Non-failure of filaments and global existence for the equations of fiber spinning
In this work, we give a global-in-time existence and uniqueness proof for solutions of the equations of isothermal fiber spinning. Fiber spinning is a widely used manufacturing process for the production of long thin filaments. In this process, a highly viscous fluid is withdrawn from a reservoir and stretched to form a long fiber. The equations modelling fiber spinning are essentially based on cross-sectional averaging of the axisymmetric Stokes equations with free boundary. They form a coupled system consisting of a non-linear mass transport equation and a non-linear momentum conservation equation for dominant viscous forces in one space dimension. Our analytical approach is based on a representation result of the fiber cross-sectional area in terms of a suitably chosen Lagrangian variable. This representation shows that viscous fibers do not break in finite time and that solutions retain their smoothness. These two results suffice to extend local-in-time solutions to global-in-time solutions. Our no-breakup result is in agreement with previous related work. © The Author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Hagen, T., & Renardy, M. (2011). Non-failure of filaments and global existence for the equations of fiber spinning. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), 76 (6), 834-846. https://doi.org/10.1093/imamat/hxq052