Tumbling of a rigid rod in a shear flow
The tumbling of a rigid rod in a shear flow is analyzed in the high viscosity limit. Following Burgers, the Master Equation is derived for the probability distribution of the orientation of the rod. The equation contains one dimensionless number, the Weissenberg number, which is the ratio of the she...
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| description | The tumbling of a rigid rod in a shear flow is analyzed in the high viscosity limit. Following Burgers, the Master Equation is derived for the probability distribution of the orientation of the rod. The equation contains one dimensionless number, the Weissenberg number, which is the ratio of the shear rate and the orientational diffusion constant. The equation is solved for the stationary state distribution for arbitrary Weissenberg numbers, in particular for the limit of high Weissenberg numbers. The stationary state gives an interesting flow pattern for the orientation of the rod, showing the interplay between flow due to the driving shear force and diffusion due to the random thermal forces of the fluid. The average tumbling time and tumbling frequency are calculated as a function of the Weissenberg number. A simple cross-over function is proposed which covers the whole regime from small to large Weissenberg numbers. |
| doi_str_mv | 10.48550/arxiv.1406.1317 |
| format | Article |
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Following Burgers, the Master Equation is derived for the probability distribution of the orientation of the rod. The equation contains one dimensionless number, the Weissenberg number, which is the ratio of the shear rate and the orientational diffusion constant. The equation is solved for the stationary state distribution for arbitrary Weissenberg numbers, in particular for the limit of high Weissenberg numbers. The stationary state gives an interesting flow pattern for the orientation of the rod, showing the interplay between flow due to the driving shear force and diffusion due to the random thermal forces of the fluid. The average tumbling time and tumbling frequency are calculated as a function of the Weissenberg number. A simple cross-over function is proposed which covers the whole regime from small to large Weissenberg numbers.</description><identifier>EISSN: 2331-8422</identifier><identifier>DOI: 10.48550/arxiv.1406.1317</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Crossovers ; Diffusion rate ; Dimensionless numbers ; Physics - Statistical Mechanics ; Shear flow ; Shear forces ; Shear rate ; Tumbling</subject><ispartof>arXiv.org, 2014-06</ispartof><rights>2014. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). 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Following Burgers, the Master Equation is derived for the probability distribution of the orientation of the rod. The equation contains one dimensionless number, the Weissenberg number, which is the ratio of the shear rate and the orientational diffusion constant. The equation is solved for the stationary state distribution for arbitrary Weissenberg numbers, in particular for the limit of high Weissenberg numbers. The stationary state gives an interesting flow pattern for the orientation of the rod, showing the interplay between flow due to the driving shear force and diffusion due to the random thermal forces of the fluid. The average tumbling time and tumbling frequency are calculated as a function of the Weissenberg number. A simple cross-over function is proposed which covers the whole regime from small to large Weissenberg numbers.</description><subject>Crossovers</subject><subject>Diffusion rate</subject><subject>Dimensionless numbers</subject><subject>Physics - Statistical Mechanics</subject><subject>Shear flow</subject><subject>Shear forces</subject><subject>Shear rate</subject><subject>Tumbling</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><sourceid>BENPR</sourceid><sourceid>GOX</sourceid><recordid>eNotj0lrwzAUhEWh0JDm3lMQ9NCT3aenp8XHErpBoBffjeQlUXDsVK67_PvaTU_DwDAzH2M3AlKySsG9i9_hMxUEOhVSmAu2QClFYgnxiq2G4QAAqA0qJRfsLh-Pvg3djvcNdzyGXah47CseuskO-9pF3rT91zW7bFw71Kt_XbL86THfvCTbt-fXzcM2cUrMG1oZ47VD0qiBSlCyhBJq0lZTppG8M6TLCitfZySlcqUXVoFF5W0l5JKtz7V_EMUphqOLP8UMU8wwU-D2HDjF_n2sh4_i0I-xmy4VOLVkxioS8he1nkjF</recordid><startdate>20140605</startdate><enddate>20140605</enddate><creator>J M J van Leeuwen</creator><creator>Blöte, H W J</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</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>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>GOX</scope></search><sort><creationdate>20140605</creationdate><title>Tumbling of a rigid rod in a shear flow</title><author>J M J van Leeuwen ; Blöte, H W J</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a511-846577b6a2462604c053c0c0e468649624ba746cd2dbe94335acb1850825b8d13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Crossovers</topic><topic>Diffusion rate</topic><topic>Dimensionless numbers</topic><topic>Physics - Statistical Mechanics</topic><topic>Shear flow</topic><topic>Shear forces</topic><topic>Shear rate</topic><topic>Tumbling</topic><toplevel>online_resources</toplevel><creatorcontrib>J M J van Leeuwen</creatorcontrib><creatorcontrib>Blöte, H W J</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</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>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</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>arXiv.org</collection><jtitle>arXiv.org</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>J M J van Leeuwen</au><au>Blöte, H W J</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Tumbling of a rigid rod in a shear flow</atitle><jtitle>arXiv.org</jtitle><date>2014-06-05</date><risdate>2014</risdate><eissn>2331-8422</eissn><abstract>The tumbling of a rigid rod in a shear flow is analyzed in the high viscosity limit. Following Burgers, the Master Equation is derived for the probability distribution of the orientation of the rod. The equation contains one dimensionless number, the Weissenberg number, which is the ratio of the shear rate and the orientational diffusion constant. The equation is solved for the stationary state distribution for arbitrary Weissenberg numbers, in particular for the limit of high Weissenberg numbers. The stationary state gives an interesting flow pattern for the orientation of the rod, showing the interplay between flow due to the driving shear force and diffusion due to the random thermal forces of the fluid. The average tumbling time and tumbling frequency are calculated as a function of the Weissenberg number. 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| subjects | Crossovers Diffusion rate Dimensionless numbers Physics - Statistical Mechanics Shear flow Shear forces Shear rate Tumbling |
| title | Tumbling of a rigid rod in a shear flow |
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