  A linear stability analysis of the hydrodynamic equations with respect to the
homogeneous cooling state is performed to study the conditions for stability of
a suspension of solid particles immersed in a viscous gas. The dissipation in
such systems arises from two different sources: inelasticity in particle
collisions and viscous friction dissipation due to the influence of gas phase
on solid particles. The starting point is a suspension model based on the
(inelastic) Enskog kinetic equation where the effect of the interstitial gas
phase on the dynamics of grains is modeled via a viscous drag force. The study
is carried out in two different steps. First, the transport coefficients of the
system are obtained by solving the Enskog equation by means of the
Chapman-Enskog method up to first order in spatial gradients. Once the
transport properties are known, then the corresponding linearized hydrodynamic
equations are solved to get the dispersion relations. In contrast to previous
studies [V. Garz\'o \emph{et al.}, Phys. Rev. E \textbf{93}, 012905 (2016)],
the hydrodynamic modes are \emph{analytically} obtained as functions of the
parameter space of the system. As expected, linear stability shows $d-1$
transversal (shear) modes (where $d$ is the dimensionality of the system) and a
longitudinal "heat" mode to be unstable with respect to long enough wavelength
excitations. The results also show that the main effect of gas phase is to
decrease the value of the critical length $L_c$ (beyond which the system
becomes unstable) with respect to its value for a dry granular fluid.
Comparison with direct numerical simulations for $L_c$ shows a good agreement
for conditions of practical interest.
