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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

Kyueon Choi, Kyungkeun Kang, Seungchan Ko

TL;DR

The paper addresses local-in-time well-posedness for unsteady flows of shear-thinning non-Newtonian fluids with a concentration-dependent power-law index in a 3D periodic domain, coupling generalized Navier–Stokes dynamics with a convection–diffusion equation for concentration. A Galerkin approximation with a regularized concentration is employed, and variable-exponent Sobolev theory is used to control the nonlinear stress through the energies $\mathcal{I}_p(c,\boldsymbol{v})$ and $\mathcal{J}_p(c,\boldsymbol{v})$, yielding uniform estimates and a local Grönwall argument to obtain a strong solution on a time interval $I'$. Uniqueness is shown under the enhanced bound $p^+<\frac{28}{15}$, and the analysis is adapted to the 2D case, where existence and uniqueness hold under the broader shear-thinning regime $1<p^-\le p(\cdot)\le p^+\le 2$. The work extends prior results by incorporating concentration-dependent rheology and addressing the extra terms arising from differentiating the stress with respect to concentration, providing a rigorous local-in-time theory with potential pathways to numerical schemes and higher-dimensional generalizations. Overall, the results offer a mathematically rigorous framework for models of synovial and other concentration-sensitive fluids, with implications for numerical analysis and applications in biomechanics.

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 $1 < p^- \le p(\cdot) \le p^+ \le 2$.

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

The paper addresses local-in-time well-posedness for unsteady flows of shear-thinning non-Newtonian fluids with a concentration-dependent power-law index in a 3D periodic domain, coupling generalized Navier–Stokes dynamics with a convection–diffusion equation for concentration. A Galerkin approximation with a regularized concentration is employed, and variable-exponent Sobolev theory is used to control the nonlinear stress through the energies and , yielding uniform estimates and a local Grönwall argument to obtain a strong solution on a time interval . Uniqueness is shown under the enhanced bound , and the analysis is adapted to the 2D case, where existence and uniqueness hold under the broader shear-thinning regime . The work extends prior results by incorporating concentration-dependent rheology and addressing the extra terms arising from differentiating the stress with respect to concentration, providing a rigorous local-in-time theory with potential pathways to numerical schemes and higher-dimensional generalizations. Overall, the results offer a mathematically rigorous framework for models of synovial and other concentration-sensitive fluids, with implications for numerical analysis and applications in biomechanics.

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 , where denotes the variable power-law index and and are its lower and upper bounds, respectively. Furthermore, we prove the uniqueness of the solution under the additional condition . In particular, our three-dimensional analysis directly implies the existence and uniqueness of solutions in the two-dimensional case under the less restrictive condition .
Paper Structure (14 sections, 20 theorems, 210 equations, 1 figure)

This paper contains 14 sections, 20 theorems, 210 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Galerkin approximation and uniform estimates
  4. Galerkin approximation for the regularized system
  5. A priori estimates
  6. Application of local Grönwall inequality
  7. Proof of Theorem \ref{['maintheorem']}
  8. Proof of Theorem \ref{['uniqueness theorem']}
  9. Two-dimensional problem
  10. Main theorems in two dimensions
  11. Construction of Galerkin system and a priori estimates
  12. Proof of Theorem \ref{['maintheorem2d']}
  13. Proof of Theorem \ref{['uniqueness theorem2d']}
  14. Conclusion

Key Result

Lemma 2.1

Let $\mathcal{I}_p(c,\boldsymbol{v})$ be the energy defined in special_energy with variable exponent $p(\cdot)$ satisfying $1<p^-\leq p(\cdot)\le p^+\le2$. Then for sufficiently smooth $\boldsymbol{v}$, there holds for a.e. $t\in I$ that

Figures (1)

  • Figure 1: The exponent $p(c)$ is plotted as a function of the concentration of hyaluronan molecules associated with the viscosity \ref{['visc_form']} for synovial fluid; see, e.g., Hron2010 for further details. Physiological values typically observed in non-pathological synovial fluid lie approximately within the range $(0.1, 0.25)$, as indicated by the shaded region in the figure. It is evident that the condition $p^-\geq \frac{d+2}{2}$ ($\frac{5}{2}$ when $d=3$) assumed in choi2024 is too restrictive, and an existence analysis for smaller values of the power-law index is required to cover the case of higher physiological concentrations.

Theorems & Definitions (29)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Local version of Grönwall's inequality
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 19 more