  The aim of the paper is to construct and justify asymptotic approximations
for solutions to quasilinear convection-diffusion problems with a predominance
of nonlinear convective flow in a thin cylinder, where an inhomogeneous
nonlinear Robin-type boundary condition involving convective and diffusive
fluxes is imposed on the lateral surface. The limit problem for vanishing
diffusion and the cylinder shrinking to an interval is a nonlinear first-order
conservation law. For a time span that allows for a classical solution of this
limit problem corresponding uniform pointwise and energy estimates are proven.
They provide precise model error estimates with respect to the small parameter
that controls the double viscosity-geometric limit. In addition, other problems
with more higher P\'eclet numbers are also considered.
