Slowing of sheet recoil by surface viscosity

Liquid sheets arise in curtain coating, polymer processing and sprays. When a fluid is ejected from a die (nozzle) to form a liquid sheet, its cross-section is rectangular albeit for the two rounded ends. The latter retract due to surface tension. The retraction dynamics is also affected by stresses owing to bulk rheology, which may be viscous and/or viscoelastic in nature, and surface rheology, which may be due to the presence of surface-active agents. We analyse theoretically and numerically the retraction dynamics of highly viscous Newtonian liquid sheets when surface viscous stresses are present. While it has been shown recently that viscoelasticity increases retraction rate, it is demonstrated that surface viscosity operates synergistically with bulk viscosity to decrease retraction rate. As the two surfaces of a retracting sheet remain flat outside of the two tip regions, an exact analytical solution is obtained for the transient sheet thickness in terms of the Lambert W function. An asymptotic solution for sheet thickness, valid for early times, is also obtained and shown to agree well with the analytical solution and simulations. An energy analysis is performed to rationalise that at early times, the rate of energy dissipation due to the action of surface viscous stresses can be dominant in slowing retraction, but it can wane in importance and be overtaken at large times by the rate at which energy is dissipated due to the action of bulk viscous stresses.