On the local well-posedness of strong solutions to the unsteady flows of shear-thinning non-Newtonian fluids with a concentration-dependent power-law index

TL;DR

This paper establishes local existence and uniqueness of strong solutions for a complex fluid flow model with variable shear-thinning properties, relevant to biological joint fluids, in both 2D and 3D settings.

Contribution

It proves the local well-posedness of a nonlinear PDE system modeling shear-thinning fluids with concentration-dependent properties, extending previous results to variable power-law indices.

Findings

Existence of local strong solutions in 3D for certain p^- and p^+ bounds.

Uniqueness of solutions under a stricter upper bound p^+ < 28/15.

Existence and uniqueness in 2D with less restrictive conditions.

Abstract

We investigate a system of nonlinear partial differential equations modeling the unsteady flow of a shear-thinning non-Newtonian fluid with a concentration-dependent power-law index. The system consists of the generalized Navier-Stokes equations coupled with a convection-diffusion equation describing the evolution of chemical concentration. This model arises from the mathematical description of the behavior of synovial fluid in the cavities of articulating joints. We prove the existence of a local-in-time strong solution in a three-dimensional spatially periodic domain, assuming that $\frac{7}{5} < p^- \le p(\cdot) \le p^+ \le 2$, where $p(\cdot)$ denotes the variable power-law index and $p^-$ and $p^+$ are its lower and upper bounds, respectively. Furthermore, we prove the uniqueness of the solution under the additional condition $p^+ < \frac{28}{15}$. In particular, our three-dimensional analysis directly implies the existence and uniqueness of solutions in the two-dimensional case under the less restrictive condition $1 < p^- \le p(\cdot) \le p^+ \le 2$.