  A microscopic analysis of the viscous energy gain of energetic particles in
(gradual) non-relativistic shear flows is presented. We extend previous work
and derive the Fokker-Planck coefficients for the average rate of momentum
change and dispersion in the general case of a momentum-dependent scattering
time $\tau(p) \propto p^{\alpha}$ with $\alpha \geq 0$. We show that in
contrast to diffusive shock acceleration the characteristic shear acceleration
timescale depends inversely on the particle mean free path which makes the
mechanism particularly attractive for high energy seed particles. Based on an
analysis of the associated Fokker-Planck equation we show that above the
injection momentum $p_0$ power-law differential particle number density spectra
$n(p) \propto p^{-(1+ \alpha)}$ are generated for $\alpha >0$ if radiative
energy losses are negligible. We discuss the modifications introduced by
synchrotron losses and determine the contribution of the accelerated particles
to the viscosity of the background flow. Possible implications for the plasma
composition in mildly relativistic extragalactic jet sources (WATs) are
addressed.
