  Spatial correlations play an important role in characterizing material
properties related to non-local effects. Inter alia, they can give rise to
fluctuation-induced forces. Equilibrium correlations in fluids provide an
extensively studied paradigmatic case, in which their range is typically
bounded by the correlation length. Out of equilibrium, conservation laws have
been found to extend correlations beyond this length, leading, instead, to
algebraic decays. In this context, here we present a systematic study of the
correlations and forces in fluids driven out of equilibrium simultaneously by
quenching and shearing, both for non-conserved as well as for conserved
Langevin-type dynamics. We identify which aspects of the correlations are due
to shear, due to quenching, and due to simultaneously applying both, and how
these properties depend on the correlation length of the system and its
compressibility. Both shearing and quenching lead to long-ranged correlations,
which, however, differ in their nature as well as in their prefactors, and
which are mixed up by applying both perturbations. These correlations are
employed to compute non-equilibrium fluctuation-induced forces in the presence
of shear, with or without quenching, thereby generalizing the framework set out
by Dean and Gopinathan. These forces can be stronger or weaker compared to
their counterparts in unsheared systems. In general, they do not point along
the axis connecting the centers of the small inclusions considered to be
embedded in the fluctuating medium. Since quenches or shearing appear to be
realizable in a variety of systems with conserved particle number, including
active matter, we expect these findings to be relevant for experimental
investigations.
