  In this article, we considered the bulk viscous fluid in the formalism of
modified gravity in which the general form of a gravitational action is $f(R,
T)$ function, where $R$ is the curvature scalar and $T$ is the trace of the
energy momentum tensor within the frame of flat FRW space time. The
cosmological model dominated by bulk viscous matter with total bulk viscous
coefficient expressed as a linear combination of the velocity and acceleration
of the expansion of the universe in such a way that
$\xi=\xi_0+\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$, where
$\xi_0$, $\xi_1$ and $\xi_2$ are 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 obtained 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 four possible scenarios for different values of
$\gamma$, such as $\gamma=0$, $\gamma=\frac{2}{3}$, $\gamma=1$ and
$\gamma=\frac{4}{3}$ with the possible positive and negative ranges of
$\lambda$ to observe the accelerated expansion history of the universe.
Finally, a big-rip singularity is observed.
