The fluid dynamics of dripping onto a substrate

Extensional flows of complex fluids are important in many industrial applications such as spraying and atomisation, and microfluidic-based drop deposition. Dripping-on-Substrate (DoS) 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 dispensed from a syringe pump onto a solid substrate. By following the filament thinning process the extensional viscosity and relaxation time of the sample can be determined. Importantly, it 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 allow us to optimise and extend the performance of this protocol. To achieve this we employ a computational rheology approach using adaptively-refined 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 provides a stable and accurate solution of the viscoelastic constitutive equation. Here we focus on understanding the role of elasticity and finite chain extensibility on controlling the Elasto-Capillary (EC) regime, as well as the perturbative effects that gravity and the wetting of the solid substrate 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 interplay of inertia and capillarity dominates and the structure of the transition region to the non-linear EC regime where the rapidly growing elastic stresses in the thread balance the capillary pressure as the filament thins towards breakup.