A SYSTEMATIC DERIVATION OF THE LEADING-ORDER EQUATIONS FOR EXTENSIONAL FLOWS IN SLENDER GEOMETRIES

We consider the extensional flow and twist of a finite, slender, nearly straight, Newtonian viscous fibre when its ends are drawn apart at prescribed velocity. The initial cross-section of the fibre may be arbitrary and may vary gradually in the axial direction. We derive the leading-order equations for the fibre's free surface and its flow velocity from a regular perturbation expansion of the full Stokes' flow problem in powers of the aspect ratio. In order to obtain these equations systematically, it is necessary to consider terms beyond the leading order in the perturbation expansion, because those obtainable from the leading-order terms give an indeterminate set of equations. Our results are a pair of well-known hyperbolic equations for the area and axial velocity, together with (i) straightness of the line of centres of mass of the cross-section and (ii) a new hyperbolic evolution equation for the twist of the cross-section. It is only through this hyperbolic equation that history effects are manifest.