Yarn Balloons, Differential Geometry and the Navier-Stokes Equations

Cave, Gregory Edward

Yarn Balloons, Differential Geometry and the Navier-Stokes Equations

Cave, Gregory Edward

Permalink

Publication Type:: Thesis
Issue Date:: 2023

Open Access

Copyright Clearance Process

Recently Added
In Progress
Open Access

This item is open access.

Adobe PDF

Download thesisAdobe PDF (1.21 MB)

View statistics

Full metadata record

Field	Value	Language
dc.contributor.author	Cave, Gregory Edward
dc.date.accessioned	2023-05-30T01:50:22Z
dc.date.available	2023-05-30T01:50:22Z
dc.date.issued	2023
dc.identifier.uri	http://hdl.handle.net/10453/170541
dc.description	University of Technology Sydney. Faculty of Science.	en_US.UTF-8
dc.description.abstract	In the textile industry, there are hundreds of millions of ring spindles or equivalent devices in use today around the world converting millions of tonnes of short staple fibers such as cotton or polyester into clothing and other textiles. The aim of this thesis is to improve the mathematical model of yarn spinning devices to potentially lead to an improvement in the efficiency of the production process. Past research in this area, referred to as the Padfield model, expressed the yarn balloon equations in cylindrical polar coordinates, in a reference frame rotating with a constant angular velocity. The air drag was assumed to be proportional to the velocity normal to the yarn squared, while the tangential component was neglected, and the drag coefficient $C_D$ was assumed to be equal to 1 for all values of the Reynolds number $Re$, because the exact functional relationship between $C_D$ and $Re$ is not known. The ultimate objective of this thesis is to improve the accuracy of the mathematical model of this system by representing the yarn balloon equations in intrinsic coordinates, and finding a more accurate representation of the air drag term in these equations along the entire length of the yarn balloon. This is achieved by finding a new exact similarity solution of the 2-D incompressible steady state Navier-Stokes equations which will enable significant progress to be made in finding the functional relationship between $C_D$ and $Re$ for $0 < Re < 47$. The process through which this similarity solution is used to find the corresponding physical solution is described in detail for $0 < Re < 47$. The viscous solution in the boundary layer is matched to the inviscid external solution so that all boundary conditions are satisfied. This is significant progress towards improving the mathematical model of yarn spinning devices through finding the relationship between $C_D$ and $Re$ for low Reynolds numbers. The solution found here also has more applications than just the yarn balloon problem, and thus a greater importance.	en_US.UTF-8
dc.format	Thesis (PhD)
dc.language.iso	en_US	en_US.UTF-8
dc.relation	https://opus.lib.uts.edu.au/bitstream/10453/170541/1/thesis.pdf
dc.rights	info:eu-repo/semantics/openAccess
dc.rights	The author owns the copyright in this thesis including all reproduction and reuse rights for the work. The work may not be altered without the permission of the copyright owner. Attribution is essential when quoting or paraphrasing from this thesis.
dc.rights	© 2023 Gregory Edward Cave
dc.rights	au.edu.uts.lib/cph
dc.title	Yarn Balloons, Differential Geometry and the Navier-Stokes Equations	en_US.UTF-8
dc.type	Thesis
utslib.copyright.status	open_access	*

Abstract:

In the textile industry, there are hundreds of millions of ring spindles or equivalent devices in use today around the world converting millions of tonnes of short staple fibers such as cotton or polyester into clothing and other textiles. The aim of this thesis is to improve the mathematical model of yarn spinning devices to potentially lead to an improvement in the efficiency of the production process. Past research in this area, referred to as the Padfield model, expressed the yarn balloon equations in cylindrical polar coordinates, in a reference frame rotating with a constant angular velocity. The air drag was assumed to be proportional to the velocity normal to the yarn squared, while the tangential component was neglected, and the drag coefficient $C_D$ was assumed to be equal to 1 for all values of the Reynolds number $Re$, because the exact functional relationship between $C_D$ and $Re$ is not known. The ultimate objective of this thesis is to improve the accuracy of the mathematical model of this system by representing the yarn balloon equations in intrinsic coordinates, and finding a more accurate representation of the air drag term in these equations along the entire length of the yarn balloon. This is achieved by finding a new exact similarity solution of the 2-D incompressible steady state Navier-Stokes equations which will enable significant progress to be made in finding the functional relationship between $C_D$ and $Re$ for $0 < Re < 47$. The process through which this similarity solution is used to find the corresponding physical solution is described in detail for $0 < Re < 47$. The viscous solution in the boundary layer is matched to the inviscid external solution so that all boundary conditions are satisfied. This is significant progress towards improving the mathematical model of yarn spinning devices through finding the relationship between $C_D$ and $Re$ for low Reynolds numbers. The solution found here also has more applications than just the yarn balloon problem, and thus a greater importance.

Please use this identifier to cite or link to this item:

http://hdl.handle.net/10453/170541