  Extensions to a $\Lambda$DM model have been explored to face current tensions
that occur within its framework, which encompasses broadening the nature of the
dark matter (DM) component to include warmness and a non-perfect fluid
description. In this paper, we investigated the late-time cosmological
evolution of an exact solution recently found in [N. Cruz, E. Gonz\'alez, and
J. Jovel, Phys. Rev. D \textbf{105}, 024047 (2022)], which describe a viscous
warm $\Lambda$DM model ($\Lambda$WDM) with a DM component that obeys a
polytropic equation of state (EoS), which experience dissipative effects with a
bulk viscosity proportional to its energy density, with proportionality
constant $\xi_{0}$. This solution has the particularity of having a very
similar behavior to the $\Lambda$CDM model for small values of $\xi_{0}$,
evolving also to a de Sitter type expansion in the very far future. We explore
firstly the thermodynamic consistences of this solution in the framework of the
Eckart's theory of non-perfect fluids, focusing on the fulfillment of the two
following conditions: (i) the near equilibrium condition and (ii) the
positiveness of the entropy production. We explore the range of parameters of
the model that allow to fulfilling these two conditions at the same time,
finding that a viscous WDM component is compatible with both ones, being in
this sense, a viable model from the thermodynamic point of view. Besides, we
constraint the free parameters of the model with the observational data coming
form supernovae Ia (SNe Ia) and the observational Hubble parameter data
(OHD).(Abridged)
