Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion: Adjoint method
File(s)
Accepted version
Author(s)
Schnitzer, Ory
Type
Journal Article
Abstract
In this Series, we study the weakly nonlinear dynamics of chemically active particles near the
threshold for spontaneous motion. In this Part, we focus on steady solutions and develop an
‘adjoint method’ for deriving the nonlinear amplitude equation governing the particle’s velocity,
first assuming the canonical model in the literature of an isotropic chemically active particle and
then considering general perturbations about that model. As in previous works, the amplitude
equation is obtained from a solvability condition on the inhomogeneous problem at second order of
a particle-scale weakly nonlinear expansion, the formulation of that problem involving asymptotic
matching with a leading-order solution in a remote region where advection and diffusion are balanced. We develop a generalised solvability condition based on a Fredholm Alternative argument,
which entails identifying the adjoint linear operator at the threshold and calculating its kernel.
This circumvents the apparent need in earlier theories to solve the second-order inhomogeneous
problem, resulting in considerable simplification and adding insight by making it possible to treat
a wide range of perturbation scenarios on a common basis. To illustrate our approach, we derive
and solve amplitude equations for a number of perturbation scenarios (external force and torque
fields, non-uniform surface properties, first-order surface kinetics and bulk absorption), demonstrating that sufficiently near the threshold weak perturbations can appreciably modify and enrich
the landscape of steady solutions.
threshold for spontaneous motion. In this Part, we focus on steady solutions and develop an
‘adjoint method’ for deriving the nonlinear amplitude equation governing the particle’s velocity,
first assuming the canonical model in the literature of an isotropic chemically active particle and
then considering general perturbations about that model. As in previous works, the amplitude
equation is obtained from a solvability condition on the inhomogeneous problem at second order of
a particle-scale weakly nonlinear expansion, the formulation of that problem involving asymptotic
matching with a leading-order solution in a remote region where advection and diffusion are balanced. We develop a generalised solvability condition based on a Fredholm Alternative argument,
which entails identifying the adjoint linear operator at the threshold and calculating its kernel.
This circumvents the apparent need in earlier theories to solve the second-order inhomogeneous
problem, resulting in considerable simplification and adding insight by making it possible to treat
a wide range of perturbation scenarios on a common basis. To illustrate our approach, we derive
and solve amplitude equations for a number of perturbation scenarios (external force and torque
fields, non-uniform surface properties, first-order surface kinetics and bulk absorption), demonstrating that sufficiently near the threshold weak perturbations can appreciably modify and enrich
the landscape of steady solutions.
Date Issued
2023-03-13
Date Acceptance
2022-07-18
Citation
Physical Review Fluids, 2023, 8 (3)
ISSN
2469-990X
Publisher
American Physical Society
Journal / Book Title
Physical Review Fluids
Volume
8
Issue
3
Copyright Statement
©2023 American Physical Society.
Publication Status
Published
Article Number
ARTN 034201