Quenched Dipole Pairs in Viscous Fluid Membranes across the Saffman Crossover: Integrable Hamiltonian Dynamics

TL;DR

This paper develops an analytic theory of quenched force-dipole interactions in viscous membranes, revealing a Saffman crossover that reorganizes the dynamics from one-dimensional to coupled two-dimensional behavior, with implications for membrane aggregation.

Contribution

It introduces a new integrable Hamiltonian framework for quenched dipole interactions across the Saffman crossover, highlighting a transition in the phase-space structure of membrane hydrodynamics.

Findings

Near-field dynamics is exactly solvable and effectively one-dimensional.

Far-field dynamics remains integrable but becomes intrinsically two-dimensional.

Pullers exhibit late-time collapse with R∼(tc−t)^{1/3} scaling.

Abstract

We investigate an analytic theory of force-dipole hydrodynamics in a viscous membrane coupled to an infinite surrounding fluid, focusing on quenched (orientation-fixed) dipoles. While the single-dipole flow exhibits the known Saffman crossover from a near-field $v\sim r^{-1}$ to a screened far-field $v\sim r^{-2}$, we show that this crossover induces a qualitatively new reorganization of dipole--dipole interactions. For two identical quenched dipoles, the near-field dynamics is exactly solvable and effectively one-dimensional, with a fixed line of centers and linear evolution of the squared separation. In the far field, the system remains integrable but becomes intrinsically two-dimensional, with coupled radial and angular dynamics and an exact first integral. For pullers, the angular dynamics drives alignment toward an attracting manifold, leading to universal late-time collapse $R\sim (t_c-t)^{1/3}$, in contrast to the near-field scaling $R\sim (t_c-t)^{1/2}$. The Saffman crossover thus reorganizes the Hamiltonian phase-space structure of dipolar interactions and produces a transition from effectively one-dimensional to fully coupled dynamics, providing a minimal framework for aggregation in viscous fluid membranes.