Nonexistence of steady solutions for rotational slender fibre spinning with surface tension

Abstract

Reduced one-dimensional equations for the stationary, isothermal rotational spinning process of slender fibers are considered for the case of large Reynolds ($\delta=3/\text{Re}\ll 1$) and small Rossby numbers ($\varepsilon \ll 1$). Surface tension is included in the model using the parameter $\kappa=\sqrt{\pi}/(2 \text{We})$ related to the inverse Weber number. The inviscid case $\delta=0$ is discussed as a reference case. For the viscous case $\delta > 0$ numerical simulations indicate, that for a certain parameter range, no physically relevant solution may exist. Transferring 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 $\delta > 3\varepsilon^2 \left( 1- \frac{3}{2}\kappa +\frac{1}{2}\kappa^2\right)$ no physical relevant stationary solution can exist.