The fluid dynamics of a viscoelastic fluid dripping onto a substrate

Extensional flows of complex fluids play an important role in many industrial applications, such as spraying and atomisation, as well as microfluidic-based drop deposition. The dripping-on-substrate (DoS) technique is a conceptually-simple, but dynamically-complex, probe of the extensional rheology of low-viscosity, non-Newtonian fluids. It incorporates the capillary-driven thinning of a liquid bridge, produced by a single drop as it is slowly dispensed from a syringe pump onto a solid partially-wetting substrate. By following the filament thinning and pinch-off process the extensional viscosity and relaxation time of the sample can be determined. Importantly, DoS allows experimentalists to measure the extensional properties of lower viscosity solutions than is possible with commercially available capillary break-up extensional rheometers. Understanding the fluid mechanics behind the operation of DoS will enable users to optimise and extend the performance of this protocol. To achieve this understanding, we employ a computational rheology approach, using adaptively-refined time-dependent axisymmetric numerical simulations with the open-source Eulerian code, Basilisk. The volume-of-fluid technique is used to capture the moving interface, and the log-conformation transformation enables a stable and accurate solution of the viscoelastic constitutive equation. Here, we focus on understanding the roles of surface tension, elasticity and finite chain extensibility in controlling the elasto-capillary (EC) regime, as well as the perturbative effects that gravity and substrate wettability play in setting the evolution of the self-similar thinning and pinch-off dynamics. To illustrate the interplay of these different forces, we construct a simple one-dimensional model that captures the initial rate of thinning when the dynamics are dominated by a balance between inertia and capillarity. This model also captures the structure of the transition region to the nonlinear EC regime in which the rapidly growing elastic stresses in the thread balance the capillary pressure as the filament thins towards breakup. Finally, we propose a fitting methodology based on the analytical solution for FENE-P fluids to improve the accuracy in determining the effective relaxation time of an unknown fluid.