  The Rossby wave instability (RWI), which depends on the density bumps and
extremum in the vortensities in the differentially rotating discs, plays an
important role in the evolution of the protoplanetary discs. In this article,
we investigate the effect of viscosity on the non-axisymmetric RWI in the
self-graviting accretion discs. For this purpose, we add the viscosity to the
work of Lovelace and Hohlfeld (2013). Consideration of viscosity complicates
the problem so that we use the numerical method to investigate the stable and
unstable modes. We consider three ranges of viscosities: high viscosity in the
ranges $0.1\leq \alpha\leq 0.4$, moderate viscosity in the ranges $0.01\leq
\alpha < 0.1$, and low viscosity in the ranges $\alpha < 0.01$. The results
show that the occurrence of the RWI is related to the value of viscosity so
that the effect of high viscosity is important, while the low viscosity is
negligible. These results may be applied for the study of the RWI role in
planet formation and angular momentum transport for different kinds of the
protoplanetary discs with different viscosities.
