Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. Part 1: Adjoint method

In this Series, we revisit the weakly nonlinear theory describing the dynamics of chemically active particles near the threshold for spontaneous motion. In this Part, we develop an 'adjoint method' for deriving nonlinear amplitude equations governing spontaneous motion for a canonical model of an isotropic chemically active particle and general perturbations thereof; we focus on steady solutions, leaving separate subtleties associated with unsteadiness to a subsequent Part. Our method is based on identifying the operator adjoint to that encountered at linear order of the weakly nonlinear expansion, the latter's kernel describing steady rectilinear motion of the particle with an arbitrary velocity vector; the kernel of the adjoint operator, in which the roles of solute transport and liquid flow are transposed, also describes steady rectilinear motion although the physical mechanism driving spontaneous motion is unfamiliar. The adjoint operator and its kernel imply a solvability condition on the inhomogeneous problem encountered at quadratic order of the weakly nonlinear expansion, which constitutes the amplitude equation. Our adjoint method circumvents the need to directly solve the inhomogeneous problem, promoting fully three-dimensional analyses and adding insight by allowing to treat a wide range of physical scenarios on a common basis. Beyond the canonical model, we apply our method to several perturbations scenarios having a leading-order effect sufficiently near the threshold: external force fields, non-uniform surface properties, first-order surface kinetics and bulk absorption. The amplitude equations we obtain in these cases present various effects relative to the nominally isotropic and singular pitchfork bifurcation familiar from the canonical model: bifurcation shift and regularisation, imperfection and restrictions of the motion to sets of direction in three dimensions.