2. 3. M. A. Lavrent'ev and B. V. Shabat, Methods in the Theory of Functions of a Complex Variable [in Rus- sian], Izd. Gostekhizdat, Moscow (1958). Yu. I. Kapranov, "Exact solutions in problems of the desalination of soils," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1 (1972). STRUCTURE OF THE DIFFUSIONAL WAKE OF AN ABSORBING PARTICLE NEAR THE CRITICAL LINES A. D. Polyanin UDC 532.72 INTRODUCTION In problems of the calculation of mass transfer between particles and a flow, it is important to evaluate the perturbing effect of an absorbing particle on the originally homogeneous field of the concentration in the flow. This effect depends to a considerable degree on the distribution of the velocities of the flow. For a particle located in a homogeneous translational flow and other flows with isolated critical points, the corre- sponding analysis has been made earlier [1-3]. In the present work, an investigation was made of the field of the concentration near an absorbing parti- cle, around which the flow is characterized by the presence of critical lines (for example, homogeneous trans- lational flow around a particle [4]). The method of combined asymptotic expansions (with respect to a large Peclet number P) is used to ob- tain the distribution of the concentration and the diffusional flux to the surface of the absorbing particle near the critical line of the outflow (the line in whose vicinity the normal component of the velocity of the liquid is directed away from the surface), for example, the boundary of the region of closed circulation at the body. The article discusses the case of the plane or axisymmetric laminar flow of a viscous incompressible liquid around a particle. A sharp difference in the distribution of the concentration and in the structure of the diffusion wake with the presence of an isolated critical point [1-3] and a critical line is demonstrated. Here, in the case of a criti- cal line, the whole change of the concentration in the wake takes place at distances on the order of p-t/~ (in the case of a critical point, at distances on the order of p1/3). w In a system of coordinates connected with the particle, we write the equation of steady-state convec- tive diffusion and the boundary conditions under the assumption of total absorption of the diffusing substance at the surface of the particle and the constancy of the concentration far from it: Oc + v~ Oc 3'03c'+ kOc i 02c k;ictg0~ ~ (i.I) c],=Rco~=O, c [ ~.**=l (P=s-3=aU/D) v__[rsinO],. ~ t 04 . vo-------trsinO]'=~ O~ r 00 ' Or Here k = 1 {k = 2) corresponds to the plane (axisymmetric) case; r, 0 is a cylindrical (spherical) system of coordinates, where the angle 0 is reckoned from the outflow line under discussion (from an arbitrary critical point on the surface of the body); c is the dimensionless concentration; a is the characteristic dimension of the particle; U is the characteristic velocity of the flow; and r = R(0) is the equation of the surface of the body. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekbanika Zhidkosti i Gaza, No. 3, pp. 82-86, May-June, 1977. Original article submitted September 15, 1976. [ This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 West 17th Street, "New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written p~mission of the publisher. A copy of this article is available from the publisher for $7.50. I 410