Transient dispersion process of active particles

Abstract

Active particles often swim in confined environments. The transport mechanisms, especially the global one as reflected by the Taylor dispersion model, are of great practical interest to various applications. For the active dispersion process in confined flows, previous analytical studies focused on the long-time asymptotic values of dispersion characteristics. Only several numerical studies preliminarily investigated the temporal evolution. Extending recent studies of Jiang & Chen (J. Fluid Mech., vol. 877, 2019, pp. 1–34; vol. 899, 2020, A18), this work makes a semi-analytical attempt to investigate the transient process. The temporal evolution of the local distribution in the confined-section–orientation space, drift, dispersivity and skewness, is explored based on moments of distributions. We introduce the biorthogonal expansion method for solutions because the classic integral transform method for passive transport problems is not applicable due to the self-propulsion effect. Two types of boundary condition, the reflective condition and the Robin condition for wall accumulation, are imposed respectively. A detailed study on spherical and ellipsoidal swimmers dispersing in a plane Poiseuille flow demonstrates the influences of the swimming, shear flow, initial condition, wall accumulation and particle shape on the transient dispersion process. The swimming-induced diffusion makes the local distribution reach its equilibrium state faster than that of passive particles. Although the wall accumulation significantly affects the evolution of the local distribution and the drift, the time scale to reach the Taylor regime is not obviously changed. The shear-induced alignment of ellipsoidal particles can enlarge the dispersivity but impacts slightly on the drift and the skewness.

Gyrotactic trapping of micro-swimmers in simple shear flows: a study directly from the fundamental Smoluchowski equation

“…Following Barton (1983) and Jiang & Chen (2021), we write the transformed local moments in the form of eigenfunction expansion: where and are the eigenvalues and eigenfunctions of operator with sorted in an ascending order. Here are the undetermined expansion coefficients, and the corresponding eigenfunction characterises the steady state, which has been discussed in Jiang & Chen (2019, 2020) and Wang et al.…”

“…The eigenvalue problem of is however, this eigenvalue problem cannot be easily solved due to the complexity of the operator . Following Jiang & Chen (2021), we approximate the solution with a Galerkin method. With the basis functions given in (2.29), can be expanded as where are the coefficients.…”

“…However, in the current situation, are not orthogonal because is not self-adjoint. Jiang & Chen (2021) used another set of eigenfunctions, namely the dual basis functions satisfying the biorthogonality relation with if both are normalised (Strand et al. 1987; Brezinski 1992) where is the Kronecker delta.…”

“…Long-time asymptotic solution can be obtained with the new GTD method, referred to as ‘one-step GTD method’. For the transient dispersion process, Jiang & Chen (2021) derived the time-dependent local moments of probability density function (p.d.f.) of micro-swimmers in the positional–orientational space, by following the expansion of moments derived by Barton (1983) and resorting to the biorthogonal expansion method to the eigenfunctions (Strand, Kim & Karrila 1987; Brezinski 1992).…”

“…Using the method in Jiang & Chen (2021), i.e. eigenfunction expansion of moments with expansion coefficients determined with the biorthogonal expansion technique, the main purpose of this work is to make an analysis of the transient transport of gyrotactic micro-swimmers in representative simple shear flows.…”