  Transport coefficients in highly ionised plasmas like the intra-cluster
medium (ICM) are still ill-constrained. They influence various processes, among
them the mixing at shear flow interfaces due to the Kelvin-Helmholtz
instability (KHI). The observed structure of potential mixing layers can be
used to infer the transport coefficients, but the data interpretation requires
a detailed knowledge of the long-term evolution of the KHI under different
conditions. Here we present the first systematic numerical study of the effect
of constant and temperature-dependent isotropic viscosity over the full range
of possible values. We show that moderate viscosities slow down the growth of
the KHI and reduce the height of the KHI rolls and their rolling-up.
Viscosities above a critical value suppress the KHI. The effect can be
quantified in terms of the Reynolds number Re = U{\lambda}/{\nu}, where U is
the shear velocity, {\lambda} the perturbation length, and {\nu} the kinematic
viscosity. We derive the critical Re for constant and temperature dependent,
Spitzer-like viscosities, an empirical relation for the viscous KHI growth time
as a function of Re and density contrast, and describe special behaviours for
Spitzer-like viscosities and high density contrasts. Finally, we briefly
discuss several astrophysical situations where the viscous KHI could play a
role, i.e., sloshing cold fronts, gas stripping from galaxies, buoyant
cavities, ICM turbulence, and high velocity clouds.
