Drawing of fibres composed of shear-thinning or shear-thickening fluid with internal holes

We explore the drawing of a shear-thinning or shear-thickening thread with an axisymmetric hole that evolves due to axial drawing, inertia and surface tension effects. The stress is assumed to be proportional to the shear rate raised to the nth power. The presence of non-Newtonian rheology and surface tension forces acting on the hole introduces radial pressure gradients that make the derivation of long-wavelength equations significantly more challenging than either a Newtonian thread with a hole or shear-thinning and shear-thickening threads without a hole. In the case of weak surface tension, we determine the steady-state profiles. Our results show that for negligible inertia the hole size at the exit becomes smaller as n is decreased (i.e. strong shear-thinning effects) above a critical draw ratio, but surprisingly the opposite is true below this critical draw ratio. We determine an accurate estimate of the critical draw ratio and also discuss how inertia affects this process. We further show that the dynamics of hole closure is dominated by a different limit, and we determine the asymptotic forms of the hole closure process for shear-thinning and shear-thickening fluids with inertia. A linear instability analysis is conducted to predict the onset of draw resonance. We show that increased shear thinning, surface tension and inlet hole size all act to destabilise the flow. We also show that increasing shear-thinning effects reduce the critical Reynolds number required for unconditional stability. Our study provides valuable insights into the drawing process and its dependence on the physical effects. © The Author(s), 2024. Published by Cambridge University Press.