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

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 task includes the ratio of...
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Autor*in:	Sergey V. Meleshko [verfasserIn] Vladislav V. Pukhnachev [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2020

Schlagwörter:	symmetry admitted Lie group invariant solution boundary layer equations

Übergeordnetes Werk:	In: Symmetry - MDPI AG, 2009, 12(2020), 7, p 1084
Übergeordnetes Werk:	volume:12 ; year:2020 ; number:7, p 1084

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.3390/sym12071084

Katalog-ID:	DOAJ02592589X

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source	In Symmetry 12(2020), 7, p 1084 volume:12 year:2020 number:7, p 1084
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callnumber-first	Q - Science
author	Sergey V. Meleshko
spellingShingle	Sergey V. Meleshko misc QA1-939 misc symmetry misc admitted Lie group misc invariant solution misc boundary layer equations misc Mathematics Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions
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title	Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions
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title_sort	group analysis of the boundary layer equations in the models of polymer solutions
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title_auth	Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions
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 task includes the ratio of two characteristic length scales, one of which is the Prandtl scale, and the other 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. In the limit case, when the ratio of these two scales tends to zero, the equations of the boundary layer are exactly integrated. One of the goals of the present paper is 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 Pavlovskii model (1971). The equations of the plane non-stationary boundary layer in the Pavlovskii model are studied in more details. The equations contain an arbitrary function depending on the longitudinal coordinate and time. This function sets the pressure gradient of the external flow. The problem of group classification with respect to this function is analyzed. All functions for which there is an extension of the kernels of admitted Lie groups are found. Among the invariant 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.
abstractGer	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 task includes the ratio of two characteristic length scales, one of which is the Prandtl scale, and the other 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. In the limit case, when the ratio of these two scales tends to zero, the equations of the boundary layer are exactly integrated. One of the goals of the present paper is 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 Pavlovskii model (1971). The equations of the plane non-stationary boundary layer in the Pavlovskii model are studied in more details. The equations contain an arbitrary function depending on the longitudinal coordinate and time. This function sets the pressure gradient of the external flow. The problem of group classification with respect to this function is analyzed. All functions for which there is an extension of the kernels of admitted Lie groups are found. Among the invariant 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.
abstract_unstemmed	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 task includes the ratio of two characteristic length scales, one of which is the Prandtl scale, and the other 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. In the limit case, when the ratio of these two scales tends to zero, the equations of the boundary layer are exactly integrated. One of the goals of the present paper is 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 Pavlovskii model (1971). The equations of the plane non-stationary boundary layer in the Pavlovskii model are studied in more details. The equations contain an arbitrary function depending on the longitudinal coordinate and time. This function sets the pressure gradient of the external flow. The problem of group classification with respect to this function is analyzed. All functions for which there is an extension of the kernels of admitted Lie groups are found. Among the invariant 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.
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title_short	Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions
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