  The Navier--Stokes transport coefficients of multicomponent granular
suspensions at moderate densities are obtained in the context of the
(inelastic) Enskog kinetic theory. The suspension is modeled as an ensemble of
solid particles where the influence of the interstitial gas on grains is via a
viscous drag force plus a stochastic Langevin-like term defined in terms of a
background temperature. In the absence of spatial gradients, it is shown first
that the system reaches a homogeneous steady state where the energy lost by
inelastic collisions and viscous friction is compensated for by the energy
injected by the stochastic force. Once the homogeneous steady state is
characterized, a \emph{normal} solution to the set of Enskog equations is
obtained by means of the Chapman--Enskog expansion around the \emph{local}
version of the homogeneous state. To first-order in spatial gradients, the
Chapman--Enskog solution allows us to identify the Navier--Stokes transport
coefficients associated with the mass, momentum, and heat fluxes. In addition,
the first-order contributions to the partial temperatures and the cooling rate
are also calculated. Explicit forms for the diffusion coefficients, the shear
and bulk viscosities, and the first-order contributions to the partial
temperatures and the cooling rate are obtained in steady-state conditions by
retaining the leading terms in a Sonine polynomial expansion. The results show
that the dependence of the transport coefficients on inelasticity is clearly
different from that found in its granular counterpart (no gas phase). The
present work extends previous theoretical results for \emph{dilute}
multicomponent granular suspensions [Khalil and Garz\'o, Phys. Rev. E
\textbf{88}, 052201 (2013)] to higher densities.

Statistical Mechanics