Working Paper Article Version 1 This version is not peer-reviewed

# Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions

Version 1 : Received: 26 May 2020 / Approved: 31 May 2020 / Online: 31 May 2020 (14:58:11 CEST)

A peer-reviewed article of this Preprint also exists.

Meleshko, S.V.; Pukhnachev, V.V. Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions. Symmetry 2020, 12, 1084. Meleshko, S.V.; Pukhnachev, V.V. Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions. Symmetry 2020, 12, 1084.

## Abstract

The famous Toms effect (1948) consists of a substantial increase of the critical Reynolds number when a small amount of soluble polymer is introduced into water. The most noticeable influence of polymer additives is manifested in the boundary layer near solid surfaces. The goal of the present paper is a group analysis of the boundary layer equations in two mathematical models of the flow of aqueous polymer solutions: the second grade fluid (Rivlin and Ericksen, 1955) and the model derived by Pavlovskii (1971). The equations of the unsteady two-dimensional boundary layer in the Pavlovskii and Rivlin-Ericksen models are analyzed for the first time here. These equations have no definite type so that finding their exact solutions is very important in order to understand the mathematical nature of the above mentioned models. The problem of group classification with respect to the arbitrary function of the longitudinal coordinate and time present in the equations, which sets the pressure gradient of the external flow, arises. All functions for which an extension of the admitted Lie group occurs are found. The task includes the ratio of two characteristic length scales. One of them is the Prandtl scale, and another is defined as the square root of the normalized coefficient of relaxation viscosity (Frolovskaya and Pukhnachev, 2018) and does not depend on the characteristics of the motion. The paper contains a number of exact solutions in the Pavlovskii model including a solution describing the flow near a critical point. Among the solutions of the new model of the boundary layer, a special place is taken by the solution of the stationary problem of flow around a rectilinear plate. Within the framework of the Prandtl theory of the boundary layer, such a solution was constructed by Blasius (1908). As is well-known, this solution has a non-removable defect: the transverse velocity near the edge of the plate increases without bound. The introduction of a relaxation term into the model makes it possible to eliminate this singularity.

## Keywords

symmetry; admitted Lie group; invariant solution; boundary layer equations; polymer solution

## Subject

MATHEMATICS & COMPUTER SCIENCE, Applied Mathematics

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