  The steady problem resulting from a mixture of two distinct fluids of
power-law type is analyzed in this work. Mathematically, the problem results
from the superposition of two power laws, one for a constant power-law index
with other for a variable one. For the associated boundary-value problem, we
prove the existence of very weak solutions, provided the variable power-law
index is bounded from above by the constant one. This result requires the
lowest possible assumptions on the variable power-law index and, as a
particular case, extends the existence result by Ladyzhenskaya dated from 1969
to the case of a variable exponent and for all zones of the pseudoplastic
region. In a distinct result, we extend a classical theorem on the existence of
weak solutions to the case of our problem.
