The Fluid Dynamics of Dripping onto a Substrate (DOS)

Extensional flows of complex fluids are important in many industrial applications such as spraying, atomization, and microfluidic-based drop deposition. The Dripping-on-Substrate (DoS) technique pioneered by the Sharma ODES Lab is a conceptually-simple, but fluid 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 partially-wettable solid substrate. By following the filament thinning process, the extensional viscosity and relaxation time of the sample can be determined. Importantly, it allows rheologists to measure the extensional properties of lower viscosity materials than is possible with commercially-available capillary break-up extensional rheometers. Understanding the fluid mechanics behind the operation of DoS will allow us to optimize and extend the application of this protocol. 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. We explore the role of elasticity and polymer finite 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 nonlinear one-dimensional model that can capture the initial rate of thinning, when the interplay of inertia and capillarity dominates, as well as 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.