The extensional flow of a thin sheet of incompressible, transversely isotropic fluid

Motivated by the aim of modelling the mechanical behaviour of biological gels (such as collagen gels) which have a fibrous microstructure, we consider the extensional flow of a thin two-dimensional film of incompressible, transversely isotropic viscous fluid. Neglecting inertia, and the effects of gravity and surface tension, leading-order equations are derived from a perturbation expansion of the full flow problem in powers of the (small) inverse aspect ratio. The existence and uniqueness of the solution of the reduced system of equations for small times is then proven. Special cases, in which the solution may be determined explicitly, are considered and we discuss the physical interpretation of the results.