First-order contributions to the partial temperatures in binary granular suspensions at low density

The Boltzmann kinetic equation is considered to evaluate the first-order contributions $T_i^{(1)}$ to the partial temperatures in binary granular suspensions at low density. The influence of the surrounding gas on the solid particles is modeled via a drag force proportional to the particle velocity plus a stochastic Langevin-like term. The Boltzmann equation is solved by means of the Chapman--Enskog expansion around the local version of the reference homogeneous base state. To first-order in spatial gradients, the coefficients $T_i^{(1)}$ are computed by considering the leading terms in a Sonine polynomial expansion. In addition, the influence of $T_i^{(1)}$ on the first-order contribution $zeta^{(1)}$ to the cooling rate is also assessed. Our results show that the magnitude of both $T_i^{(1)}$ and $zeta^{(1)}$ can be relevant for some values of the parameter space of the system.