The dynamics of a moving sheet of liquid, part I: derivation of the 2D equations of motion

In order to investigate the transient response of a planar liquid sheet, subjected to deflection due to inertia forces, a mathematical description of the time-dependent behaviour of a liquid sheet is needed. This paper discusses the derivation of the equations of motion governing the time-dependent deflection of a moving sheet of liquid. The magnitude of the deflection considered is on the order of magnitude of the sheet's length, and the Reynolds number considered is small. The equations are expressed within a 2D orthogonal curvilinear co-ordinate system, moving and changing shape along with the sheet. The co-ordinate system's motion is described by a specific velocity component, and is imposed by the motion of the slot from which the sheet emanates. The derived equations include viscous contributions to the sheet's dynamics and are general enough to allow comparison with special cases found in previous literature works. The influence of viscosity on the sheet's dynamics is assessed by numerical computations of the derived equations.