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Orientation dynamics of a spheroid in the simple shear flow of a weakly elastic fluid

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Weak elasticity turns closed Jeffery orbits into four regimes of spheroid orientation.

desk verdict A careful, mostly convincing analytical derivation of the small-De spheroid phase diagram, but the unresolved Sharma & Koch discrepancy in the slender-fiber limit is a real correctness risk, not a footnote. read the letter →

arxiv 2505.23361 v1 pith:MSCKIMCL submitted 2025-05-29 physics.flu-dyn

Pavan Kumar Singeetham , Deepak Madival , Piyush Garg , Ganesh Subramanian This is my paper
classification physics.flu-dyn
keywords spheroidorientationdynamicsJefferyorbitssecond-orderfluidDeborahnumberorbitaldriftkayakingmoderotationarrestviscoelasticsuspension
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 asks how weak fluid elasticity changes the way an elongated or flattened particle spins in a simple shear flow. In a Newtonian fluid the particle would circle forever on one of a family of closed Jeffery orbits; the paper shows that a small Deborah number (weak elasticity) breaks this closed-trajectory structure and produces a slow drift between orbits. Using a reciprocal theorem with vector spheroidal harmonics, the authors compute the O(De) angular velocity for a spheroid of any aspect ratio, and a multiple-scales analysis reduces the dynamics to the drift of the Jeffery orbit constant. That drift organizes orientation into four regimes on the aspect-ratio versus fluid-parameter plane: stable spinning, stable tumbling, stable kayaking, or an unstable kayaking limit cycle separating basins. For polymer-like fluids, prolate spheroids of every aspect ratio drift to spinning, oblate spheroids tumble unless they are flatter than a threshold aspect ratio (between 0.2 and 0.454), in which case they kayak, and extreme-aspect-ratio spheroids stop rotating entirely above a threshold Deborah number.

What carries the argument

The central object is the orbit-averaged drift ΔC(C; κ, ε): the change in the Jeffery orbit constant over one rotation period, computed from the O(De) angular velocity by a multiple-scales (fast/slow time) expansion. The angular velocity is first written in an invariant quadratic form with four aspect-ratio functions β_i(κ, ε), which are evaluated in closed form in spheroidal coordinates using the reciprocal theorem and vector spheroidal harmonics; the fluid rheology enters only through the single parameter ε of the second-order fluid. The sign and zero-crossings of ΔC/(C²+1) as a function of C/(C+1) select one of four regimes, and the near-sphere, slender-fiber, and flat-disk asymptotes of ΔC supply the bifurcation loci on the κ-ε plane. For rotation arrest, the aligned-phase equation for a slender prolate spheroid gives the threshold De_c(κ), while for a thin oblate spheroid two further thresholds mark a saddle-node tri-furcation cascade that ends in stable flow-gradient-plane arrested states.

What would settle it

Integrate the full orientation equations for a prolate spheroid with κ=20 and ε=-0.6: the paper predicts rotation arrest only above De_c≈0.422 and predicts that arrested states in the flow-gradient plane are unstable, so any off-plane perturbation leads to a spiralling approach to the vorticity axis. Observing permanent arrest for De<0.422, or stable arrest for De>0.422 without the initially monotonic then spiralling transient, would falsify the threshold (4.2) and the stability analysis.

Watch

Extended reading notes

Core claim

For a neutrally buoyant spheroid of any aspect ratio in simple shear of a second-order fluid, weak elasticity breaks the closed Jeffery-orbit topology and produces an O(De) drift in the orbit constant C. The orbit-averaged drift ΔC(C; κ, ε) plotted against C/(C+1) has four possible shapes: always negative (spiralling to the spinning mode, Regime 1), always positive (spiralling to tumbling, Regime 2), a negative-slope zero crossing (stable kayaking limit cycle, Regime 3), or a positive-slope zero crossing (unstable limit cycle separating basins of spinning and tumbling, Regime 4). On the κ-ε plane these regimes are organized by bifurcation loci that asymptote to ε=-1/2 and ε=-1 in the slender-fiber limit, to ε=-1/3 in the flat-disk limit, and that intersect the sphere line at ε=-3/8, enclosing an island of Regime 3 near the prolate side. In the polymeric range ε∈[-0.7,-0.5], prolate spheroids of every aspect ratio drift to spinning; oblate spheroids drift to tumbling for κ > κ_c(ε), with κ_c increasing from 0.2 to 0.454 as ε varies from -0.7 to -0.5, and to a stable kayaking mode for κ < κ_c(ε). For slender prolate and thin oblate spheroids the drift picture breaks down and rotation about the vorticity axis is arrested above De_c(κ): an O(1/κ) threshold for prolate spheroids whose arrested states are unstable to off-plane perturbations, and for oblate spheroids a primary saddle-node at De_c1=12κ/(1+6ε), followed by secondary and tertiary bifurcations that create stable rotation-arrested states in the flow-gradient plane.

Load-bearing premise

The entire regime map assumes that the viscoelastic drift is always much slower than one Jeffery rotation, a separation that fails for extreme aspect ratios, so the phase diagram is only valid for De values that shrink toward zero as the spheroid becomes very slender or very flat.

Editorial extensions

If this is right

  • In the polymeric range ε∈[-0.7,-0.5], a dilute suspension of rod-like spheroids should acquire a vorticity-aligned orientation distribution at long times, independent of initial orientation.
  • Disk-like spheroids should switch from flow-gradient-plane tumbling to a kayaking limit cycle when their aspect ratio drops below κ_c(ε), with κ_c increasing from 0.2 to 0.454 as ε goes from -0.7 to -0.5.
  • Slender fibers should exhibit rotation arrest in the flow-gradient plane above De_c = O(1/κ) with arrested states that are unstable to off-plane perturbations, whereas thin disks should show a narrow Deborah-number window where stable arrested states coexist with the kayaking limit cycle before the limit cycle is destroyed.
  • The O(De) angular-velocity formula, with its closed-form aspect-ratio functions, can serve as the input for computing the leading-order elastic rheology and shear-induced migration of dilute spheroid suspensions.
  • Because the drift coefficients scale linearly with (1+2ε) in the slender limit, Boger fluids near ε=-0.5 should display a logarithmically smaller drift, making the predicted regimes hardest to observe there.

Reading between the lines

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

  • If the sign of ΔC is read from the closed-form functions, the regime boundaries at ε=-1/2 and ε=-1/3 suggest that for fluids near these parameter values the leading-order drift changes sign in a vanishingly small neighborhood of the sphere or of the flat-disk limit; finite-De corrections may then dominate and shift or destroy the predicted regime transitions.
  • The paper's conclusion that no stable flow-vorticity-plane equilibria exist at small De implies that the stationary orientations seen in experiments and finite-De simulations of prolate spheroids require fluid-memory effects beyond the second-order fluid; a testable extension would be to see whether such equilibria vanish as De→0 in an ε=-0.6 fluid.
  • Combining the tabulated viscoelastic β_i with the known inertial shape functions would yield an inertio-elastic regime diagram, since the two contributions are additive at small Re and De—a natural extension that the paper notes is planned for future work.
  • The predicted oblate cascade with three closely spaced thresholds (for example 0.2, 0.212, and 0.215 for κ=0.05 and ε=-2/3) means that modest uncertainty in either De or κ could move a disk between qualitatively different long-time states; experiments will need precise control of both to test the phase diagram.
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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 / 5 minor

Summary. The paper analyzes the orientation dynamics of a neutrally buoyant spheroid of arbitrary aspect ratio in simple shear flow of a weakly elastic second-order fluid, working to first order in the Deborah number. Using a reciprocal theorem formulation with vector spheroidal harmonics, the authors derive closed-form expressions for the O(De) correction to the angular velocity, then use a multiple-scales analysis to reduce the dynamics to a slow drift of the Jeffery orbit constant C. The drift curves are organized into four regimes on the κ–ε plane: stable spinning, stable tumbling, stable kayaking, and unstable kayaking. For the polymeric range ε∈[−0.7,−0.5], the paper claims that prolate spheroids always drift to the spinning mode, while oblate spheroids drift to tumbling for κ>κ_c(ε) and to a stable kayaking mode for κ<κ_c(ε). It also analyzes rotation arrest for extreme aspect ratios, deriving thresholds De_c∼1/κ for slender prolate spheroids and De_c∼κ for thin oblate spheroids, with secondary and tertiary bifurcations on the oblate side. Appendix B provides an independent Fourier-space slender-body derivation that reproduces the large-κ limit up to a factor of 4 compared with Leal (1975).

Significance. If correct, this is a substantial contribution: it gives a parameter-free, closed-form description of the leading viscoelastic correction to spheroid rotation across the full aspect-ratio range, identifies a stable kayaking mode for oblate spheroids in the polymeric range, and predicts arrest scalings that connect to earlier experiments and computations. The paper's strengths include a systematic reciprocal-theorem derivation, closed-form aspect-ratio functions in Appendix A, an independent slender-body check in Appendix B, and qualitative agreement with D'Avino et al. (2014) and with the early experiments of Gauthier and Bartram. The main risk is the slender-prolate corner, where the paper itself concedes an unresolved disagreement with Sharma & Koch (2023); because the abstract's 'prolate always drifts to spinning' claim covers that corner, this is a load-bearing issue rather than a peripheral discrepancy.

major comments (3)
  1. [§5 and Appendix B] The central claim that a prolate spheroid of any aspect ratio drifts to the spinning mode in the polymeric range is not settled in the slender-fiber limit. In §5 the authors state that Sharma & Koch (2023), in the limit c*De→0, found a stable limit cycle near the vorticity axis, that this disagrees with both Leal (1975) and the present results, and that 'the reasons for this discrepancy are not clear at the moment.' The independent derivation in Appendix B uses only the leading-order line distribution of Stokeslets (Eqs. B4–B7) and explicitly neglects the O(κ^-2) higher-order singularities that Sharma & Koch identify as controlling the long near-aligned phase. The spheroidal-harmonic expansion in §2.2 formally contains those singularities, but the paper never shows explicitly that its large-κ limit reproduces or refutes Sharma & Koch's aligned-phase torque balance. To support the 'always' claim, the authors should either retain the higher-order singularities in the large-κ reduction and show the sign of the drift near the flow-gradient plane, or explicitly restrict the claim to aspect ratios below the range where the Sharma–Koch mechanism operates.
  2. [§3 and §3.3, Figure 8] The global regime classification in Figure 8 is an asymptotic statement, not a fixed-De phase map. As the authors correctly note at the start of §3 and in §3.3, the time-scale separation underlying the multiple-scales reduction breaks down for slender prolate and thin oblate spheroids near their aligned phases, so De must be taken increasingly small as κ→∞ (De≪1/κ) or κ→0 (De≪κ). For any fixed experimental De in the extreme-aspect-ratio limits, rotation arrest and the secondary bifurcations of §4 intervene before the drift picture applies. This caveat appears in the text but is absent from the abstract and from the Figure 8 caption, where the four regimes are presented on the κ–ε plane without the De constraint. The paper should carry this qualification into the abstract and figure so that readers do not interpret the diagram as a fixed-De prediction.
  3. [§4, Eq. (4.1)–(4.2)] The prolate rotation-arrest threshold De_c(κ) is derived from the aligned-phase equation (4.1), which is truncated at leading order in κ^-1. The paper is appropriately cautious in Figure 10 by drawing the moderate-κ portions as dashed, but it does not give a quantitative criterion for where the asymptotic expression is reliable. Since the paper uses κ=20 as a representative case (Figure 11) and compares with computations for κ up to 16, an estimate of the neglected O(κ^-2) terms in (4.1) is needed to justify the precision implied by the quoted threshold values, especially near the turning-around region of the dashed curves where the prediction is acknowledged to be unphysical.
minor comments (5)
  1. [Abstract and §5] The phrase 'our results are exact' in §5 should be qualified: the results are exact within the second-order-fluid truncation and the regular O(De) expansion for a spheroid, whereas Appendix B itself is a leading-logarithmic slender-body calculation. As written, the sentence invites confusion between the spheroidal-harmonic calculation and the asymptotic slender-body check.
  2. [§2.3, after Eq. (2.42)] The list of relations between the β_i and the trigonometric coefficients appears to contain a notational slip, with one relation seemingly repeated. Please check that all coefficients are defined consistently, since the subsequent drift plots depend on these definitions.
  3. [§3.3 and Figures 6–7] The insets in Figures 6(b) and 6(f) show that the convergence to the slender-fiber asymptote is logarithmic and that the exact curves visibly differ even at κ=10^4. This is consistent with the text, but in the figure captions it would help to state explicitly that the plotted asymptote is the leading-logarithmic result from Appendix B, not the full large-κ limit of the exact expressions.
  4. [Appendix B, Eq. (B1)] The reciprocal-theorem prefactor in (B1) is stated for a slender fiber with the Oseen-Burgers tensor, but the derivation would be easier to follow if the paper stated explicitly that the longitudinal force profile is that of a slender spheroid, R(x1)=κ^-1√(1−x1^2), before quoting I1=2lnκ/3.
  5. [Throughout] There are numerous typographical artifacts in the rendered text (e.g., stray 'u1D' glyphs and inconsistent spacing in equations). A careful proofread of the final manuscript is needed before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the O(De) angular velocity and orbital drift are derived from the reciprocal theorem and vector spheroidal harmonics, with prior same-group Stokesian fields used as published parameter-free inputs rather than as fits to the paper's own predictions.

full rationale

The paper's central derivation is self-contained against its own equations. The O(De) angular velocity follows from the generalized reciprocal theorem in Eq. (2.16), with the disturbance fields evaluated from Stokesian spheroidal-harmonic solutions quoted from Dabade et al. (2016); those are published, parameter-free analytical inputs that do not themselves contain the viscoelastic drift prediction. The orbit-constant drift is then obtained independently via a multiple-scales average in Eq. (3.6), and the closed-form aspect-ratio functions are tabulated in Appendix A. The slender-fiber limit is checked by an independent Fourier-space slender-body calculation in Appendix B, which reproduces Leal (1975) up to a factor of four and matches the large-aspect-ratio limit of the main expressions. No free parameter is fitted to the target predictions, and the regime diagram in Figure 8 is organized by the computed sign and zero crossings of the drift rather than imposed by construction. The acknowledged discrepancy with Sharma & Koch (2023) in Section 5 (“the reasons for this discrepancy are not clear at the moment”) is a correctness and validation risk, not a circularity: the paper does not redefine Sharma & Koch's result as its own prediction, and the disagreement leaves the central derivation unchanged. Self-citations of the authors' prior work are used only for standard Stokesian building blocks and are not load-bearing circular justification.

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

The derivation is a first-principles reciprocal-theorem calculation. It introduces no fitted constants and no new physical entities. The main inputs are the second-order fluid constitutive assumption with material parameter epsilon, the Stokesian approximation of disturbance fields in the O(De) integrals, the completeness of vector spheroidal harmonics borrowed from Dabade et al. (2016), and the multiple-scales time-separation assumption. None of these are tuned to match the paper's target predictions; they are standard tools or domain assumptions.

assumptions (5)
  • domain assumption The suspending fluid is modeled as a second-order fluid, so viscoelasticity is captured by the O(De) stress terms (2.5)-(2.6) with material parameter epsilon.
    Invoked in Section 2.1; all O(De) results are conditional on this constitutive truncation being valid for small De.
  • domain assumption The disturbance velocity and pressure fields in the O(De) integrals can be replaced by their Stokesian (De = 0) approximations.
    Used around Eq. (2.16); standard for a regular perturbation, but not proven in detail in the paper.
  • standard math The structure of the O(De) angular velocity is exactly the four-function invariant form (2.40).
    Derived from linearity, the unit-length constraint, and affine response in solid-body rotation in Section 2.3; a symmetry argument, not an empirical fit.
  • domain assumption The multiple-scales reduction in Eq. (3.6) is valid because Jeffery rotation is fast compared with viscoelastic drift.
    The paper restricts to De well below the arrest threshold and states in Section 3 that the separation breaks down near aligned phases for extreme kappa.
  • domain assumption Polymer solutions correspond to epsilon in [-0.7, -0.5].
    Taken from Subramanian and Koch (2006, 2007); used to frame the predictions for the polymeric range.

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Pith. "Pith review of Orientation dynamics of a spheroid in the simple shear flow of a weakly elastic fluid." pith.science (2026). https://pith.science/paper/MSCKIMCL

@misc{pith2026250523361,
  author       = {Pith},
  title        = {Pith review of: Orientation dynamics of a spheroid in the simple shear flow of a weakly elastic fluid},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSCKIMCL}},
  note         = {Machine review of arXiv:2505.23361}
}
abstract

We investigate the orientation dynamics of a neutrally buoyant spheroid, of an arbitrary aspect ratio ($\kappa$), freely rotating in a weakly viscoelastic fluid undergoing simple shear flow. Weak elasticity is characterized by a small but finite Deborah number ($De$), and the suspending fluid rheology is therefore modeled as a second-order fluid, with the constitutive equation involving a material parameter $\epsilon$ related to the ratio of the first and second normal stress differences; polymer solutions correspond to $\epsilon\in[-0.7,-0.5]$. Employing a reciprocal theorem formulation, along with expressions for the relevant disturbance fields in terms of vector spheroidal harmonics, we obtain the spheroid angular velocity to $O(De)$. In the Newtonian limit, a spheroid rotates along Jeffery orbits parametrized by an orbit constant $C$, although this closed-trajectory topology is structurally unstable, being susceptible to weak perturbations. For $De$ well below a threshold, $De_c(\kappa)$, weak viscoelasticity transforms the closed-trajectory topology into a tightly spiralling one. A multiple-scales analysis is used to interpret the resulting orientation dynamics in terms of an $O(De)$ orbital drift. The drift in orbit constant over a Jeffery period $\Delta C$, when plotted as a function of $C$, identifies four different orientation dynamics regimes on the $\kappa-\epsilon$ plane. For $\epsilon$ in the polymeric range, prolate spheroids always drift towards the spinning mode. Oblate spheroids drift towards the tumbling mode for $\kappa > \kappa_c(\epsilon)$, but towards an intermediate kayaking mode for $\kappa < \kappa_c(\epsilon)$. The rotation of spheroids of extreme aspect ratios, either slender prolate spheroids ($\kappa \gg 1$) or thin oblate ones ($\kappa \ll 1$), about the vorticity axis, is arrested for $De \geq De_c(\kappa)$

Figures

Figures reproduced from arXiv: 2505.23361 by the authors.

Figure 1
Figure 1. Jeffery orbits on the unit hemisphere for spheroids of different aspect ratios; = (a) 0.045 (b) 0.6 (c) 1 (d) 1.67 and (e) 22.37. = 0 and = ∞ correspond to the spinning (log-rolling) and tumbling modes, respectively; the interval 0 < < ∞ correspond to three-dimensional kayaking modes. restrict ourselves to reviewing those that have examined viscoelasticity of the suspending fluid as the underlying cause for irrevers… view at source ↗
Figure 2
Figure 2. The body-fixed ( ) and space-fixed ( ′ ′ ′ ) coordinate systems for a spheroid in a simple shear flow; the -axis being constrained to lie in the ′ ′ -plane at all times. The polar angle between the spheroid symmetry axis and ambient vorticity ( ), and the dihedral angle between the flow-vorticity and orientation-vorticity planes () define the spheroid orientation in the space-fixed reference frame. On using the boun… view at source ↗
Figure 3
Figure 3. The orbital coordinate as a function of fast time scale starting from ( , ) = (23/48, 0) for spheroid of aspect ratio = 4 for (a) = −2 and (b) = 2. The calculations are based on (1) obtained from the numerical integration of the governing equations of and (solid line), (2) orbit drift alone obtained from solving (3.6) (dash-dotted line), and (3) the drift together with the fluctuations (dashed line). + 6 6 (0, )  +… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: The drift due to elasticity, −1Δ/( 2 + 1), as characterized by the normalized change in the orbit constant in a single Jeffery period, plotted as a function of /( + 1), for a prolate spheroid; /( + 1) = 0 and /( + 1) = 1 correspond to the spinning and tumbling modes, r…
Figure 5
Figure 5. Figure 5: The drift due to elasticity, −1Δ/( 2 + 1), as characterized by the normalized change in the orbit constant in a single Jeffery period, plotted as a function of /( + 1), for a prolate spheroid with = 5.0. (a) Δ has a zero crossing with a negative slope, and (c) has a ze…
Figure 6
Figure 6. Figure 6: The scaled orbital drift, −1Δ3/2 0 p 0 − 1/( 2 + 1), plotted as a function of /( + 1), for prolate spheroids of a varying aspect-ratio with fixed; the black dashed and dash-dotted curves denote the near-sphere and slender-fiber asymptotes, respectively. (a)-(b) = −2, (…
Figure 7
Figure 7. Figure 7: The scaled orbital drift, −1Δ3/2 0 p 0 − 1/( 2 + 1), plotted as a function of /( + 1), for oblate spheroids of a varying aspect-ratio with fixed. The black dashed and dash-dotted curves denote the near-sphere and flat-disk asymptotes, respectively; the latter exhibits …
Figure 8
Figure 8. Figure 8: The orientation dynamics regimes on the - plane, for small but finite ; the four different regimes are numbered as follows: 1 - stable spinning; 2 - stable tumbling; 3 - stable kayaking; 4 -unstable kayaking sandwiched between stable spinning and tumbling modes. The ho…
Figure 9
Figure 9. Figure 9: The scaled orbital drift, −1Δ3/2 0 p 0 − 1/( 2 + 1), plotted as a function of /( + 1), for spheroids with increasing from zero to infinity; = −0.36. Figures (a) and (b) pertain to oblate spheroids, and (d)-(f) to prolate spheroids; (c) shows the transition of the drift…
Figure 10
Figure 10. Figure 10: The threshold Deborah number, () for rotation arrest of a prolate spheroid (in the flow-gradient plane) in an ambient simple shear flow: (a) vs for different , and (b) vs for different . The continuous curves in (a), either solid or dashed, denote the prediction based…
Figure 11
Figure 11. Figure 11: The orbit constant as a function of the slow time , for a spheroid starting from ( , ) ≡ (23/48, 0) - with = 20, = −0.6 for (a) = 0.25, (b) = 0.45, and with = 0.05, = −0.6 for (c) = 0.2, (d) = 0.3. The blue curves are obtained from numerical integration of the governi…
Figure 12
Figure 12. Figure 12: Unit sphere trajectory topologies for below and above rotation arrest, for a prolate spheroid with = 20, for = −0.6. (a) = 0.3 and (c) = 0.45; () ≈ 0.422. The plot in (b) shows the chosen ’s in (a) and (c), in relation to the threshold curve for rotation arrest. The t…
Figure 13
Figure 13. Figure 13: Unit sphere trajectory topologies before and after rotation arrest for an oblate spheroid with = 0.05, for = −2/3, for three different Deborah numbers: (a) = 0.1, (c) = 0.21, (d) = 0.213 and (e) = 0.22. The stable limit cycle appears as a red curve in (a), (c) and (d)…
Figure 14
Figure 14. Figure 14: Projected views of the unit hemisphere illustrate the boundaries (in black) demarcating the basins of attraction, corresponding to the different invariant sets, for (a) = 0.214 ∈ (2, 3) and (b) = 0.23 > 3; the views correspond to an oblate spheroid with = 0.05, for = …
Figure 15
Figure 15. Figure 15: The viscoelastic aspect-ratio functions plotted against /(1 − ), where = 1/0 represents the eccentricity of the prolate spheroid. (a) Functions , and , which contribute to (¤ ), and (b) functions , and ,, which contribute to (¤ ). The black dashed lines correspond to …
Figure 16
Figure 16. Figure 16: The viscoelastic shape-dependent functions are plotted against /(1 − ), where = 1/0 represents the eccentricity of the oblate spheroid. (a) Functions , and , which contribute to (¤ ), and (b) functions , and ,, which contributes to (¤ ). The black dashed lines indicat…

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Pith tools

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