Numerical evidence for the non-existence of solutions of the equations describing rotational fiber spinning

Technical Report, Berichte des Fraunhofer ITWM, Nr. 108, 2007

Authors

• T. Götz
• Axel Klar
• Andreas Unterreiter
• Raimund Wegener

Abstract

The stationary, isothermal rotational spinning process of fibers is considered. The investigations are concerned with the case of large Reynolds ($\delta=3/Re$) and small Rossby numbers ($\eps = Rb$). Modelling the fibers as a Newtonian fluid and applying slender body approximations, the process is described by a two--point boundary value problem of ODEs. The involved quantities are the coordinates of the fiber's centerline, the fluid velocity and viscous stress. The inviscid case $\delta=0$ is discussed as a reference case. For the viscous case $\delta\gr 0$ numerical simulations are carried out. Transfering some properties of the inviscid limit to the viscous case, analytical bounds for the initial viscous stress of the fiber are obtained. A good agreement with the numerical results is found. These bounds give strong evidence, that for $\delta > 3\eps^2$ no physical relevant solution can exist. A possible interpretation of the above coupling of $\delta$ and $\eps$ related to the die--swell phenomenon is given.

BibTeX

 
@TechReport{ Goetz.Klar.EA07numerical, 
   title = { Numerical evidence for the non-existence of solutions of the equations describing rotational fiber spinning }, 
   author = { T. Götz and Axel Klar and Andreas Unterreiter and Raimund Wegener },     
   series = { Berichte des Fraunhofer ITWM, Nr. 108 },            
   year = 2007, 
} 

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This publication belongs to the project Fimod.