Existence for the steady problem of a mixture of two power-law fluids

TL;DR

This paper proves the existence of very weak solutions for a steady mixture of two power-law fluids with variable and constant exponents, extending classical results to more general variable exponent cases.

Contribution

It establishes the existence of solutions for a mixed power-law fluid problem with minimal assumptions on the variable exponent, extending Ladyzhenskaya's classical theorem.

Findings

Existence of very weak solutions under bounded variable exponent

Extension of Ladyzhenskaya's theorem to variable exponent case

Applicability to pseudoplastic fluid regions

Abstract

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.