On the asymptotic analysis of the JWKB method via change of dependent variable in the first-order Bessel’s equation

The first-order Jeffreys–Wentzel–Kramers–Brillouin method (called (JWKB)1) is a conventional semi-classical approximation method used in quantum mechanical systems for accurate solutions. It is known to give accurate energy and wave-function in the classically accessible region of the related quantum mechanical system defined by Schroedinger’s equation whereas the solutions in the classically inaccessible region require special treatment, conventionally known as the asymptotic matching rules. In this work, (JWKB)1 solution of the Bessel differential equation of the first order (called (BDE)1), chosen as a mathematical model, is studied by being transformed into the normal form via the change of dependent variable. General JWKB solution of the initial value problem where appropriately chosen initial values are applied is studied in both normal and standard form representations to be analyzed by the generalized JWKB asymptotic matching rules regarding the Sij matrix elements defined in the literature. Consequently, regions requiring first-order and zeroth-order JWKB approximations are determined successfully.

SOLUSI ASIMTOTIK PADA PERSAMAAN DIFUSI DENGAN WAKTU SINGKAT

Abstrak. Pada makalah ini akan ditentukan solusi asimtotik dari persamaan difusidengan waktu singkat pada kasus satu dimensi. Untuk mengatasi syarat awal yang takkontinuloncat di x = 0, digunakan metode pencocokan asimtotik (asymptotic matching)yang dianalisis pada tiga daerah yang berbeda, yaitu daerah I (x < 0), daerah II (x > 0),dan daerah III (jxj 1). Hasil yang diperoleh dapat menjelaskan bagaimana perilakusolusi sesaat setelah kondisi awal mulai berdifusi.Kata Kunci: Solusi asimtotik, persamaan difusi, pencocokan asimtotik

Two-dimensional phoretic swimmers: the singular weak-advection limits

Because of the associated far-field logarithmic divergence, the transport problem governing two-dimensional phoretic self-propulsion lacks a steady solution when the Péclet number $\mathit{Pe}$ vanishes. This indeterminacy, which has no counterpart in three dimensions, is remedied by introducing a non-zero value of $\mathit{Pe}$, however small. We consider that problem employing a first-order kinetic model of solute absorption, where the ratio of the characteristic magnitudes of reaction and diffusion is quantified by the Damköhler number $\mathit{Da}$. As $\mathit{Pe}\rightarrow 0$ the dominance of diffusion breaks down at distances that scale inversely with $\mathit{Pe}$; at these distances, the leading-order transport represents a two-dimensional point source in a uniform stream. Asymptotic matching between the latter region and the diffusion-dominated near-particle region provides the leading-order particle velocity as an implicit function of $\log \mathit{Pe}$. Another scenario involving weak advection takes place under strong reactions, where $\mathit{Pe}$ and $\mathit{Da}$ are large and comparable. In that limit, the breakdown of diffusive dominance occurs at distances that scale as $\mathit{Da}^{2}/\mathit{Pe}$.

Boundary-induced autophoresis of isotropic colloids: anomalous repulsion in the lubrication limit

When suspended in a liquid solution, chemically active colloids may self-propel due to an asymmetry in either particle shape or the interfacial distribution of solute absorption. We here consider a chemically homogeneous spherical particle which undergoes self-diffusiophoresis due to the presence of nearby inert wall. In particular, we focus upon the near-contact limit where it was recently observed (Yariv, Phys. Rev. Fluids, vol. 1 (3), 2016, 032101) that the solute-concentration profile within the narrow gap separating the particle and the wall cannot be uniquely determined by a gap-scale analysis. We here revisit this near-contact limit using matched asymptotic expansions, the inner region being the gap domain and the outer region being on the particle scale. Asymptotic matching with the Hankel-transform representation of the outer distribution of solute concentration serves to determine both the scaling and magnitude of the corresponding inner profile. The ensuing gap-scale pressure field, set by a lubrication mechanism, gives rise to an anomalous particle–wall interaction, scaling as an irrational power of the gap clearance.