Extensional Fall of a Very Viscous Fluid Drop

A finite drop of fluid with large viscosity μ and density ρ is initially at rest hanging under gravity g from the underside of a solid boundary. The initial configuration may be of a general boundary shape, with (vertical) maximum length L(0) = L0 and (horizontal) maximum width w0. The subsequent motion, drop length L(t) as a function of time t , and boundary shape is determined both by a slender-drop approximate theory (for w0 L0) and by an exact finite-element calculation. The slender-drop theory is derived both by Lagrangian and Eulerian methods. A wall boundary layer is identified, and empirical corrections made to the Trouton viscosity appearing in the slender-drop theory to account for this layer. When inertia is neglected, there is a crisis at a finite time t = t∗ = O(μ/(ρgL0)), such that L(t) → ∞ as t → t∗, this time being related to the time of break-off and entry of the drop into free fall. When the break point falls outside the wall boundary layer, its location and hence the fraction of the original drop which falls can be obtained directly from the slender-drop theory, and is confirmed by the finite-element computations.