  We consider the acceleration of charged particles in relativistic shearing
flows, with Lorentz factor up to $\Gamma_0 \sim 20$. We present numerical
solutions to the particle transport equation and compare these with results
from analytical calculations. We show that in the highly relativistic limit the
particle energy spectrum that results from acceleration approaches a power law,
$N(E)\propto E^{-\tilde{q}}$, with a universal value $\tilde{q}=(1+\alpha)$ for
the slope of this power law, where $\alpha$ parameterizes the power-law
momentum dependence of the particle mean free path. At mildly relativistic flow
speeds, the energy spectrum becomes softer and sensitive to the underlying flow
profile. We explore different flow examples, including Gaussian and
power-law-type velocity profiles, showing that the latter yield comparatively
harder spectra, producing $\tilde{q}\simeq 2$ for $\Gamma_0 \simeq 3$ and
Kolmogorov turbulence. We provide a comparison with a simplified leaky-box
approach and derive an approximate relation for estimating the spectral index
as a function of the maximum shear flow speed. These results are of relevance
for jetted, high-energy astrophysical sources such as active galactic nuclei,
since shear acceleration is a promising mechanism for the acceleration of
charged particles to relativistic energies and is likely to contribute to the
high-energy radiation observed.
