In this paper a cosmological solution of polynomial type $H \approx ( t + const. )^{-1}$ for the causal thermodynamical approach of Isarel-Stewart, found in \cite{MCruz:2017, Cruz2017}, is constrained using the joint of the latest measurements of the Hubble parameter (OHD) and Type Ia Supernovae (SNIa)... Since the expansion described by this solution does not present a transition from a decelerated phase to an accelerated one, both phases can be well modeled connecting both phases by requiring the continuity of the Hubble parameter at $z=z_{t}$, the accelerated-decelerated transition redshift. Our best fit constrains the main free parameters of the model to be $A_1= 1.58^{+0.08}_{-0.07}$ ($A_2=0.84^{+0.02}_{-0.02}$) for the accelerated (decelerated) phase. For both phases we obtain $q=-0.37^{+0.03}_{-0.03}$ ($0.19^{+0.03}_{-0.03}$) and $\omega_{eff} = -0.58^{+0.02}_{-0.02}$ ($-0.21^{+0.02}_{-0.02}$) for the deceleration parameter and the effective equation of state, respectively. Comparing our model and LCDM statistically through the Akaike information criterion and the Bayesian information criterion we obtain that the LCDM model is preferred by the OHD+SNIa data. Finally, it is shown that the constrained parameters values satisfy the criterion for a consistent fluid description of a dissipative dark matter component, but with a high value of the speed of sound within the fluid, which is a drawback for a consistent description of the structure formation. We briefly discuss the possibilities to overcome this problem with a non-linear generalization of the causal linear thermodynamics of bulk viscosity and also with the inclusion of some form of dark energy. (read more)