Couette flow in channels with wavy walls
Summary Three–dimensional Couette flows enclosed by a plane and by a wavy wall are addressed; the wave amplitude is proportional to the mean clearance of the channel multiplied by a small dimensionless parameter ɛ. A perturbation expansion in terms of the powers of ɛ of the full steady Navier–Stokes...
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Veröffentlicht in: | Acta mechanica 2008-05, Vol.197 (3-4), p.247-283 |
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description | Summary
Three–dimensional Couette flows enclosed by a plane and by a wavy wall are addressed; the wave amplitude is proportional to the mean clearance of the channel multiplied by a small dimensionless parameter ɛ. A perturbation expansion in terms of the powers of ɛ of the full steady Navier–Stokes equations yields a cascade of boundary value problems which are solved at each step in closed form. The supremum value of ɛ for which the expansion converges, is determined as a function of the Reynolds number
The analytical-numerical algorithm is applied to compute the velocity in the channel to
O
(ɛ
4
). Even in the first order approximation
O
(ɛ), new results are obtained which complement the triple deck theory and its modifications. In particular, the incipient separation–detachment is discussed using the Prandtl-Schlichting criterion of starting eddies. The value ɛ
e
for which eddies start in the channel, is analytically deduced as a function of
as well as analytical formulas for the coordinates of the separation points. These analytical formulas show that ɛ
e
in 3D channels is always less than ɛ
e
in 2D channels. For non-smooth channels, a criterion of infinitesimally small ɛ
e
is deduced. The critical value of ɛ up to which bifurcation of the solutions can occur is estimated. |
doi_str_mv | 10.1007/s00707-007-0507-z |
format | Article |
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Three–dimensional Couette flows enclosed by a plane and by a wavy wall are addressed; the wave amplitude is proportional to the mean clearance of the channel multiplied by a small dimensionless parameter ɛ. A perturbation expansion in terms of the powers of ɛ of the full steady Navier–Stokes equations yields a cascade of boundary value problems which are solved at each step in closed form. The supremum value of ɛ for which the expansion converges, is determined as a function of the Reynolds number
The analytical-numerical algorithm is applied to compute the velocity in the channel to
O
(ɛ
4
). Even in the first order approximation
O
(ɛ), new results are obtained which complement the triple deck theory and its modifications. In particular, the incipient separation–detachment is discussed using the Prandtl-Schlichting criterion of starting eddies. The value ɛ
e
for which eddies start in the channel, is analytically deduced as a function of
as well as analytical formulas for the coordinates of the separation points. These analytical formulas show that ɛ
e
in 3D channels is always less than ɛ
e
in 2D channels. For non-smooth channels, a criterion of infinitesimally small ɛ
e
is deduced. The critical value of ɛ up to which bifurcation of the solutions can occur is estimated.</description><identifier>ISSN: 0001-5970</identifier><identifier>EISSN: 1619-6937</identifier><identifier>DOI: 10.1007/s00707-007-0507-z</identifier><identifier>CODEN: AMHCAP</identifier><language>eng</language><publisher>Vienna: Springer-Verlag</publisher><subject>Algorithms ; Channels ; Classical and Continuum Physics ; Control ; Couette flow ; Criteria ; Dynamical Systems ; Engineering ; Engineering Thermodynamics ; Exact sciences and technology ; Flows in ducts, channels, nozzles, and conduits ; Fluid dynamics ; Fundamental areas of phenomenology (including applications) ; General theory ; Heat and Mass Transfer ; Mathematical analysis ; Mathematical models ; Mathematics ; Mechanical engineering ; Navier-Stokes equations ; Physics ; Reynolds number ; Rotational flow and vorticity ; Solid Mechanics ; Theoretical and Applied Mechanics ; Theory ; Three dimensional ; Vibration ; Walls</subject><ispartof>Acta mechanica, 2008-05, Vol.197 (3-4), p.247-283</ispartof><rights>Springer-Verlag Wien 2007</rights><rights>2008 INIST-CNRS</rights><rights>Springer-Verlag Wien 2008</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c442t-b543865acbc06b7b9d4bb6c4f1783456143d86fff65506e9c020dd211572f0f83</citedby><cites>FETCH-LOGICAL-c442t-b543865acbc06b7b9d4bb6c4f1783456143d86fff65506e9c020dd211572f0f83</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00707-007-0507-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00707-007-0507-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=20361788$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Malevich, A. E.</creatorcontrib><creatorcontrib>Mityushev, V. V.</creatorcontrib><creatorcontrib>Adler, Pierre M.</creatorcontrib><title>Couette flow in channels with wavy walls</title><title>Acta mechanica</title><addtitle>Acta Mech</addtitle><description>Summary
Three–dimensional Couette flows enclosed by a plane and by a wavy wall are addressed; the wave amplitude is proportional to the mean clearance of the channel multiplied by a small dimensionless parameter ɛ. A perturbation expansion in terms of the powers of ɛ of the full steady Navier–Stokes equations yields a cascade of boundary value problems which are solved at each step in closed form. The supremum value of ɛ for which the expansion converges, is determined as a function of the Reynolds number
The analytical-numerical algorithm is applied to compute the velocity in the channel to
O
(ɛ
4
). Even in the first order approximation
O
(ɛ), new results are obtained which complement the triple deck theory and its modifications. In particular, the incipient separation–detachment is discussed using the Prandtl-Schlichting criterion of starting eddies. The value ɛ
e
for which eddies start in the channel, is analytically deduced as a function of
as well as analytical formulas for the coordinates of the separation points. These analytical formulas show that ɛ
e
in 3D channels is always less than ɛ
e
in 2D channels. For non-smooth channels, a criterion of infinitesimally small ɛ
e
is deduced. The critical value of ɛ up to which bifurcation of the solutions can occur is estimated.</description><subject>Algorithms</subject><subject>Channels</subject><subject>Classical and Continuum Physics</subject><subject>Control</subject><subject>Couette flow</subject><subject>Criteria</subject><subject>Dynamical Systems</subject><subject>Engineering</subject><subject>Engineering Thermodynamics</subject><subject>Exact sciences and technology</subject><subject>Flows in ducts, channels, nozzles, and conduits</subject><subject>Fluid dynamics</subject><subject>Fundamental areas of phenomenology (including applications)</subject><subject>General theory</subject><subject>Heat and Mass Transfer</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Mechanical engineering</subject><subject>Navier-Stokes equations</subject><subject>Physics</subject><subject>Reynolds number</subject><subject>Rotational flow and vorticity</subject><subject>Solid Mechanics</subject><subject>Theoretical and Applied Mechanics</subject><subject>Theory</subject><subject>Three dimensional</subject><subject>Vibration</subject><subject>Walls</subject><issn>0001-5970</issn><issn>1619-6937</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2008</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>BENPR</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNqNkV1LwzAUhoMoOKc_wLsiKLupnqT56qUMv2DgjV6XNEtcR9bOpHVsv96UDgVB8SInJ-Q5L-flRegcwzUGEDchFhAp9IfFsjtAI8xxnvI8E4doBAA4ZbmAY3QSwjK-iKB4hCbTpjNtaxLrmk1S1YleqLo2LiSbql0kG_WxjcW5cIqOrHLBnO3vMXq9v3uZPqaz54en6e0s1ZSSNi0ZzSRnSpcaeCnKfE7LkmtqsZAZZRzTbC65tZYzBtzkGgjM5wRjJogFK7Mxuhp0175570xoi1UVtHFO1abpQpFlhEseXY3R5E8QgyQECM3pv1Ccy7hDRC9-oMum83V0XBCSxfUl7SE8QNo3IXhji7WvVspvo1LRx1EMcRR928dR7OLM5V5YBa2c9arWVfgaJBDFhez9k4EL8at-M_57gd_FPwFtfZbx</recordid><startdate>20080501</startdate><enddate>20080501</enddate><creator>Malevich, A. E.</creator><creator>Mityushev, V. V.</creator><creator>Adler, Pierre M.</creator><general>Springer-Verlag</general><general>Springer</general><general>Springer Nature B.V</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7TB</scope><scope>7XB</scope><scope>88I</scope><scope>8AO</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8G5</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FR3</scope><scope>GNUQQ</scope><scope>GUQSH</scope><scope>HCIFZ</scope><scope>KR7</scope><scope>L6V</scope><scope>M2O</scope><scope>M2P</scope><scope>M7S</scope><scope>MBDVC</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope><scope>S0W</scope></search><sort><creationdate>20080501</creationdate><title>Couette flow in channels with wavy walls</title><author>Malevich, A. E. ; Mityushev, V. V. ; Adler, Pierre M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c442t-b543865acbc06b7b9d4bb6c4f1783456143d86fff65506e9c020dd211572f0f83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2008</creationdate><topic>Algorithms</topic><topic>Channels</topic><topic>Classical and Continuum Physics</topic><topic>Control</topic><topic>Couette flow</topic><topic>Criteria</topic><topic>Dynamical Systems</topic><topic>Engineering</topic><topic>Engineering Thermodynamics</topic><topic>Exact sciences and technology</topic><topic>Flows in ducts, channels, nozzles, and conduits</topic><topic>Fluid dynamics</topic><topic>Fundamental areas of phenomenology (including applications)</topic><topic>General theory</topic><topic>Heat and Mass Transfer</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Mechanical engineering</topic><topic>Navier-Stokes equations</topic><topic>Physics</topic><topic>Reynolds number</topic><topic>Rotational flow and vorticity</topic><topic>Solid Mechanics</topic><topic>Theoretical and Applied Mechanics</topic><topic>Theory</topic><topic>Three dimensional</topic><topic>Vibration</topic><topic>Walls</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Malevich, A. E.</creatorcontrib><creatorcontrib>Mityushev, V. V.</creatorcontrib><creatorcontrib>Adler, Pierre M.</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Science Database (Alumni Edition)</collection><collection>ProQuest Pharma Collection</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>Research Library (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Engineering Research Database</collection><collection>ProQuest Central Student</collection><collection>Research Library Prep</collection><collection>SciTech Premium Collection</collection><collection>Civil Engineering Abstracts</collection><collection>ProQuest Engineering Collection</collection><collection>Research Library</collection><collection>Science Database</collection><collection>Engineering Database</collection><collection>Research Library (Corporate)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><collection>DELNET Engineering & Technology Collection</collection><jtitle>Acta mechanica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Malevich, A. E.</au><au>Mityushev, V. V.</au><au>Adler, Pierre M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Couette flow in channels with wavy walls</atitle><jtitle>Acta mechanica</jtitle><stitle>Acta Mech</stitle><date>2008-05-01</date><risdate>2008</risdate><volume>197</volume><issue>3-4</issue><spage>247</spage><epage>283</epage><pages>247-283</pages><issn>0001-5970</issn><eissn>1619-6937</eissn><coden>AMHCAP</coden><abstract>Summary
Three–dimensional Couette flows enclosed by a plane and by a wavy wall are addressed; the wave amplitude is proportional to the mean clearance of the channel multiplied by a small dimensionless parameter ɛ. A perturbation expansion in terms of the powers of ɛ of the full steady Navier–Stokes equations yields a cascade of boundary value problems which are solved at each step in closed form. The supremum value of ɛ for which the expansion converges, is determined as a function of the Reynolds number
The analytical-numerical algorithm is applied to compute the velocity in the channel to
O
(ɛ
4
). Even in the first order approximation
O
(ɛ), new results are obtained which complement the triple deck theory and its modifications. In particular, the incipient separation–detachment is discussed using the Prandtl-Schlichting criterion of starting eddies. The value ɛ
e
for which eddies start in the channel, is analytically deduced as a function of
as well as analytical formulas for the coordinates of the separation points. These analytical formulas show that ɛ
e
in 3D channels is always less than ɛ
e
in 2D channels. For non-smooth channels, a criterion of infinitesimally small ɛ
e
is deduced. The critical value of ɛ up to which bifurcation of the solutions can occur is estimated.</abstract><cop>Vienna</cop><pub>Springer-Verlag</pub><doi>10.1007/s00707-007-0507-z</doi><tpages>37</tpages></addata></record> |
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source | Springer Nature - Complete Springer Journals |
subjects | Algorithms Channels Classical and Continuum Physics Control Couette flow Criteria Dynamical Systems Engineering Engineering Thermodynamics Exact sciences and technology Flows in ducts, channels, nozzles, and conduits Fluid dynamics Fundamental areas of phenomenology (including applications) General theory Heat and Mass Transfer Mathematical analysis Mathematical models Mathematics Mechanical engineering Navier-Stokes equations Physics Reynolds number Rotational flow and vorticity Solid Mechanics Theoretical and Applied Mechanics Theory Three dimensional Vibration Walls |
title | Couette flow in channels with wavy walls |
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