Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. I. Adjoint method

Ory Schnitzer

Phys. Rev. Fluids 8, 034201 – Published 13 March 2023

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 generalized 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, nonuniform 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.

Received 13 May 2022
Accepted 18 July 2022

DOI:https://doi.org/10.1103/PhysRevFluids.8.034201

Physics Subject Headings (PhySH)

Self-propelled particles

Perturbative methods

Fluid Dynamics

Authors & Affiliations

Ory Schnitzer

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. II. History-dependent motion

Gunnar G. Peng and Ory Schnitzer

Phys. Rev. Fluids 8, 033602 (2023)

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Vol. 8, Iss. 3 — March 2023

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Figure 1
Steady solutions: $U \sim ε U_{1}$ as $ε ↘ 0$ , with $Pe = 4 + ε χ$ (a) Canonical isotropic model (Sec. 2). $U_{1}$ satisfies the amplitude equation (2.37) describing a singular-pitchfork bifurcation from a symmetric-stationary state, which exists for all $χ$ , to spontaneous-motion states of magnitude $| U_{1} | = χ / 16$ and arbitrary direction, which exist for $χ > 0$ . (In this case, it suffices to consider $χ = \pm 1$ , without loss of generality.) Stability is indicated based on linear stability analysis of the stationary-symmetric state [12] and numerical simulations [11]. (b) External force field $6 π ε^{2} \hat{ı}$ , where $\hat{ı}$ is a unit vector (Sec. 4a1). Here $U_{1} = U_{∥} \hat{ı}$ , with $U_{∥}$ satisfying the amplitude equation (4.4) describing an imperfect pitchfork bifurcation of spontaneous-motion states, which are restricted to the directions parallel or anti-parallel to the force field. (c) External torque field $8 π ε \hat{ı}$ (Sec. 4a2). We have $U_{1} = U_{∥} \hat{ı}$ , with $U_{∥}$ satisfying the amplitude equation (4.15) describing a singular-pitchfork bifurcation of spontaneous-motion states restricted to the directions parallel or anti-parallel to the torque field; note that the spontaneous speed is identical to that in the isotropic canonical case.
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Figure 2
Steady solutions $U \sim ε U / \sqrt{8}$ as $ε ↘ 0$ , with $Pe = 4 + 16 ε X / \sqrt{8}$ , in the scenario of a force field $6 π ε^{2} \hat{ı}$ perpendicular to a torque field $(80 π \sqrt{2} / 11) ε W \hat{ȷ}$ (Sec. 4a3). (a) There are up to two solutions perpendicular to the force, depending on the sign of $X W - 1$ . As $W ↗ \infty$ , these solutions tend to the nontrivial states in the torque scenario. (b) There are between one and three solutions perpendicular to the torque. As $W ↘ 0$ , these solutions limit to the solutions in the force scenario. As $W ↗ \infty$ , there is one such solution that approaches the trivial state.
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Figure 3
Steady solutions $U \sim ε (U_{∥} \hat{p} + U_{⊥})$ as $ε ↘ 0$ , with $Pe = 4 + ε χ$ and $U_{⊥} \cdot \hat{p} = 0$ , in the scenario of flux and slip-coefficient perturbations symmetric about an axis defined by the unit vector $\hat{p}$ and the particle's centroid (Sec. 4b2). Nonlongitudinal solutions with $U_{⊥} \neq 0$ are only consistent in the case $β_{r} = 0$ [cf. Eq. (4.35)]; for such solutions, the direction of $U_{⊥}$ normal to $\hat{p}$ is arbitrary. Solution branches are shown as a function of the shifted bifurcation parameter $\bar{χ} = χ + β_{I}$ , for $\bar{α} = 0$ (solid lines), $\bar{α} = 0.2$ (dashed curves) and $\bar{α} = 0.4$ (dash-dotted curves), in the two cases (a): $\bar{β} = - 1$ , and (b): $\bar{β} = 1$ . For definitions of the lumped perturbation parameters $\bar{α}, \bar{χ}$ and $\bar{β}$ , see Eqs. (4.38) and (4.39a).
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Figure 4
Effect of bulk absorption with Damkohler number ${Da}_{b} ≪ 1$ (Sec. 4d). Steady solutions $U \sim \sqrt{{Da}_{b}} U_{1}$ , with $Pe = 4 + χ \sqrt{{Da}_{b}}$ , $U_{1}$ having an arbitrary direction and magnitude as plotted as a function of $χ$ ; as $χ \to \infty$ , the solutions approach those in the canonical isotropic model.
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