Kaluza-Klein bulk viscous fluid cosmological models and the validity of   the second law of thermodynamics in $f(R, T)$ gravity
  The authors considered the bulk viscous fluid in $f(R, T)$ gravity within the framework of Kaluza-Klein space time. The bulk viscous coefficient $(\xi)$ expressed as $\xi=\xi_0+\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$, where $\xi_0$, $\xi_1$ and $\xi_2$ are positive constants. We take $p=(\gamma-1)\rho$, where $0\le\gamma\le2$ as an equation of state for perfect fluid. The exact solutions to the corresponding field equations are given by assuming a particular model of the form of $f(R, T)=R+2f(T)$, where $f(T)=\lambda T$, $\lambda$ is constant. We studied the cosmological model in two stages, in first stage: we studied the model with no viscosity, and in second stage: we studied the model involve with viscosity. The cosmological model involve with viscosity is studied by five possible scenarios for bulk viscous fluid coefficient $(\xi)$. The total bulk viscous coefficient seems to be negative, when the bulk viscous coefficient is proportional to $\xi_2\frac{\ddot{a}}{\dot{a}}$, hence the second law of thermodynamics is not valid, however, it is valid with the generalized second law of thermodynamics. The total bulk viscous coefficient seems to be positive, when, the bulk viscous coefficient is proportional to $\xi=\xi_1\frac{\dot{a}}{a}$, $\xi=\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$ and $\xi=\xi_0+\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$, so the second law of thermodynamics and the generalized second law of thermodynamics is satisfied throughout the evolution. We calculate statefinder parameters of the model and observed that, it is different from the $\wedge$CDM model. Finally, some physical and geometrical properties of the models are discussed. 
G. C. Samanta, R. Myrzakulov and Parth Shah
gr-qc