  We propose a theoretical method to decompose the solution of a Stokes flow
past a body immersed in a confined fluid in two simpler problems, related
separately to the two geometrical elements of these systems: (i) the body
immersed in the unbounded fluid (represented by its Fax\'en operators), and
(ii) the domain of the confinement (represented by its Stokesian multipoles).
Specifically, by using a reflection method, and assuming linear and reciprocal
boundary conditions \citep{procopio-giona_pof}, we provide the expression for
the velocity field, the forces, torques and higher-order moments acting on the
body in terms of: (i) the volume moments of the body in the unbounded ambient
flow; (ii) the multipoles in the domain of the confinement; (iii) the
collection of all the volumetric moments on the body immersed in all the
regular parts of the multipoles considered as ambient flows. A detailed
convergence analysis of the reflection method is developed. In the light of
practical applications, we estimate the truncation error committed by
considering only the lower order moments (thus truncating the matrices) and the
errors associated with the approximated expressions available in the literature
for force and torques. We apply the theoretical results to the archetypal
hydrodynamic system of a sphere with Navier-slip boundary conditions near a
plane wall with no-slip boundary conditions, to determine forces and torques on
a translating and rotating sphere as a function of the slip length and of the
distance of the sphere from the plane. The hydromechanics of a spheroid is also
addressed.
