  We investigate a new class of LRS Bianchi type-II cosmological models by
revisiting in the paper of Mishra {\it et al} (2013) by considering a new
deceleration parameter (DP) depending on the time in string cosmology for the
modified gravity theory suggested by S$\acute{a}$ez \& Ballester (1986). We
have considered the energy-momentum tensor proposed by Leteliar (1983) for bulk
viscous and perfect fluid under some assumptions. To make our models consistent
with recent astronomical observations, we have used scale factor (Sharma {\it
et al} 2018; Garg {\it et al} 2019) $ a(t)=\exp{[\frac{1}{\beta}\sqrt{2 \beta t
+ k}]}$, where $\beta $ and $k$ are positive constants and it provides a
time-varying DP. By using the recent constraints ($H_{0}=73.8$, and $q_{0} =
-0.54$) from SN Ia data in combination with BAO and CMB observations (Giostri
{\it et al}, arXiv:1203.3213v2[astro-ph.CO]), we affirm $\beta = 0.0062$ and $k
= 0.000016$. For these constraints, we have substantiated a new class of
cosmological transit models for which the expansion takes place from early
decelerated phase to the current accelerated phase. Also, we have studied some
physical, kinematic and geometric behavior of the models, and have found them
consistent with observations and well established theoretical results . We have
also compared our present results with those of Mishra {\it et al} (2013) and
observed that the results in this paper are much better, stable under
perturbation and in good agreement with cosmological reflections.
