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Force Dipole Interactions in Membranes with Odd Viscosity

Sneha Krishnan, Udaya Maurya, Rickmoy Samanta

Abstract

We develop a hydrodynamic framework for the interactions and collective dynamics of force dipoles embedded in a compressible fluid membrane supported by a shallow viscous subphase. Starting from the generalized two-dimensional Stokes equations with shear, dilatational, and odd (Hall) viscosities, we derive an exact real-space Green tensor using Hankel transforms. The resulting tensor is characterized by three hydrodynamic screening scales associated with shear, compressional, and odd-viscous modes, and smoothly reduces to the standard limiting cases of incompressible membranes and compressible parity-symmetric membranes, while also capturing the chiral response generated by odd viscosity. Using this Green tensor we obtain the velocity and vorticity fields generated by a force dipole and formulate the dynamical system governing interacting dipoles. The analysis reveals several distinct dynamical regimes and identifies observables that isolate the antisymmetric odd-viscous contribution to dipole interactions, including transverse drift and chiral relative motion.

Force Dipole Interactions in Membranes with Odd Viscosity

Abstract

We develop a hydrodynamic framework for the interactions and collective dynamics of force dipoles embedded in a compressible fluid membrane supported by a shallow viscous subphase. Starting from the generalized two-dimensional Stokes equations with shear, dilatational, and odd (Hall) viscosities, we derive an exact real-space Green tensor using Hankel transforms. The resulting tensor is characterized by three hydrodynamic screening scales associated with shear, compressional, and odd-viscous modes, and smoothly reduces to the standard limiting cases of incompressible membranes and compressible parity-symmetric membranes, while also capturing the chiral response generated by odd viscosity. Using this Green tensor we obtain the velocity and vorticity fields generated by a force dipole and formulate the dynamical system governing interacting dipoles. The analysis reveals several distinct dynamical regimes and identifies observables that isolate the antisymmetric odd-viscous contribution to dipole interactions, including transverse drift and chiral relative motion.
Paper Structure (26 sections, 313 equations, 13 figures)

This paper contains 26 sections, 313 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Basic setup and analytic solution using Hankel transform
  4. Comments on screening lengths
  5. Useful limits of the membrane Green tensor, fluid velocity and vorticity
  6. Compressible limit with vanishing odd viscosity
  7. Incompressible parity--symmetric limit of the Green tensor
  8. Degenerate identical-screening limit
  9. Near Field
  10. Far field
  11. Dynamical system for interacting force dipoles on the membrane
  12. COM drift and Polarization evolution equations
  13. Pair Geometry
  14. Specialization to the two--dipole system
  15. Two--dipole dynamics in the near--field degenerate limit
  16. ...and 11 more sections

Figures (13)

  • Figure 1: Schematic illustration of interacting force dipole motors confined to a fluid interface. Each yellow ellipse represents a particle, with the double-headed arrow indicating its orientation. The light blue layer corresponds to a two-dimensional membrane with viscosity $\eta_s, \eta_d, \eta_o$, supported by a fluid subphase of viscosity $\eta$ above a rigid substrate.
  • Figure 2: Near–zone dynamics of pusher and puller motors in an incompressible supported membrane with dynamical orientations. Rows: axial, side-by-side, perpendicular, random pair, 12-dipole cluster. Columns: trajectories and mean pair separation for pusher (left) and puller (right) dipoles.
  • Figure 3: Far–zone dynamics of pusher and puller dipoles in an incompressible supported membrane with dynamical orientations. Rows correspond to axial, side-by-side, perpendicular, random, and cluster configurations. Columns show trajectories and mean pair separations for pushers (left) and pullers (right).
  • Figure 4: Near–zone dynamics of pusher and puller dipoles in an incompressible supported membrane with quenched orientations. Rows correspond to axial, side-by-side, perpendicular, random, and cluster configurations. Columns show trajectories and mean separations for pushers (left) and pullers (right).
  • Figure 5: Far–zone dynamics of pusher and puller dipoles in an incompressible supported membrane with quenched orientations. Rows correspond to axial, side-by-side, perpendicular, random, and cluster configurations. Columns show trajectories and mean separations for pushers (left) and pullers (right).
  • ...and 8 more figures