Viscous stabilisation of Rayleigh-Plateau modes on a cylindrical filament through radial oscillatory forcing

Faraday waves are standing wave patterns appearing on a liquid surface subject to an oscillatory body force and have been observed in base states of Cartesian, spherical or cylindrical coordinate systems (Benjamin et. al., Proc. Roy. Soc. Lond., 1954; Adou et. al., J. Fluid Mech., 2016; Patankar et. al., J. Fluid Mech. 2018). In these geometries, under inviscid irrotational approximation, small-amplitude standing waves are governed by the Mathieu equation. Taking viscosity into account produces a memory term leading to an integro-differential, damped, Mathieu equation. In Cartesian geometry such an equation was derived by Beyer et. al., Phys. Rev. E, 1995. Here, we derive an analogous integro-differential equation governing Faraday waves on a cylindrical filament of a viscous liquid utilising the toroidal-poloidal decomposition (Boronski et. al., J. Comput. Phys. 2007). We demonstrate perpetual stabilisation of an unstable axisymmetric Rayleigh$-$Plateau mode $(kR_0<1)$ perturbation in the viscous case by choosing the strength of forcing appropriately. Linearised theory is compared with Direct Numerical Simulations using the open-source code Gerris showing good agreement. We also show the possibility of generating droplets in a controlled manner by fragmenting the filament.