  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.
