  Orbital sloshing is a technique to gently mix a container's liquid content
and it is commonly used in fermentation and cell cultivation processes. Besides
the rich wave dynamics observed at the interface, Bouvard et al. (2017) [1]
unveiled the structure of the Lagrangian mean flow hiding in the fluid bulk.
The latter flow shows a global toroidal (azimuthal) rotation co-directed with
the wave and nontrivial poloidal vortices near the contact line. Rotating
sloshing waves are known to induce a net motion of fluid particles and hence a
wave-averaged difference between the Eulerian flow - viscous streaming - and
the Lagrangian flow, that is commonly referred to as Stokes drift.
Nevertheless, discerning these two components in an experiment is challenging
as they scale similarly with the forcing amplitude and frequency. Their
relative contributions remain therefore unquantified, particularly for highly
viscous fluids, for which prior analysis, based on inviscid arguments, fails.
In this work, we construct a truncated asymptotic approximation of the problem,
where the solution at each order is computed numerically to describe as
accurately as possible the contact line region, otherwise analytically
untractable. The results of this weakly nonlinear analysis in terms of
first-order wave and second-order mean flow are then thoroughly compared with
the experiments by Bouvard et al. (2017) [1], showing a remarkable agreement
for off-resonance frequencies. When viscosity matters, our findings suggest
that it is incorrect to attribute the poloidal patterns solely to the Eulerian
streaming flow and that viscous corrections to the Stokes drift are equally
important in the resulting mean Lagrangian flow.
