arxiv: 2605.07754 · v1 · submitted 2026-05-08 · ❄️ cond-mat.soft · physics.flu-dyn

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Droplet Deformation and Emulsion Rheology in Two-Dimensional Odd Stokes Flow

Hugo Fran\c{c}a, Maziyar Jalaal, Thomas Appleford

Authors on Pith no claims yet

Pith reviewed 2026-05-11 03:24 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn

keywords odd viscositydroplet deformationemulsion rheologyStokes flowchiral fluidsTaylor deformationsurface tensionapparent viscosity

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The pith

The leading-order deformation of a droplet in two-dimensional shear flow remains unchanged by odd viscosity, while the emulsion's apparent viscosities depend on the odd-viscosity difference.

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

This paper examines how a viscous droplet deforms under simple shear when the surrounding fluid possesses odd viscosity, a feature of chiral fluids that breaks time-reversal symmetry. An analytical solution is obtained for the shape and flow, valid for moderate differences in both even and odd viscosities inside and outside the droplet. The result shows that the steady droplet shape at small shear rates matches the classical Taylor prediction exactly at leading order, while the macroscopic even and odd viscosities of a dilute suspension of such droplets acquire explicit dependence on the odd-viscosity contrast. Because many active and biological fluids exhibit odd viscosity, the analysis supplies a concrete way to predict emulsion behavior and to interpret droplet measurements as probes of chirality.

Core claim

Within the framework of two-dimensional odd Stokes flow, the steady-state Taylor deformation parameter of the droplet satisfies D_T^∞ = Ca + O(Ca²), so the leading-order deformation is identical to the classical even-viscous result. Closed-form expressions are derived for the apparent even and odd viscosities of a dilute emulsion, and it is shown that the flow field depends solely on the difference in odd viscosity across the interface rather than on the individual values. Direct numerical simulations further demonstrate that odd-viscous corrections to the droplet deformation appear at higher orders in the capillary number.

What carries the argument

The analytical solution of the odd Stokes equations for a nearly circular droplet interface, obtained by expanding the shape in powers of the capillary number and matching even and odd stress and velocity boundary conditions.

Load-bearing premise

Viscosity differences between droplet and exterior remain moderate so that the droplet stays close to circular and the expansion in capillary number remains valid.

What would settle it

Measurement of the steady-state droplet aspect ratio at small capillary number Ca ≈ 0.05 in a fluid with known odd viscosity; the claim holds if the deformation parameter matches Ca to within a few percent, with deviations appearing only at larger Ca or larger viscosity contrasts.

Figures

Figures reproduced from arXiv: 2605.07754 by Hugo Fran\c{c}a, Maziyar Jalaal, Thomas Appleford.

**Figure 2.** Figure 2: (a) The pressure field p for an odd-viscous droplet in an odd-viscous simple shear flow according to the analytical solution. The Laplace pressure, σ/R, has been subtracted from the droplet pressure field. The parameters are Ca = 0.01, λ = 1 and L/R = 20. Panels show different combinations of the oddness parameters (βD, βM) ∈ [−1, 1] × [−1, 1]. The plotted region is [−2.5R, 2.5R] × [−2.5R, 2.5R]. The grey … view at source ↗

**Figure 3.** Figure 3: Comparison between numerical and analytical solutions for the flow around an odd-viscous droplet. Shown are the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: The pressure field p for an odd-viscous droplet in an odd-viscous simple shear flow according to direct numerical simulations. The Laplace pressure, σ/R, has been subtracted from the droplet pressure field. The parameters are Re = 0.01, Ca = 0.3, λ = 1 and L/R = 20. Panels show different combinations of the oddness parameters (βD, βM) ∈ [−1, 1] × [−1, 1]. The plotted region is [−2.5R, 2.5R] × [−2.5R, 2.5R]… view at source ↗

**Figure 5.** Figure 5: Effect of odd viscosity on steady-state droplet deformation. (a) [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Apparent even and odd viscosities of a dilute droplet suspension from the analytical expression Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: Initial configuration of the droplet under shear DNS. A single droplet of radius [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: The effect of odd viscosity on time-dependent droplet deformation. All simulations use Re = 0 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

**Figure 9.** Figure 9: The effect of the oddness parameters βD and βM on the time-dependent droplet deformation. All simulations use Re = 0.01 λ = 1 and L/R = 20. Shown are the evolutions of the Taylor deformation parameter DT (t) for (a) δ = 0.0, Ca = 0.1, (b) δ = 0.5, Ca = 0.1, (c) δ = 1.0, Ca = 0.1, (d) δ = 0.0, Ca = 0.3, (e) δ = 0.5, Ca = 0.3 and (f) δ = 1.0, Ca = 0.3, each for several combinations βD and βM consistent with … view at source ↗

**Figure 10.** Figure 10: The effect of the oddness parameters βD and βM on the time-dependent droplet deformation. All simulations use Re = 0.01, λ = 1, Ca = 1.0 and L/R = 20. Shown are the evolutions of the Taylor deformation parameter DT (t) for (a) δ = −0.5, (b) δ = 0.0 and (c) δ = 0.5, each for at least two combinations βD and βM consistent with the chosen value of δ. The black dotted line shows the leading order theory D∞ T … view at source ↗

**Figure 11.** Figure 11: (a) The steady state deformation parameter [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗

**Figure 12.** Figure 12: The pressure field p for an odd-viscous droplet in an odd-viscous simple shear flow according to direct numerical simulations. The Laplace pressure, σ/R, has been subtracted from the droplet pressure field. The parameters are Re = 0.01, Ca = 1.0, λ = 1 and L/R = 20. Panels show different combinations of the oddness parameters (βD, βM) ∈ [−0.5, 0.5] × [−0.5, 0.5]. The plotted region is [−5R, 5R] × [−5R, 5R… view at source ↗

**Figure 13.** Figure 13: Comparison between numerical and analytical solutions for the flow around an odd-viscous droplet. Shown are the [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗

read the original abstract

We study the deformation of a two-dimensional viscous droplet in simple shear in the presence of odd viscosity. We derive an analytical solution for the droplet shape and surrounding flow field within the framework of odd Stokes flow, allowing for differences in both even and odd viscosity between the droplet and the surrounding fluid. This solution yields closed-form expressions for the macroscopic apparent even and odd viscosities of a dilute emulsion. We show that, provided all viscosity differences remain moderate, the steady-state Taylor deformation parameter satisfies $D_T^\infty = \text{Ca} + \mathcal{O}(\text{Ca}^2)$ so that the leading-order droplet deformation is unchanged from the classical (even-viscous) result. Nevertheless, pronounced effects emerges beyond leading order, where our direct numerical simulations reveal odd-viscous differences to the droplet deformation. In addition, we show that the flow is influenced only by the difference in odd viscosity between the droplet and the medium and not on their individual values. Our analysis clarifies how odd viscosity might modify the effective rheology of dilute emulsions and provides a framework for interpreting droplet-based measurements of odd-viscous response. Key words: odd viscosity $|$ droplets $|$ emulsions $|$ surface tension $|$ chiral fluids

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.

Desk Editor's Note private letter to a colleague

Odd viscosity leaves the leading-order droplet deformation the same as classical Taylor flow but changes higher-order shape and sets the emulsion's apparent viscosities through the odd contrast alone.

read the letter

The main takeaway is that the leading-order Taylor deformation stays exactly D_T^∞ = Ca + O(Ca²) even when odd viscosity is present, provided the viscosity contrasts stay moderate. Beyond that order the shape does shift, and the effective even and odd viscosities of a dilute emulsion come out in closed form from the far-field flow. The flow itself turns out to depend only on the difference in odd viscosity between droplet and medium, not on their separate values. That last point is a clean simplification that follows directly from the Stokes equations in 2D odd hydrodynamics. The authors get an analytical solution for both the interior and exterior flow plus the interface shape at linear order in Ca, then use DNS to illustrate the higher-order corrections. The fact that the result reduces to the known even-viscous case when the odd parts vanish is a useful consistency check. This is the sort of calculation that can help people interpret droplet experiments in chiral or active fluids where odd viscosity appears. The assumptions are standard for this class of problem: small Ca so the droplet stays nearly circular at leading order, and moderate viscosity ratios so the perturbative expansion holds. The 2D setting is explicit and limits direct mapping to real emulsions, but that is not a flaw in the work itself. Without the full derivation in front of me I cannot audit every algebra step, yet nothing in the abstract or stress-test outline suggests hidden fitting or circular reasoning. The numerics are presented as confirmation rather than the main evidence. This is aimed at the soft-matter hydrodynamics crowd working on odd-viscous or chiral systems. It is worth sending to a serious referee; the analytical expressions are new relative to the classical Taylor problem and the cited odd-viscosity papers, and the claims are narrow enough that a referee can check them without needing massive new data. I would recommend review rather than desk rejection.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates the deformation of a two-dimensional viscous droplet in simple shear flow within the odd Stokes framework, allowing differences in both even and odd viscosities between the droplet and the surrounding fluid. It derives an analytical solution for the droplet shape and flow field, yielding closed-form expressions for the apparent even and odd viscosities of a dilute emulsion. The central result is that, for moderate viscosity differences, the steady-state Taylor deformation parameter satisfies D_T^∞ = Ca + O(Ca²), leaving the leading-order deformation unchanged from the classical even-viscous case. Direct numerical simulations illustrate pronounced odd-viscous effects at higher orders, and the flow is shown to depend only on the difference in odd viscosity.

Significance. If the derivations hold, this work is significant for extending classical emulsion rheology to chiral fluids with odd viscosity. The leading-order invariance of the Taylor deformation provides a clean, parameter-free result that simplifies interpretation and reduces to the known even-viscous limit without redefinition. The closed-form rheology expressions, the independence of the flow from individual odd-viscosity values, and the DNS evidence for higher-order corrections offer a useful framework for droplet-based measurements of odd-viscous response. These elements strengthen the contribution to cond-mat.soft.

minor comments (3)

Abstract: grammatical error in 'pronounced effects emerges beyond leading order' (should be 'emerge').
Abstract: 'Key words' should be 'Keywords'.
The assumption of 'moderate' viscosity differences is load-bearing for the perturbative result but is stated without quantitative bounds or error estimates on the O(Ca²) remainder; this is a presentation issue that can be addressed by adding a brief validity discussion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The referee summary accurately reflects the manuscript's central findings on droplet deformation in odd Stokes flow, including the leading-order invariance of the Taylor deformation parameter and the role of odd-viscosity differences. We appreciate the recognition of the work's significance for chiral fluid rheology. As no specific major comments were listed in the report, we provide no point-by-point responses below.

Circularity Check

0 steps flagged

Derivation self-contained from governing equations

full rationale

The central result D_T^∞ = Ca + O(Ca²) follows from an analytical solution of the 2D odd Stokes equations with even/odd viscosity jumps and surface tension; the leading-order shape correction is obtained by balancing the viscous traction discontinuity against capillary pressure in the small-Ca expansion, reproducing the classical even-viscous case without redefinition or parameter fitting. Closed-form emulsion viscosities are likewise direct outputs of the same boundary-value solution. DNS are used only to illustrate O(Ca²) odd-viscous corrections, not to close the leading-order claim. No self-citation chains, ansatzes, or fitted-input predictions appear in the load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on the standard 2D Stokes equations augmented by odd-viscosity terms, the assumption of small capillary number, and the dilute-emulsion limit. No new free parameters or invented entities are introduced beyond the viscosity contrasts already present in the problem statement.

axioms (2)

domain assumption The flow obeys the odd Stokes equations with constant even and odd viscosities inside and outside the droplet.
Invoked to obtain the analytical solution for shape and flow field.
domain assumption Viscosity differences remain moderate so that the droplet stays nearly circular at leading order.
Required for the perturbative expansion D_T^∞ = Ca + O(Ca²).

pith-pipeline@v0.9.0 · 5532 in / 1485 out tokens · 38061 ms · 2026-05-11T03:24:21.740837+00:00 · methodology

discussion (0)

Reference graph

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