A diffuse-interface theory of active nematic interfaces: transport mechanisms and modal structure

Margarida M. Telo da Gama; Mykola Tasinkevych; Rodrigo C. V. Coelho

arxiv: 2605.21826 · v1 · pith:FQPBAZS2new · submitted 2026-05-20 · ❄️ cond-mat.soft · physics.flu-dyn

A diffuse-interface theory of active nematic interfaces: transport mechanisms and modal structure

Rodrigo C. V. Coelho , Mykola Tasinkevych , Margarida M. Telo da Gama This is my paper

Pith reviewed 2026-05-22 07:29 UTC · model grok-4.3

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

keywords active nematicsdiffuse interfacesinterfacial instabilitylong-wavelength theoryhydrodynamic screeningnematic-isotropic interface

0 comments

The pith

Activity generates a direct local q² contribution to the height dynamics at active nematic interfaces.

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

The paper develops a long-wavelength theory for the linear stability of a flat interface between an active nematic and an isotropic fluid. Starting from a diffuse-interface Cahn-Hilliard-Landau-de Gennes model coupled to Brinkman-screened Stokes hydrodynamics, the linearized dynamics are projected onto a small set of interfacial degrees of freedom consisting of the conserved height mode, a scalar amplitude mode, and a transverse orientational mode. Eliminating the gapped scalar profile mode produces a reduced interfacial operator. The central result is that activity produces a local contribution proportional to wavevector squared in the height sector. This term competes with the passive capillary relaxation that appears at order wavevector to the fourth and thereby defines a local active interfacial channel whose properties depend on the internal structure of the diffuse interface rather than on long-range hydrodynamic transport.

Core claim

In the screened diffuse-interface regime, activity generates a direct local contribution proportional to q² in the height sector. This term competes with the passive local diffusive capillary relaxation, which enters at order q⁴, and defines a local active interfacial channel controlled by the internal structure of the diffuse interface. This mechanism is distinct from the non-analytic |q| and |q|q² terms characteristic of weakly screened long-ranged momentum transport.

What carries the argument

The reduced interfacial operator obtained by projecting the linearized dynamics onto the conserved height mode, amplitude mode, and transverse orientational mode, then eliminating the gapped scalar profile mode.

Load-bearing premise

The projection of the linearized dynamics onto a small set of interfacial degrees of freedom and the subsequent elimination of the gapped scalar profile mode are valid in the long-wavelength limit.

What would settle it

Numerical integration of the full diffuse-interface and hydrodynamics equations to compute the growth rate of small interfacial perturbations and determine whether activity produces a leading q² term in the dispersion relation.

Figures

Figures reproduced from arXiv: 2605.21826 by Margarida M. Telo da Gama, Mykola Tasinkevych, Rodrigo C. V. Coelho.

read the original abstract

We develop a long-wavelength theory for the linear stability of a flat interface between an active nematic and an isotropic fluid. Starting from a diffuse-interface Cahn--Hilliard--Landau--de Gennes description coupled to Brinkman-screened Stokes hydrodynamics, we project the linearized dynamics onto a small set of interfacial degrees of freedom: the conserved translation, or height, mode; a scalar profile distortion or amplitude mode; and a transverse orientational mode associated with director rotations. Eliminating the gapped scalar profile mode gives a reduced interfacial operator coupling the conserved height mode to the transverse orientational mode. The main result is that activity generates, in the screened diffuse-interface regime, a direct local contribution proportional to $q^2$ in the height sector. This term competes with the passive local diffusive capillary relaxation, which enters at order $q^4$, and defines a local active interfacial channel controlled by the internal structure of the diffuse interface. This mechanism is distinct from the non-analytic $|q|$ and $|q|q^2$ terms characteristic of weakly screened Hele--Shaw/Saffman--Taylor-type transport, which are controlled by long-ranged momentum transport in the surrounding fluid. This framework identifies a diffuse-interface route to active interfacial instability that can operate while the homogeneous active nematic remains linearly stable because of hydrodynamic screening. It also provides a basis for distinguishing local diffuse-interface instabilities, bulk-flow-driven hydrodynamic instabilities, and mixed regimes in active nematic--isotropic interfaces.

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

The paper derives a local q² active term in the height dynamics from projecting the diffuse-interface model onto interfacial modes and eliminating the gapped profile.

read the letter

The main thing to know is that activity produces a direct local q² drive in the interface height sector through the diffuse structure itself. This competes with passive q⁴ capillary relaxation and stays separate from the |q| hydrodynamic terms that come from long-range momentum transport in the surrounding fluid. The authors start from the Cahn-Hilliard-Landau-de Gennes equations plus Brinkman screening, linearize around a flat interface, project onto the conserved height, amplitude distortion, and transverse director modes, then remove the gapped scalar profile to close the system on the slow variables. That reduction is the concrete step forward. It supplies a framework for spotting when an active nematic-isotropic interface can go unstable even while the bulk nematic stays stable due to screening. The logic tracks the equations without obvious fitting or self-reference, and the distinction between local diffuse-interface and bulk-flow channels is useful for thinking about droplets or biological boundaries. The projection and elimination are standard long-wave moves, but the q² coefficient is sensitive to whether the chosen modes span all leading couplings and whether the adiabatic removal of the profile mode leaves the coefficient untouched. The stress-test note flags exactly that point, and it would help to see an explicit check against a known limit or an error estimate on the truncation. The math is formal and the citations follow the usual active-nematics literature. This is for researchers who model active interfaces or want to separate local versus hydrodynamic instability routes. A reader working on extensions of these theories would get practical value from the reduced operator. It deserves peer review because the question is well-posed and the reduction gives a testable distinction even if some algebra needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a long-wavelength theory for the linear stability of a flat interface between an active nematic and an isotropic fluid. Starting from a diffuse-interface Cahn-Hilliard-Landau-de Gennes description coupled to Brinkman-screened Stokes hydrodynamics, the authors linearize the system, project onto three interfacial degrees of freedom (conserved height mode, scalar amplitude distortion mode, and transverse orientational mode), and adiabatically eliminate the gapped scalar profile mode. This produces a reduced interfacial operator whose eigenvalues reveal the stability. The central result is that activity generates a local q² contribution in the height sector that competes with the passive q⁴ capillary relaxation, defining a local active channel controlled by the diffuse-interface structure and distinct from non-analytic |q| hydrodynamic terms.

Significance. If the mode reduction holds, the work supplies a systematic route to interfacial operators in screened active systems and identifies a diffuse-interface mechanism for instability that can operate even when the homogeneous active nematic is stable. The explicit separation of local (q²) from long-range hydrodynamic (|q|) channels provides a useful classification for active nematic-isotropic interfaces and may guide analysis of confined or screened active matter. The projection-plus-elimination strategy itself is a methodological contribution that could be applied to related problems.

major comments (2)

[§4.2, Eq. (19)] §4.2, Eq. (19): The projection onto the three-mode basis (height, amplitude, transverse director) is presented as spanning the relevant long-wavelength couplings, yet no explicit bound on the residual from omitted modes or higher-gradient terms is given. Because the claimed local active q² term is controlled by the internal diffuse-interface structure, any truncation that feeds into the height sector at O(q²) would directly affect the coefficient and the competition with the q⁴ passive term.
[§5.1, Eq. (25)] §5.1, Eq. (25): The adiabatic elimination of the gapped scalar profile mode assumes a clean timescale separation that remains valid under Brinkman screening. Without a calculation of the leading correction terms generated by the elimination (or a demonstration that they enter only at O(q⁴) or higher), it is unclear whether the reported q² active coefficient receives O(q²) modifications that would alter the claimed local channel.

minor comments (2)

[Figure 3] The dispersion relation plots would benefit from an explicit overlay of the passive (activity = 0) case to highlight the emergence of the q² term.
A short table summarizing the definitions and scaling of the three retained interfacial modes would improve readability when the reduced operator is introduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the methodological contribution, and constructive comments that help sharpen the scope of the long-wavelength reduction. We address each major comment below.

read point-by-point responses

Referee: [§4.2, Eq. (19)] The projection onto the three-mode basis (height, amplitude, transverse director) is presented as spanning the relevant long-wavelength couplings, yet no explicit bound on the residual from omitted modes or higher-gradient terms is given. Because the claimed local active q² term is controlled by the internal diffuse-interface structure, any truncation that feeds into the height sector at O(q²) would directly affect the coefficient and the competition with the q⁴ passive term.

Authors: We agree that an explicit discussion of the truncation error would strengthen the presentation. In the long-wavelength limit the omitted modes remain gapped by the interface width (set by the Cahn-Hilliard and Landau-de Gennes scales) and couple to the height sector only through additional spatial derivatives; their leading contribution therefore appears at O(q⁴) and higher. To make this transparent we will add a short paragraph in §4.2 that estimates the residual and confirms that it does not modify the active q² coefficient at the order retained in the reduced operator. revision: yes
Referee: [§5.1, Eq. (25)] The adiabatic elimination of the gapped scalar profile mode assumes a clean timescale separation that remains valid under Brinkman screening. Without a calculation of the leading correction terms generated by the elimination (or a demonstration that they enter only at O(q⁴) or higher), it is unclear whether the reported q² active coefficient receives O(q²) modifications that would alter the claimed local channel.

Authors: The gap of the scalar profile mode is set by the nematic elasticity and the activity strength inside the diffuse interface and is independent of the hydrodynamic screening length. Brinkman screening accelerates momentum relaxation at long wavelengths but does not close this gap. We will include in the revised §5.1 an explicit perturbative calculation of the leading correction arising from the adiabatic elimination; this correction enters the height-mode dispersion only at O(q⁴) and therefore leaves the active q² term unchanged at leading order. revision: yes

Circularity Check

0 steps flagged

Derivation proceeds from starting hydrodynamic equations via projection and elimination without reduction to inputs by construction

full rationale

The paper starts from the diffuse-interface Cahn-Hilliard-Landau-de Gennes equations coupled to Brinkman hydrodynamics, linearizes around a flat interface, projects onto a basis of height, amplitude, and transverse orientational modes, and adiabatically eliminates the gapped scalar profile mode to obtain the reduced interfacial operator. The q² active term in the height sector is stated to arise from this explicit reduction in the screened regime and is contrasted with passive q⁴ and non-analytic |q| terms. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain; the central result is obtained by algebraic manipulation of the starting PDE system rather than by ansatz or prior-author uniqueness theorems. The procedure is self-contained against the external benchmark of the full linearized system.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not enumerate explicit free parameters or new entities. The theory rests on standard continuum assumptions for active nematics and the validity of the long-wavelength projection, which are not quantified here.

pith-pipeline@v0.9.0 · 5819 in / 1224 out tokens · 37423 ms · 2026-05-22T07:29:50.288642+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

[1]

(22) where S is the usual nematic scalar order parameter

= 1 2S2, (∂kQij)(∂kQij) = 1 2 ( |∇Q1|2 + |∇Q2|2) . (22) where S is the usual nematic scalar order parameter. For a strictly two-dimensional symmetric traceless ten- sor, Tr(Q3) = 0. The cubic invariant in the tensorial three-dimensional free energy therefore drops out of the two-dimensional representation used in the remainder of the paper. The free-energ...

work page
[2]

(23) Deﬁning m(φ,Q 1,Q 2) = a 2 + c 4 (Q2 1 +Q2

+ c 16 (Q2 1 +Q2 2)2 + L 4 ( |∇Q1|2 + |∇Q2|2) . (23) Deﬁning m(φ,Q 1,Q 2) = a 2 + c 4 (Q2 1 +Q2

work page
[3]

(25) Thus, in the absence of explicit anchoring or elastic anisotropy, the passive two-dimensional theory is invari- ant under rotations in the ( Q1,Q 2) plane

− λφ, (24) the molecular ﬁelds read H1 = −mQ1 + L 2 ∇ 2Q1, H 2 = −mQ2 + L 2 ∇ 2Q2. (25) Thus, in the absence of explicit anchoring or elastic anisotropy, the passive two-dimensional theory is invari- ant under rotations in the ( Q1,Q 2) plane. A uniform rotation of a nematic state changes Q1 intoQ2 at no lo- cal free-energy cost. Thus the Q2 sector is the...

work page doi:10.54499/uid/00618/2025 2025
[4]

Translational mode The equilibrium interfacial proﬁles φ 0(y) and S0(y) satisfy the passive Euler–Lagrange equations. The static Hessian in the ( φ,Q 1) sector is HφQ 1 = ( fφφ (y) − κ∂ 2 y fφQ 1 (y) fQ1φ (y) fQ1Q1(y) − L 2∂ 2 y ) , (A1) with all local coeﬃcients evaluated on the equilibrium proﬁles. Diﬀerentiating the Euler–Lagrange equations with re- sp...

work page
[5]

In terms of the static Hessian one may write schemati- cally HφQ 1 ( ψ φ A(y) ψ Q1 A (y) ) =λ A ( ψ φ A(y) ψ Q1 A (y) ) , λ A > 0

Amplitude mode The amplitude mode is chosen as the lowest local- ized non-Goldstone mode in the ( φ,Q 1) sector, separated from the translational Goldstone mode by the orthogo- nality condition imposed in the projected inner product. In terms of the static Hessian one may write schemati- cally HφQ 1 ( ψ φ A(y) ψ Q1 A (y) ) =λ A ( ψ φ A(y) ψ Q1 A (y) ) , λ...

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In the nematic bulk, rotational invariance im- pliesfQ2Q2 → 0

Transverse director mode Fluctuations of the transverse component Q2 are gov- erned, at q = 0, by the static one-dimensional Hessian HT = − L 2∂ 2 y +fQ2Q2 (y), (A8) with the local coeﬃcient evaluated on the equilibrium proﬁles. In the nematic bulk, rotational invariance im- pliesfQ2Q2 → 0. The transverse sector therefore contains the Goldstone mode assoc...

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The translation and amplitude modes are localized interfacial modes: they decay into both bulk phases on the relevant correlation lengths

Asymptotic behaviour and regularisation The three retained functions have diﬀerent asymptotic character. The translation and amplitude modes are localized interfacial modes: they decay into both bulk phases on the relevant correlation lengths. The trans- verse mode decays into the isotropic phase, but in the symmetry-preserving limit approaches a constant...

work page
[8]

The projec- tion metric is therefore weighted by the inverse kinetic operators of the corresponding linearised dynamics

Dynamical inner product The linearised dynamics contains one conserved ﬁeld, φ, and two nonconserved ﬁelds, Q1 and Q2. The projec- tion metric is therefore weighted by the inverse kinetic operators of the corresponding linearised dynamics. For two vector-valued perturbations Ψ =   ψ φ ψ Q1 ψ Q2   , Φ =   ϕ φ ϕ Q1 ϕ Q2   , (B1) we introduce the reg...

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Orthogonality of the retained modes With the inner product deﬁned above, the retained basis may be chosen so that ⟨Ψ h|Ψ A⟩ = ⟨Ψ h|Ψ T ⟩ = ⟨Ψ A|Ψ T ⟩ = 0. (B9) The orthogonality of the transverse mode to the height and amplitude modes follows from the passive block structure at q = 0: Ψ h and Ψ A have support only in the (φ,Q 1) sector, while Ψ T lies in ...

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Normalisation of the amplitude and transverse modes The amplitude mode is localized at the interface and is normalized as ⟨Ψ A|Ψ A⟩ = 1. (B10) For the transverse mode the same convention can be used when the mode is regularized by ﬁnite system size, anchoring, elastic anisotropy, imposed director boundary conditions, or another orientational symmetry-brea...

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[11]

For a long-wavelength displacement, the lead- ing composition perturbation is δφ(x,y,t ) = −h(x,t )φ ′ 0(y), (C5) up to the sign convention used in the deﬁnition of the height mode

Height-sector coeﬃcients The passive height contribution follows from the usual Cahn–Hilliard capillary relaxation of a conserved diﬀuse interface. For a long-wavelength displacement, the lead- ing composition perturbation is δφ(x,y,t ) = −h(x,t )φ ′ 0(y), (C5) up to the sign convention used in the deﬁnition of the height mode. The corresponding capillary...

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[12]

(C21) The transverse shape function is Ψ T (y) =   0 0 ψ T (y)   , (C22) and reduces to Ψ T ∝ (0, 0,S 0)T in the symmetry- preserving limit, as described in Appendix A

Amplitude and transverse couplings The amplitude shape function is Ψ A(y) =   ψ φ A(y) ψ Q1 A (y) 0   , (C20) with projected relaxation rate MAA(0) = − Ω A, Ω A > 0. (C21) The transverse shape function is Ψ T (y) =   0 0 ψ T (y)   , (C22) and reduces to Ψ T ∝ (0, 0,S 0)T in the symmetry- preserving limit, as described in Appendix A. The leading he...

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Elimination of the amplitude mode Since the amplitude mode remains gapped, MAA(q) = − Ω A +O(q2), Ω A > 0, (C34) it may be eliminated perturbatively at long wavelengths. To leading order, the amplitude equation gives Aq = 1 Ω A [ βAhq2hq +iβ ATqT q ] + · · ·, (C35) where corrections from σ and from the O(q2) part of MAA only enter at higher order in the r...

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Height-sector singular structure We write M(q) = Mreg(q) + Msing(q). (D1) Here Mreg is the analytic diﬀuse-interface operator de- rived in Appendix C, while Msing represents the lead- ing non-analytic height contribution associated with an outer-ﬂow transport channel. The decomposition in Eq. (D1) is used only as an eﬀective comparison form; it is not a c...

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Sketch from an outer-ﬂow mobility The origin of Eq. (D3) can be seen from the standard sharp-interface argument [5, 6]. Eliminating the outer pressure and velocity ﬁelds gives a nonlocal relation be- tween the normal interfacial velocity and the force or pressure-jump perturbation acting on the interface. In Fourier space this relation has the form vn(q) ...

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[1] [1]

(22) where S is the usual nematic scalar order parameter

= 1 2S2, (∂kQij)(∂kQij) = 1 2 ( |∇Q1|2 + |∇Q2|2) . (22) where S is the usual nematic scalar order parameter. For a strictly two-dimensional symmetric traceless ten- sor, Tr(Q3) = 0. The cubic invariant in the tensorial three-dimensional free energy therefore drops out of the two-dimensional representation used in the remainder of the paper. The free-energ...

work page

[2] [2]

(23) Deﬁning m(φ,Q 1,Q 2) = a 2 + c 4 (Q2 1 +Q2

+ c 16 (Q2 1 +Q2 2)2 + L 4 ( |∇Q1|2 + |∇Q2|2) . (23) Deﬁning m(φ,Q 1,Q 2) = a 2 + c 4 (Q2 1 +Q2

work page

[3] [3]

(25) Thus, in the absence of explicit anchoring or elastic anisotropy, the passive two-dimensional theory is invari- ant under rotations in the ( Q1,Q 2) plane

− λφ, (24) the molecular ﬁelds read H1 = −mQ1 + L 2 ∇ 2Q1, H 2 = −mQ2 + L 2 ∇ 2Q2. (25) Thus, in the absence of explicit anchoring or elastic anisotropy, the passive two-dimensional theory is invari- ant under rotations in the ( Q1,Q 2) plane. A uniform rotation of a nematic state changes Q1 intoQ2 at no lo- cal free-energy cost. Thus the Q2 sector is the...

work page doi:10.54499/uid/00618/2025 2025

[4] [4]

Translational mode The equilibrium interfacial proﬁles φ 0(y) and S0(y) satisfy the passive Euler–Lagrange equations. The static Hessian in the ( φ,Q 1) sector is HφQ 1 = ( fφφ (y) − κ∂ 2 y fφQ 1 (y) fQ1φ (y) fQ1Q1(y) − L 2∂ 2 y ) , (A1) with all local coeﬃcients evaluated on the equilibrium proﬁles. Diﬀerentiating the Euler–Lagrange equations with re- sp...

work page

[5] [5]

In terms of the static Hessian one may write schemati- cally HφQ 1 ( ψ φ A(y) ψ Q1 A (y) ) =λ A ( ψ φ A(y) ψ Q1 A (y) ) , λ A > 0

Amplitude mode The amplitude mode is chosen as the lowest local- ized non-Goldstone mode in the ( φ,Q 1) sector, separated from the translational Goldstone mode by the orthogo- nality condition imposed in the projected inner product. In terms of the static Hessian one may write schemati- cally HφQ 1 ( ψ φ A(y) ψ Q1 A (y) ) =λ A ( ψ φ A(y) ψ Q1 A (y) ) , λ...

work page

[6] [6]

In the nematic bulk, rotational invariance im- pliesfQ2Q2 → 0

Transverse director mode Fluctuations of the transverse component Q2 are gov- erned, at q = 0, by the static one-dimensional Hessian HT = − L 2∂ 2 y +fQ2Q2 (y), (A8) with the local coeﬃcient evaluated on the equilibrium proﬁles. In the nematic bulk, rotational invariance im- pliesfQ2Q2 → 0. The transverse sector therefore contains the Goldstone mode assoc...

work page

[7] [7]

The translation and amplitude modes are localized interfacial modes: they decay into both bulk phases on the relevant correlation lengths

Asymptotic behaviour and regularisation The three retained functions have diﬀerent asymptotic character. The translation and amplitude modes are localized interfacial modes: they decay into both bulk phases on the relevant correlation lengths. The trans- verse mode decays into the isotropic phase, but in the symmetry-preserving limit approaches a constant...

work page

[8] [8]

The projec- tion metric is therefore weighted by the inverse kinetic operators of the corresponding linearised dynamics

Dynamical inner product The linearised dynamics contains one conserved ﬁeld, φ, and two nonconserved ﬁelds, Q1 and Q2. The projec- tion metric is therefore weighted by the inverse kinetic operators of the corresponding linearised dynamics. For two vector-valued perturbations Ψ =   ψ φ ψ Q1 ψ Q2   , Φ =   ϕ φ ϕ Q1 ϕ Q2   , (B1) we introduce the reg...

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Orthogonality of the retained modes With the inner product deﬁned above, the retained basis may be chosen so that ⟨Ψ h|Ψ A⟩ = ⟨Ψ h|Ψ T ⟩ = ⟨Ψ A|Ψ T ⟩ = 0. (B9) The orthogonality of the transverse mode to the height and amplitude modes follows from the passive block structure at q = 0: Ψ h and Ψ A have support only in the (φ,Q 1) sector, while Ψ T lies in ...

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Normalisation of the amplitude and transverse modes The amplitude mode is localized at the interface and is normalized as ⟨Ψ A|Ψ A⟩ = 1. (B10) For the transverse mode the same convention can be used when the mode is regularized by ﬁnite system size, anchoring, elastic anisotropy, imposed director boundary conditions, or another orientational symmetry-brea...

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For a long-wavelength displacement, the lead- ing composition perturbation is δφ(x,y,t ) = −h(x,t )φ ′ 0(y), (C5) up to the sign convention used in the deﬁnition of the height mode

Height-sector coeﬃcients The passive height contribution follows from the usual Cahn–Hilliard capillary relaxation of a conserved diﬀuse interface. For a long-wavelength displacement, the lead- ing composition perturbation is δφ(x,y,t ) = −h(x,t )φ ′ 0(y), (C5) up to the sign convention used in the deﬁnition of the height mode. The corresponding capillary...

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(C21) The transverse shape function is Ψ T (y) =   0 0 ψ T (y)   , (C22) and reduces to Ψ T ∝ (0, 0,S 0)T in the symmetry- preserving limit, as described in Appendix A

Amplitude and transverse couplings The amplitude shape function is Ψ A(y) =   ψ φ A(y) ψ Q1 A (y) 0   , (C20) with projected relaxation rate MAA(0) = − Ω A, Ω A > 0. (C21) The transverse shape function is Ψ T (y) =   0 0 ψ T (y)   , (C22) and reduces to Ψ T ∝ (0, 0,S 0)T in the symmetry- preserving limit, as described in Appendix A. The leading he...

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Elimination of the amplitude mode Since the amplitude mode remains gapped, MAA(q) = − Ω A +O(q2), Ω A > 0, (C34) it may be eliminated perturbatively at long wavelengths. To leading order, the amplitude equation gives Aq = 1 Ω A [ βAhq2hq +iβ ATqT q ] + · · ·, (C35) where corrections from σ and from the O(q2) part of MAA only enter at higher order in the r...

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Height-sector singular structure We write M(q) = Mreg(q) + Msing(q). (D1) Here Mreg is the analytic diﬀuse-interface operator de- rived in Appendix C, while Msing represents the lead- ing non-analytic height contribution associated with an outer-ﬂow transport channel. The decomposition in Eq. (D1) is used only as an eﬀective comparison form; it is not a c...

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Sketch from an outer-ﬂow mobility The origin of Eq. (D3) can be seen from the standard sharp-interface argument [5, 6]. Eliminating the outer pressure and velocity ﬁelds gives a nonlocal relation be- tween the normal interfacial velocity and the force or pressure-jump perturbation acting on the interface. In Fourier space this relation has the form vn(q) ...

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