On the planar extensional motion of an inertially driven liquid sheet

We derive a time-dependent exact solution of the free surface problem for the Navier–Stokes equations that describes the planar extensional motion of a viscous sheet driven by inertia. The linear stability of the exact solution to oneand two-dimensional symmetric perturbations is examined in the inviscid and viscous limits within the framework of the long-wave or slender body approximation. Both transient growth and long-time asymptotic stability are considered. For one-dimensional perturbations in the axial direction, viscous and inviscid sheets are asymptotically marginally stable, though depending on the Reynolds and Weber numbers transient growth can have an important effect. For one-dimensional perturbations in the transverse direction, inviscid sheets are asymptotically unstable to perturbations of all wavelengths. For two-dimensional perturbations, inviscid sheets are unstable to perturbations of all wavelengths with the transient dynamics controlled by axial perturbations and the long-time dynamics controlled by transverse perturbations. The asymptotic stability of viscous sheets to one-dimensional transverse perturbations and to two-dimensional perturbations depends on the capillary number Ca ; in both cases, the sheet is unstable to long-wave transverse perturbations for any finite Ca. © 2009 American Institute of Physics. DOI: 10.1063/1.3094026