Extensional fall of a very viscous fluid drop

A finite drop of fluid with large viscosity mu and density rho is initially at rest hanging under gravity g from the underside of a solid boundary. Th e initial configuration may be of a general boundary shape, with (vertical) maximum length L(0) = L-0 and (horizontal) maximum width w(0). The subsequ ent motion, drop length L(t) as a function of time t, and boundary shape is determined both by a slender-drop approximate theory (for w(0) << L-0) and by an exact finite-element calculation. The slender-drop theory is derived both by Lagrangian and Eulerian methods. A wall boundary layer is identifi ed, and empirical corrections made to the Trouton viscosity appearing in th e slender-drop theory to account for this layer. When inertia is neglected, there is a crisis at a finite time t = t(*) = O(mu/(rho gL(0))), such that L(t) --> infinity as t --> t(*), this time being related to the time of br eak-off and entry of the drop into free fall. When the break point falls ou tside the wall boundary layer, its location and hence the fraction of the o riginal drop which falls can be obtained directly from the slender-drop the ory, and is confirmed by the finite-element computations.