  Extensional flows of complex fluids are pivotal in industrial applications
like spraying, atomisation, and microfluidic 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. DoS involves capillary-driven thinning of a liquid bridge
formed by a slowly dispensed drop onto a partially-wetting solid substrate. By
following the filament thinning and pinch-off, the extensional viscosity and
relaxation time can be determined. Importantly, DoS enables measurements for
lower viscosity solutions than commercially available capillary break-up
extensional rheometers. To understand DoS operation, we employ a computational
rheology approach via adaptively-refined, time-dependent axisymmetric
simulations using the open-source Eulerian code, \textit{Basilisk}. The
volume-of-fluid technique is used to capture the moving interface, and the
log-conformation transformation enables a stable viscoelastic solution. We
focus on understanding the roles of surface tension, elasticity, and finite
chain extensibility in the Elasto-Capillary (EC) regime. Additionally, we
explore perturbative effects of gravity and substrate wettability 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 capturing the initial thinning rates, balancing inertia
and capillarity. This model also describes the structure of the transition
region to the nonlinear EC regime, where elastic stresses counteract capillary
pressure in the thread as the filament thins toward breakup. Finally, we
propose a fitting methodology based on the analytical solutions for FENE-P
fluids to enhance accuracy in determining the effective relaxation time for
unknown fluids.
