\times3200$ grids.

Editorial extensions

If this is right

  • Four-roll mill experiments conducted below $Re_c$ were not in simple extensional flow, so published deformation and breakup data obtained in the quadrifoliate regime may need to be re-interpreted with the vortex circulation taken into account.
  • The critical Reynolds number can be determined from two independent checks — sign reversal of $\partial_x u_x$ and $\partial_y u_y$ at the center, and vanishing of $l_x$ and $l_y$ — giving experimentalists a direct way to verify which flow mode they are in.
  • Roller radius, not container gap, is the design lever: small radii (around $r=4$) eliminate the vortex mode, while large radii (around $r=14$) can keep the quadrifoliate or dumbbell modes present up to $Re=50$.
  • Shear-thickening fluids will pass through an extra intermediate state, so rheological measurements or particle trapping in such fluids should avoid the dumbbell-shaped regime or treat it as a distinct flow condition.
  • The weak dependence on $\delta$ means compact four-roll mill configurations can be built without significantly shifting the flow-mode boundaries.

Reading between the lines

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

  • Because the same stagnation-point topology appears in cross-slot microchannels, the quadrifoliate and dumbbell modes may also exist there; if so, microfluidic extensional rheometry would need its own phase diagram in $(Re, n)$ before deformation measurements can be trusted.
  • The transition type could be used as a passive rheological indicator: measuring the critical Reynolds number and whether an intermediate dumbbell state appears gives a coarse read on the power-law index $n$ without any force measurement.
  • The mechanism behind Transition II may be the local viscosity increase in shear-thickening fluids near the stagnation point, which would alter the balance of extension and compression in the neighborhood of $O$; a targeted simulation with an artificially fixed local viscosity could isolate that effect.
  • If the bifurcation is robust in 2D, the 3D roller-length effect (short rollers) may break the assumed symmetry and shift $Re_c$, so extending the computation to finite-length rollers is a natural next test.
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.

Referee Report

3 major / 7 minor

Summary. This paper reports two-dimensional lattice Boltzmann simulations of generalized Newtonian power-law fluids in a four-roll mill, spanning Re in [1,50], power-law index n in [0.7,1.3], and variations of roller radius r and roller-container gap delta. The TRT-RLB solver is validated against analytic solutions for force-driven power-law channel flow and Taylor-Couette flow, and grid convergence of velocity and shear rate is documented at Re=10. The authors discover two flow configurations in the central region in addition to the classical simple extensional flow: a quadrifoliate vortex mode (four symmetric counter-rotating vortices around the stagnation point, with horizontal extension and vertical compression at O) at low Re, and, for n greater than about 1, a dumbbell-shaped quad-vortex mode in which vortices detach laterally from the stagnation point. Vortex lengths lx and ly decrease monotonically with Re and vanish at critical Reynolds numbers Rec that roughly coincide with sign reversal of the velocity gradients dux/dx and duy/dy at the stagnation point. The paper labels these transitions 'supercritical bifurcations' in the title, abstract, and conclusion, presents phase diagrams in (n, Re), (delta, n), and (r, n) space, and finds that r controls the transition while delta has minimal influence.

Significance. The finding that the central region of the four-roll mill is not always simple extensional flow is genuinely interesting: the device is a standard tool for extensional rheometry and droplet and cell deformation studies, and the identification of a low-Re vortex regime in which the stagnation-point deformation axes are exchanged (du/dx and du/dy both reverse sign relative to the classical state) bears directly on the interpretation of low-speed experiments. The numerical work is careful in several respects that deserve explicit credit: validation against two analytic benchmarks (power-law channel flow for n=0.5, 1.0, 1.5 and Taylor-Couette flow over beta=0.1-0.8), a documented grid-convergence study, and honest reporting of resolution-dependent nonphysical states near the transition for n=0.7-0.9 together with their resolution by mesh refinement to 1600 and 3200 grid points. If the mode classification is accepted, the phase diagrams (Figs. 20, 23, 25) provide a useful engineering map of the flow regimes.

major comments (3)
  1. [Abstract; Section IV A; Section IV B; Section V] The term 'supercritical bifurcation' is used in the title, abstract, and conclusion, but the analysis establishes only that forward-converged steady states switch between two modes at a Reynolds-number threshold. A supercritical bifurcation in the dynamical-systems sense requires a stability statement: the emerging branch must be the locally unique stable branch, and the transition must be continuous in the order parameter. None of the required evidence is provided: (i) there is no reverse continuation (for example, initializing at Re=50 with the converged extensional solution and decreasing Re through 36 to test whether the quadrifoliate mode reappears, which would rule out hysteresis); (ii) there are no perturbation experiments near Rec measuring growth or decay rates; and (iii) there is no scaling analysis of any order parameter (a pitchfork would give lx, ly proportional to (Rec-Re)^(1/2), a transcritical bifurcation a linear dependence, and so on). A subcritical or transcritical structure would be completely invisible to this forward-only procedure. The manuscript itself concedes this in the closing paragraph of Section V: 'Future work should explore the stability characteristics of these newly discovered flow modes.' Either the missing stability, continuation, and scaling analysis must be added, or the claim must be re-scoped to 'mode transitions at critical Reynolds numbers' throughout, including the title and abstract.
  2. [Section IV B; Section V, item (c); Section IV C 1] The critical-Reynolds-number detection is internally inconsistent with the claims built on it. At n=1.3 the reported values are Rec=36 from dux/dx, Rec=36 from duy/dy, Rec=38 from lx, and Rec=36 from ly; the 2-unit discrepancy is then absorbed by introducing the dumbbell-shaped quad-vortex mode as an intermediate state, even though at Re=36 and 37 the velocity gradients already have the simple-extensional signs. At n=0.7 the four criteria give 32, 31, 32, and 31. Yet conclusion item (c) states that in Transition I 'all characteristic parameters ... yield identical critical Reynolds numbers,' and Section IV C 1 repeats the coincidence claim for the delta-sweep. The paper should report a single grid-converged Rec per case and state which criterion defined each phase boundary; with the phase diagram of Fig. 20 claimed at Delta-Re=0.1 resolution near transitions, a 1-2-unit spread among the defining criteria needs a quantitative explanation, not a narrative one. I note that the grid-independence check of the intermediate state (Fig. 19) supports its physical reality; the open question is whether it is a distinct dynamical branch or a continuous deformation, which again requires the stability and continuation analysis of Major Comment 1. A related definitional coupling should also be acknowledged: the quadrifoliate mode is defined by the sign pair (dux/dx greater than 0, duy/dy less than 0), and the transition is then detected from the sign reversal of those same gradients; the vortex-length criterion is the independent evidence and should be given equal prominence in the mode definitions.
  3. [Section III D; Section IV B; Figs. 15, 16, 20, 23, 25] The grid-convergence study (Table I) is performed at Re=10, far from the transition region, and monitors only ux at a midpoint and |gamma-dot| at O. The transition diagnostic, namely the signs of dux/dx and duy/dy at the stagnation point, is exactly the quantity shown in Figs. 15-16 to be unconverged on the 800x800 grid: at n=0.7 and Re=31, the 800x800 solution exhibits the nonphysical state dux/dx<0 and duy/dy<0 (locally compressible flow), and dux/dx flips sign when the grid is refined to 1600x1600 and 3200x3200. The text states that this 'grid-dependent behavior occurs for 0.7 < n < 0.9,' which is precisely the range feeding into the phase diagrams, yet the paper never specifies which resolution was used for the points of Figs. 20, 23, and 25 or whether the reported Rec values are the resolution-converged ones. Without this policy, the transition points and phase boundaries for shear-thinning fluids are not reproducible. The authors should state the resolution used for each regime of the phase diagrams and demonstrate convergence of Rec itself (not just bulk velocity) at least along the phase boundary.
minor comments (7)
  1. [Abstract; Section V] The phrase 'supercritical bifurcation modes' in the abstract conflates steady flow modes with transitions between modes; after the re-scoping requested in Major Comment 1, the modes themselves should not be described as bifurcations.
  2. [Section III A] The TRT-RLB method is cited as Yu et al. (2023), arXiv:2312.10318; if an archival version exists, it should be cited, since the method description is delegated to this unpublished preprint.
  3. [Section I] The priority claim that this is 'the first successful extension of the TRT-RLB methodology to power-law fluid dynamics' should be checked against the existing regularized-LBM literature for non-Newtonian fluids and softened if related work exists.
  4. [Figs. 20, 23, 25; Section IV B] The construction of the phase diagrams should specify the operational criterion (gradient signs, vortex lengths, or streamline topology) used to assign each point to a mode, so that the boundaries are reproducible.
  5. [Section IV C 2; Fig. 25] The exclusion of the n=1.3, r=13 and 14 cases 'due to numerical convergence issues' appears only in the running text; it should also be stated in the caption of Fig. 25, together with the number of excluded points.
  6. [References] The reference 'V ona, M. and Lauga, E.' should read 'Vonna, M. and Lauga, E.'; hyphenation of 'four-roll mill' is also inconsistent in a few places.
  7. [Section IV A; Fig. 6] The eight vortices visible in Fig. 6a at Re=50 are likely the wall-attached vortices reported by Lagnado and Leal (1990); a sentence connecting the two observations would help situate the new central-region modes relative to the known high-Re behavior.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the flow modes and transition criteria are emergent outputs of the governing equations, not quantities defined from the target result, and the paper's self-citations are to numerical methods independently validated against analytical benchmarks.

full rationale

The paper's derivation chain runs from the Navier-Stokes and power-law constitutive equations (Eqs. 2-8), through the TRT-RLB discretization (Eq. 9) and the one-point second-order curved boundary scheme (Eq. 20), to validation against analytical power-law channel flow and Taylor-Couette flow (Figs. 3 and 5), and then to steady-state simulations whose streamline topology and stagnation-point gradients are post-processed. The mode labels (quadrifoliate, dumbbell-shaped, simple extensional) are operational classifications of the simulated velocity field; for example, Section IV A states: 'This marks the transition from the quadrifoliate vortex mode (∂xux > 0 and ∂yuy < 0) to simple extensional flow (∂xux < 0 and ∂yuy > 0).' The critical Reynolds numbers are read off where those simulated gradients reverse or where the computed vortex lengths lx and ly vanish. This is a diagnostic coupling, not a circular one: no target mode or critical Reynolds number is inserted into the solver, and no parameter is fitted to force the phase diagram. The self-citations to Yu et al. (2023) for the TRT-RLB model and for the choice τ2 = 1.6 are method citations whose accuracy is checked in this paper against external analytical benchmarks, so they constitute independent support rather than load-bearing self-citation. One legitimate gap exists, but it is a correctness issue, not a circularity: the title and abstract call the transitions 'supercritical bifurcations,' while the conclusion admits 'Future work should explore the stability characteristics of these newly discovered flow modes.' Stability, hysteresis, and branch scaling were not tested, so the supercritical label is under-supported; however, that under-support does not make the derivation equivalent to its inputs. The central mode-transition results are self-contained numerical discoveries, so the circularity score is zero.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The free parameters are numerical tuning choices; the key inputs (Re, n, r, delta, mu0) are scanned parameters, not fitted to the target result. The central result rests on the TRT-RLB model and curved boundary scheme taken from prior literature, with the steady-state and two-dimensional assumptions as additional load-bearing premises.

free parameters (1)
  • TRT relaxation time tau2 = 1.6
    Set to 1.6 as in Yu et al. (2023) to maintain stability over wide viscosity variations (Section III A). It is chosen by hand rather than fitted to the target flow modes, but it affects all simulated results.
assumptions (5)
  • domain assumption The lattice Boltzmann model recovers the incompressible Navier-Stokes equations for generalized Newtonian fluids through the Hermite expansion and forcing/source terms in Eqs. (9)-(15).
    Section III A relies on the TRT-RLB model from Yu et al. (2023); validation against channel and Taylor-Couette flows provides partial support, but the exact recovery for power-law fluids with this curved boundary scheme is assumed.
  • domain assumption The flow is two-dimensional with no axial variation or axial velocity, justified by sufficiently long rollers.
    Section II A states partial u / partial z = 0 and u_z = 0; real four-roll mills have finite length and three-dimensional end effects that are not captured.
  • domain assumption The power-law constitutive model mu = mu0 |gamma|^(n-1) describes the fluid rheology over the simulated shear-rate range.
    Section II B, Eq. (4); the model neglects elasticity, thixotropy, temperature dependence, and other non-Newtonian effects.
  • domain assumption A steady-state solution is meaningful and is reached by time-marching, and the transition is studied by continuation in Re without testing hysteresis.
    The convergence criterion Eq. (21) defines steady state; no forward and backward sweeps in Re are reported, so the supercritical label is an inference from monotonic amplitude growth.
  • domain assumption The one-point second-order curved boundary scheme (Eq. 20) provides accurate no-slip conditions on moving circular boundaries without the magic parameter constraint.
    Section III B; validated on Taylor-Couette flow and power-law channel flow, but not directly on the four-roll mill geometry at the near-stagnation low-Re conditions where the new modes appear.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Lattice Boltzmann simulation reveals supercritical bifurcation in flow mode transitions of power-law fluids in the four-roll mill." pith.science (2026). https://pith.science/paper/GRUUB42E

@misc{pith2026250104655,
  author       = {Pith},
  title        = {Pith review of: Lattice Boltzmann simulation reveals supercritical bifurcation in flow mode transitions of power-law fluids in the four-roll mill},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRUUB42E}},
  note         = {Machine review of arXiv:2501.04655}
}
abstract

The four-roll mill has been traditionally viewed as a device generating simple extensional flow with a central stagnation point. Our systematic investigation using a two-relaxation-time regularized lattice Boltzmann (TRT-RLB) model reveals unexpected richness in the flow physics, identifying two previously unreported supercritical bifurcation modes: a quadrifoliate vortex mode featuring four symmetrical counter-rotating vortices, and a dumbbell-shaped quad-vortex mode where vortices detach from but remain symmetric about the stagnation point. The numerical framework, representing the first successful extension of TRT-RLB method to power-law fluid dynamics, enables comprehensive mapping of flow characteristics across Reynolds numbers ($1 \leq Re \leq 50$), power-law indices ($0.7 \leq n \leq 1.3$), and geometric configurations. The transition from quadrifoliate vortex mode exhibits distinct pathways depending on the power-law index: at relatively small $n$, the flow undergoes a direct supercritical bifurcation to simple extensional flow, while at relatively large $n$, it evolves through an intermediate dumbbell-shaped state. Among geometric parameters, the roller radius $r$ emerges as the dominant factor controlling bifurcation points and vortex dimensions, whereas the roller-container gap $\delta$ exerts minimal influence on flow regimes. The transitions between flow modes can be precisely characterized through the evolution of vortex dimensions and velocity gradients at the stagnation point, providing quantitative criteria for flow regime identification. These findings enrich our fundamental understanding of bifurcation phenomena in extensional devices and provide quantitative guidelines for achieving desired flow patterns in four-roll mill applications.

Figures

Figures reproduced from arXiv: 2501.04655 by the authors.

Figure 1
Figure 1. FIG. 1: Cross-sectional schematic of the four-roll mill configuration. The rollers of length [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Schematic of the one-point second-order curved boundary scheme, where [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Comparison between numerical results (symbols) and analytical solutions (lines) for force-driven power-law channel flow [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Schematic of Taylor-Couette flow: an inner cylinder of radius [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Velocity profiles of Taylor-Couette flow for different radius ratios ( [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Streamline patterns in the four-roll mill for Newtonian fluid ( [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Velocity distributions characterizing simple extensional flow in the four-roll mill at [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Discovery of the quadrifoliate vortex mode: streamline patterns in the central region of the four-roll mill for Newtonian [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Velocity distributions in the quadrifoliate vortex mode: profiles of [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Variation of shear rate and velocity gradients at point [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Vortex dimensions ( [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Velocity gradients (a) [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Variations of vortex lengths ( [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Variations of velocity gradients and vortex lengths with [PITH_FULL_IMAGE:figures/full_fig_p012_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Central streamline in the four roll mill for [PITH_FULL_IMAGE:figures/full_fig_p012_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: The velocity profiles ( [PITH_FULL_IMAGE:figures/full_fig_p013_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17: Variations of velocity gradients and vortex lengths with [PITH_FULL_IMAGE:figures/full_fig_p013_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18: Central streamline in the four roll mill at ( [PITH_FULL_IMAGE:figures/full_fig_p014_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19: The velocity profiles ( [PITH_FULL_IMAGE:figures/full_fig_p015_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20: A phase diagram that illustrates the different flow modes in simulation of the four-roll mill as a function of power-law [PITH_FULL_IMAGE:figures/full_fig_p015_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21: Schematic representation of two distinct transition pathways in the four-roll mill. (a) Transition I: direct transformation [PITH_FULL_IMAGE:figures/full_fig_p016_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22: Variation of velocity gradients ( [PITH_FULL_IMAGE:figures/full_fig_p017_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23: A phase diagram that illustrates the different mode transitions at [PITH_FULL_IMAGE:figures/full_fig_p018_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24: Variations of velocity gradients ( [PITH_FULL_IMAGE:figures/full_fig_p018_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25: A phase diagram that illustrates the different mode transitions at [PITH_FULL_IMAGE:figures/full_fig_p019_25.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 36 canonical work pages

  1. [1]

    , author Philip , J

    author author Akbaridoust , F. , author Philip , J. , \ and\ author Marusic , I. ,\ title title Assessment of a miniature four-roll mill and a cross-slot microchannel for high-strain-rate stagnation point flows , \ 10.1088/1361-6501/aaacf3 journal journal Measurement Science and Technology \ volume 29 ,\ pages 045302 ( year 2018 ) NoStop

    work page doi:10.1088/1361-6501/aaacf3

  2. [2]

    , author Douady , S

    author author Andreotti , B. , author Douady , S. , \ and\ author Couder , Y. ,\ title title An experiment on two aspects of the interaction between strain and vorticity , \ 10.1017/S0022112001005353 journal journal Journal of Fluid Mechanics \ volume 444 ,\ pages 151--174 ( year 2001 ) NoStop

    work page doi:10.1017/s0022112001005353

  3. [3]

    author author Arratia , P. E. , author Thomas , C. C. , author Diorio , J. , \ and\ author Gollub , J. P. ,\ title title Elastic instabilities of polymer solutions in cross-channel flow , \ 10.1103/PhysRevLett.96.144502 journal journal Phys. Rev. Lett. \ volume 96 ,\ pages 144502 ( year 2006 ) NoStop

    doi:10.1103/physrevlett.96.144502

  4. [4]

    \ and\ author Padberg-Gehle , K

    author author Balasuriya , S. \ and\ author Padberg-Gehle , K. ,\ title title Controlling the Unsteady Analogue of Saddle Stagnation Points , \ 10.1137/120886042 journal journal SIAM Journal on Applied Mathematics \ volume 73 ,\ pages 1038--1057 ( year 2013 ) NoStop

    work page doi:10.1137/120886042

  5. [5]

    ,\ title Drop Deformation and Burst in Two-Dimensional Flows ,\ @noop Ph.D

    author author Bentley , B. ,\ title Drop Deformation and Burst in Two-Dimensional Flows ,\ @noop Ph.D. thesis ,\ school California Institute of Technology ( year 1985 ) NoStop

    work page 1985

  6. [6]

    author author Bentley , B. J. \ and\ author Leal , L. G. ,\ title title A computer-controlled four-roll mill for investigations of particle and drop dynamics in two-dimensional linear shear flows , \ 10.1017/S002211208600280X journal journal Journal of Fluid Mechanics \ volume 167 ,\ pages 219 ( year 1986 ) NoStop

    work page doi:10.1017/s002211208600280x

  7. [7]

    author author Bryngelson , S. H. \ and\ author Freund , J. B. ,\ title title Non-modal floquet stability of capsules in large-amplitude oscillatory extensional flow , \ https://doi.org/10.1016/j.euromechflu.2019.04.012 journal journal European Journal of Mechanics - B/Fluids \ volume 77 ,\ pages 171--176 ( year 2019 ) NoStop

    doi:10.1016/j.euromechflu.2019.04.012 2019

  8. [8]

    author author Crowley , D. G. , author Frank , F. C. , author Mackley , M. R. , \ and\ author Stephenson , R. G. ,\ title title Localized flow birefringence of polyethylene oxide solutions in a four roll mill , \ 10.1002/pol.1976.180140613 journal journal Journal of Polymer Science: Polymer Physics Edition \ volume 14 ,\ pages 1111--1119 ( year 1976 ) NoStop

    work page arXiv

Show all 44 references
  1. [9]

    \ and\ author Maffettone , P

    author author D'Avino , G. \ and\ author Maffettone , P. L. ,\ title title Numerical simulations of dispersive mixing of viscoelastic suspensions in a four-roll mill , \ 10.1007/s00397-017-1028-x journal journal Rheologica Acta \ volume 56 ,\ pages 695--706 ( year 2017 ) NoStop

    doi

  2. [10]

    \ and\ author Leal , L

    author author Dunlap , P. \ and\ author Leal , L. ,\ title title Dilute polystyrene solutions in extensional flows: Birefringence and flow modification , \ 10.1016/0377-0257(87)80009-5 journal journal Journal of Non-Newtonian Fluid Mechanics \ volume 23 ,\ pages 5--48 ( year 1...

    doi

  3. [11]

    \ and\ author Leal , L

    author author Feng , J. \ and\ author Leal , L. ,\ title title Numerical simulations of the flow of dilute polymer solutions in a four-roll mill , \ https://doi.org/10.1016/S0377-0257(97)00027-X journal journal Journal of Non-Newtonian Fluid Mechanics \ volume 72 ,\ pages 187-...

    doi

  4. [12]

    author author Fuller , G. G. \ and\ author Leal , L. G. ,\ title title Flow birefringence of dilute polymer solutions in two-dimensional flows , \ 10.1007/BF01517512 journal journal Rheologica Acta \ volume 19 ,\ pages 580--600 ( year 1980 ) NoStop

    doi

  5. [13]

    author author Fuller , G. G. \ and\ author Leal , L. G. ,\ title title Flow birefringence of concentrated polymer solutions in two‐dimensional flows , \ 10.1002/pol.1981.180190402 journal journal Journal of Polymer Science: Polymer Physics Edition \ volume 19 ,\ pages 557--587...

    1981

  6. [14]

    , author Verhaeghe , F

    author author Ginzburg , I. , author Verhaeghe , F. , \ and\ author d'Humi \`e res , D. ,\ title title Two-relaxation-time lattice boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions , \ https://api.semanticscholar.org/CorpusID:123717273 j...

    2008

  7. [15]

    \ and\ author Leal , L

    author author Ha , J.-W. \ and\ author Leal , L. G. ,\ title title An experimental study of drop deformation and breakup in extensional flow at high capillary number , \ 10.1063/1.1358306 journal journal Physics of Fluids \ volume 13 ,\ pages 1568--1576 ( year 2001 ) NoStop

    doi

  8. [16]

    author author Haward , S. J. , author McKinley , G. H. , \ and\ author Shen , A. Q. ,\ title title Elastic instabilities in planar elongational flow of monodisperse polymer solutions , \ 10.1038/srep33029 journal journal Scientific Reports \ volume 6 ,\ pages 33029 ( year 2016...

    doi

  9. [17]

    author author Haward , S. J. , author Ober , T. J. , author Oliveira , M. S. A. , author Alves , M. A. , \ and\ author McKinley , G. H. ,\ title title Extensional rheology and elastic instabilities of a wormlike micellar solution in a microfluidic cross-slot device , \ https:/...

    2012

  10. [18]

    author author Haward , S. J. , author Oliveira , M. S. A. , author Alves , M. A. , \ and\ author McKinley , G. H. ,\ title title Optimized cross-slot flow geometry for microfluidic extensional rheometry. \ https://api.semanticscholar.org/CorpusID:2096605 journal journal Physic...

    2012

  11. [19]

    author author Higdon , J. J. L. ,\ title title The kinematics of the four-roll mill , \ 10.1063/1.858782 journal journal Physics of Fluids A: Fluid Dynamics \ volume 5 ,\ pages 274--276 ( year 1993 ) NoStop

    1993 doi

  12. [20]

    , author Sasmal , C

    author author Hsiao , K.-W. , author Sasmal , C. , author Ravi Prakash , J. , \ and\ author Schroeder , C. M. ,\ title title Direct observation of DNA dynamics in semidilute solutions in extensional flow , \ 10.1122/1.4972236 journal journal Journal of Rheology \ volume 61 ,\ ...

    doi

  13. [21]

    author author Hymel , S. J. , author Lan , H. , \ and\ author Khismatullin , D. B. ,\ title title Elongation index as a sensitive measure of cell deformation in high-throughput microfluidic systems , \ https://doi.org/10.1016/j.bpj.2020.06.027 journal journal Biophysical Journ...

    doi

  14. [22]

    , author Walkama , D

    author author Kumar , M. , author Walkama , D. M. , author Guasto , J. S. , \ and\ author Ardekani , A. M. ,\ title title Flow-induced buckling dynamics of sperm flagella , \ 10.1103/PhysRevE.100.063107 journal journal Phys. Rev. E \ volume 100 ,\ pages 063107 ( year 2019 ) NoStop

    doi

  15. [23]

    author author Lagnado , R. R. \ and\ author Leal , L. G. ,\ title title Visualization of three-dimensional flow in a four-roll mill , \ 10.1007/BF00575332 journal journal Experiments in Fluids \ volume 9 ,\ pages 25--32 ( year 1990 ) NoStop

    doi

  16. [24]

    author author Lee , E. C. \ and\ author Muller , S. J. ,\ title title Flow Light Scattering Studies of Polymer Coil Conformation in Solutions in Extensional Flow , \ 10.1021/ma981277a journal journal Macromolecules \ volume 32 ,\ pages 3295--3305 ( year 1999 ) NoStop

    doi

  17. [25]

    \ and\ author Leal , L

    author author Milliken , W. \ and\ author Leal , L. ,\ title title Deformation and breakup of viscoelastic drops in planar extensional flows , \ 10.1016/0377-0257(91)87018-S journal journal Journal of Non-Newtonian Fluid Mechanics \ volume 40 ,\ pages 355--379 ( year 1991 ) NoStop

    doi

  18. [26]

    , author Moravec , D

    author author Narayan , S. , author Moravec , D. B. , author Dallas , A. J. , \ and\ author Dutcher , C. S. ,\ title title Droplet shape relaxation in a four-channel microfluidic hydrodynamic trap , \ 10.1103/PhysRevFluids.5.113603 journal journal Phys. Rev. Fluids \ volume 5 ...

    doi

  19. [27]

    , author Ubertini , S

    author author Pontrelli , G. , author Ubertini , S. , \ and\ author Succi , S. ,\ title title The unstructured lattice Boltzmann method for non- Newtonian flows , \ 10.1088/1742-5468/2009/06/P06005 journal journal Journal of Statistical Mechanics: Theory and Experiment \ volum...

    2009 doi

  20. [28]

    author author Poole , R. J. , author Alves , M. A. , \ and\ author Oliveira , P. J. ,\ title title Purely elastic flow asymmetries , \ 10.1103/PhysRevLett.99.164503 journal journal Phys. Rev. Lett. \ volume 99 ,\ pages 164503 ( year 2007 ) NoStop

    doi

  21. [29]

    author author Reis , T. ,\ title title On the Lattice Boltzmann Deviatoric Stress : Analysis , Boundary Conditions , and Optimal Relaxation Times , \ 10.1137/19M1244846 journal journal SIAM Journal on Scientific Computing \ volume 42 ,\ pages B397--B424 ( year 2020 ) NoStop

    doi

  22. [30]

    , author Hsiao , K.-W

    author author Sasmal , C. , author Hsiao , K.-W. , author Schroeder , C. M. , \ and\ author Ravi Prakash , J. ,\ title title Parameter-free prediction of DNA dynamics in planar extensional flow of semidilute solutions , \ 10.1122/1.4972237 journal journal Journal of Rheology \...

    doi

  23. [32]

    author author Schroeder , C. M. , author Babcock , H. P. , author Shaqfeh , E. S. G. , \ and\ author Chu , S. ,\ title title Observation of Polymer Conformation Hysteresis in Extensional Flow , \ 10.1126/science.1086070 journal journal Science \ volume 301 ,\ pages 1515--1519 ...

    doi

  24. [33]

    , author He , Q

    author author Tao , S. , author He , Q. , author Chen , B. , author Yang , X. , \ and\ author Huang , S. ,\ title title One-point second-order curved boundary condition for lattice Boltzmann simulation of suspended particles , \ 10.1016/j.camwa.2018.07.013 journal journal Comp...

    doi

  25. [34]

    author author Taylor , G. I. ,\ title title The formation of emulsions in definable fields of flow , \ https://doi.org/10.1098/rspa.1934.0169 journal journal Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character \ ,\ p...

    1934

  26. [35]

    author author Tretheway , D. C. \ and\ author Leal , L. ,\ title title Deformation and relaxation of Newtonian drops in planar extensional flows of a Boger fluid , \ 10.1016/S0377-0257(01)00123-9 journal journal Journal of Non-Newtonian Fluid Mechanics \ volume 99 ,\ pages 81-...

    doi

  27. [36]

    author author Tu , M. Q. , author Nguyen , H. V. , author Foley , E. , author Jacobs , M. I. , \ and\ author Schroeder , C. M. ,\ title title 3D manipulation and dynamics of soft materials in 3D flows , \ 10.1122/8.0000600 journal journal Journal of Rheology \ volume 67 ,\ pag...

    doi

  28. [37]

    \ and\ author Lauga , E

    author author Vona , M. \ and\ author Lauga , E. ,\ title title Stabilizing viscous extensional flows using reinforcement learning , \ 10.1103/PhysRevE.104.055108 journal journal Physical Review E \ volume 104 ,\ pages 055108 ( year 2021 ) NoStop

    doi

  29. [38]

    , author Han , J

    author author Wang , J. , author Han , J. , \ and\ author Yu , D. ,\ title title Numerical studies of geometry effects of a two-dimensional microfluidic four-roll mill on droplet elongation and rotation , \ @noop journal journal Engineering analysis with boundary elements \ vo...

    2012

  30. [39]

    \ and\ author Leonetti , M

    author author Xie , K. \ and\ author Leonetti , M. ,\ title title Mechanical characterization of core-shell microcapsules , \ 10.5802/crmeca.148 journal journal Comptes Rendus. M\'ecanique \ volume 351 ,\ pages 163--182 ( year 2023 ) NoStop

    doi

  31. [40]

    , author Qin , Z

    author author Yu , Y. , author Qin , Z. , author Chen , S. , author Shu , S. , \ and\ author Yuan , H.-Z. ,\ title title Two-relaxation-time regularized lattice boltzmann model for navier-stokes equations , \ @noop journal journal arXiv preprint arXiv:2312.10318 \ ( year 2023 ) NoStop

    2023 arXiv

  32. [41]

    , author Qin , Z

    author author Yu , Y. , author Qin , Z. , author Yuan , H. , \ and\ author Shu , S. ,\ title title Two-relaxation-time regularized lattice boltzmann model for convection-diffusion equation with spatially dependent coefficients , \ https://doi.org/10.1016/j.amc.2024.129135 jour...

  33. [42]

    \ and\ author Yong , W.-A

    author author Zhao , W. \ and\ author Yong , W.-A. ,\ title title Single-node second-order boundary schemes for the lattice Boltzmann method , \ 10.1016/j.jcp.2016.10.049 journal journal Journal of Computational Physics \ volume 329 ,\ pages 1--15 ( year 2017 ) NoStop

    2016 doi

  34. [43]

    \ and\ author Schroeder , C

    author author Zhou , Y. \ and\ author Schroeder , C. M. ,\ title title Single polymer dynamics under large amplitude oscillatory extension , \ 10.1103/PhysRevFluids.1.053301 journal journal Phys. Rev. Fluids \ volume 1 ,\ pages 053301 ( year 2016 ) NoStop

    doi

  35. [44]

    merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number orga...

    2010

  36. [45]

    merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number orga...

    2010

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.