{"title":"Curl Equation in Viscous Hydrodynamics in a Channel of Complex Geometry","authors":"S. A. Vasyutkin, A. P. Chupakhin","doi":"10.1134/S1990478923040166","DOIUrl":null,"url":null,"abstract":"<p> We consider the Navier–Stokes equations for the plane steady motion of a viscous\nincompressible fluid in an orthogonal coordinate system in which the fluid streamlines coincide\nwith the coordinate lines of one of the families of the orthogonal coordinate system. In this\ncoordinate system, the velocity vector has only the tangential component and the system of three\nNavier–Stokes equations is an overdetermined system for two functions—the tangential\ncomponent of velocity and pressure. In the present paper, the system is brought to involution, and\nthe consistency conditions are obtained, which are the equations for the curl of the velocity in this\ncoordinate system. The coefficients of these equations include the curvatures of the coordinate\nlines and their derivatives up to the second order. The equations obtained are significantly more\ncomplicated than the curl equations in a channel of simple geometry.\n</p>","PeriodicalId":607,"journal":{"name":"Journal of Applied and Industrial Mathematics","volume":"17 4","pages":"892 - 900"},"PeriodicalIF":0.5800,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied and Industrial Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1134/S1990478923040166","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Engineering","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the Navier–Stokes equations for the plane steady motion of a viscous
incompressible fluid in an orthogonal coordinate system in which the fluid streamlines coincide
with the coordinate lines of one of the families of the orthogonal coordinate system. In this
coordinate system, the velocity vector has only the tangential component and the system of three
Navier–Stokes equations is an overdetermined system for two functions—the tangential
component of velocity and pressure. In the present paper, the system is brought to involution, and
the consistency conditions are obtained, which are the equations for the curl of the velocity in this
coordinate system. The coefficients of these equations include the curvatures of the coordinate
lines and their derivatives up to the second order. The equations obtained are significantly more
complicated than the curl equations in a channel of simple geometry.
期刊介绍：
Journal of Applied and Industrial Mathematics is a journal that publishes original and review articles containing theoretical results and those of interest for applications in various branches of industry. The journal topics include the qualitative theory of differential equations in application to mechanics, physics, chemistry, biology, technical and natural processes; mathematical modeling in mechanics, physics, engineering, chemistry, biology, ecology, medicine, etc.; control theory; discrete optimization; discrete structures and extremum problems; combinatorics; control and reliability of discrete circuits; mathematical programming; mathematical models and methods for making optimal decisions; models of theory of scheduling, location and replacement of equipment; modeling the control processes; development and analysis of algorithms; synthesis and complexity of control systems; automata theory; graph theory; game theory and its applications; coding theory; scheduling theory; and theory of circuits.
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