Addendum to: “Dynamics of incompressible fluids with incompatible distortion rates” [International Journal of Engineering Science 168C (2021)]

Abstract

Fosdick and Fried (2021) proposed a generalized Navier–Stokes theory for studying the dynamics of incompressible fluids which, under certain flow conditions, may support incompatible distortion rates. Herein, we complete the development of a comprehensive boundary condition, at a fixed wall, for the incompatibility tensor $G$ of that theory; we clarify the physical conditions which express the presence of incompatibility at a wall and, thus, its transmission into the adjacent fluid. The final condition incorporates a constitutively prescribed threshold ${\tau }_c$ for the magnitude of the shear stress vector $s$ at the wall. For $\left|s\right|<{\tau }_c$, $G=O$. For $\left|s\right|\ge {\tau }_c$, $G=\gamma (1-t\otimes t)+G_{nt}n\otimes t$, where $\gamma$ is a material constant, $t$ and $n$ are appropriately defined orthonormal tangent vectors to the wall and $G_{nt}$ is a possibly non-zero component of $G$ at the wall.