  The analysis of the dynamics of tracer particles in a complex bath can
provide valuable information about the microscopic behaviour of the bath. In
this work, we study the dynamics of a forced tracer in a colloidal bath by
means of Langevin dynamics simulations and a theory model within continuum
mechanics. In the simulations, the bath is comprised by quasi-hard spheres with
a volume fraction of 50% immersed in a featureless quiescent solvent, and the
tracer is pulled with a constant small force (within the linear regime). The
theoretical analysis is based on the Navier Stokes equation, where a term
proportional to the velocity arises from coarse-graining the friction of the
colloidal particles with the solvent. As a result, the final equation is
similar to the Brinkman model, although the interpretation is different. A
length scale appears in the model, 1/k_0, where the transverse momentum
transport crosses over to friction with the solvent. The effective friction
coefficient experienced by the tracer grows with the tracer size faster than
the prediction from Stokes law. Additionally, the velocity profiles in the bath
decay faster than in a Newtonian fluid. The comparison between simulations and
theory points to a boundary condition of effective partial slip at the tracer
surface. We also study the fluctuations in the tracer position, showing that it
reaches diffusion at long times, with a subdiffusive regime at intermediate
times. The diffusion coefficient, obtained from the long-time slope of the mean
squared displacement, fulfills the Stokes-Einstein relation with the friction
coefficient calculated from the steady tracer velocity, confirming the validity
of the linear response formalism.
