  This paper deals with the small oscillations of two circular cylinders
immersed in a viscous stagnant fluid. A new theoretical approach based on an
Helmholtz expansion and a bipolar coordinate system is presented to estimate
the fluid forces acting on the two bodies. We show that these forces are linear
combinations of the {\textcolor{black}{cylinder accelerations}} and velocities,
through viscous fluid added coefficients. {\textcolor{black}{To assess the
validity of this theory, we consider the case of two equal size cylinders, one
of them being stationary while the other one is forced sinusoidally}}. The
self-added mass and damping coefficients are shown to decrease with both the
Stokes number and the separation distance. The cross-added mass and damping
coefficients tend to increase with the Stokes number and the separation
distance. Compared to the inviscid results, the effect of viscosity is to add a
correction term which scales as $Sk^{-1/2}$. When the separation distance is
sufficiently large, the two cylinders behave as if they were independent and
the Stokes predictions for an isolated cylinder are recovered. Compared to
previous works, the present theory offers a simple and flexible alternative for
an easy determination of the fluid forces and related added coefficients. To
our knowledge, this is also the first time that a numerical approach based on a
penalization method is presented in the context of fluid-structure interactions
for relatively small Stokes numbers, and successfully compared to theoretical
predictions.
