  In this paper, bulk viscosity is introduced in new holographic dark energy
(HDE) to describe the effects of cosmic non-perfect fluid on the evolution of
the universe in modified $f(R,T)$ gravity. Assuming $f(R,T)=R+\lambda T$, where
$R$ is the Ricci scalar, $T$, the trace of energy-momentum tensor and $\lambda$
is a constant, we derive a general function of Hubble parameter with bulk
viscous form to provide a procedure for the viscous new HDE model building.
Especially, we assume the total bulk viscosity coefficient proportional to the
velocity of the expansion of the universe in such a way that
$\zeta=\zeta_0+\zeta_1 H$. We obtain the solutions of the scale factor and
deceleration parameter and classify all the possible scenarios (deceleration,
acceleration and their transition) with different parameter regions chosen
properly for positive and negative ranges of $\lambda$, and $\zeta_0$ and
$\zeta_1$ to analyze the evolution of the universe. It is observed that there
is a transition from decelerated phase to accelerated phase at early or late
time depending on the values of viscous terms. For large values of viscous
terms it always accelerates through out the evolution. Furthermore, we also
investigate the statefinder pair $\{r, s\}$ and $\textit{Om}$ diagnostics for
the viscous new HDE model to discriminate with other existing DE models. Our
viscous model is also different from other existing dark energy models. The
evolution of effective equation of state parameter is -0.9745 which is very
close to the observation. The entropy and generalized second law of
thermodynamics are valid for this model under some constraints of bulk viscous
coefficient. The analysis shows that the dark energy phenomena may be explained
as the effect of bulk viscosity in the cosmic medium.
