  We discuss in detail the shear viscosity coefficient eta and the viscosity to
entropy density ratio eta/s of a hadronic gas comprised of pions and nucleons.
In particular, we study the effects of baryon chemical potential on eta and
eta/s. We solve the relativistic quantum Boltzmann equations with binary
collisions (pi pi, pi N, and NN) for a state slightly deviated from thermal
equilibrium at temperature T and baryon chemical potential mu. The use of
phenomenological amplitudes in the collision terms, which are constructed to
reproduce experimental data, greatly helps to extend the validity region in the
T-mu plane. The total viscosity coefficient eta(T,mu)=eta^pi + eta^N increases
as a function of T and mu, indirectly reflecting energy dependences of binary
cross sections. The increase in mu direction is due to enhancement of the
nucleon contribution eta^N while the pion contribution eta^pi diminishes with
increasing mu. On the other hand, due to rapid growth of entropy density, the
ratio eta/s becomes a decreasing function of T and mu in a wide region of the
T-mu plane. In the kinematical region we investigated T < 180MeV, mu < 1GeV,
the smallest value of eta/s is about 0.3. Thus, it never violates the
conjectured lower bound eta/s= 1/4pi ~ 0.1. The smallness of eta/s in the
hadronic phase and its continuity at T ~ T_c (at least for crossover at small
mu) implies that the ratio will be small enough in the deconfined phase T >
T_c. There is a nontrivial structure at low temperature and at around normal
nuclear density. We examine its possible interpretation as the liquid-gas phase
transition.
