Smooth material configurations and Eulerian fields

Let $I\subseteq \mathbb{R}$ be a nonempty open interval containing $0$. For an integer $r\ge 1$, $C^r$ means that all coordinate partial derivatives through total order $r$ exist and are continuous. Let $B\subseteq {\mathbb{R}}^3$ be a nonempty bounded open reference region whose boundary is, near each boundary point and in coordinates $y=Qx+c$ for some $Q\in {\mathbb{R}}^{3\times 3}$ and $c\in {\mathbb{R}}^3$ with $Q^{\mathsf{T}}Q=I_3$, the graph of a $C^1$ function; let $U\subseteq {\mathbb{R}}^3$ be open with $\bar{B}\subset U$. A smooth motion is a map $\chi :I\times U\to {\mathbb{R}}^3$ of class $C^2$ such that, for every $t\in I$, the map ${\chi }_t(a):=\chi (t,a)$ is a bijection from $U$ onto an open set ${\chi }_t(U)$ with a $C^2$ inverse and with $\det D_a\chi (t,a)>0$ for all $a\in U$. The coordinate $a$ labels a reference point and $x=\chi (t,a)$ is its current position; $\mathrm{\Omega }:=\{(t,x):t\in I,x\in {\chi }_t(U)\}$ is the swept open set and ${\mathrm{\Omega }}_t:={\chi }_t(B)$ is the current image of $B$.

For $(t,x)\in \mathrm{\Omega }$, define the inverse label $A(t,x)$ by $\chi (t,A(t,x))=x$, and put $F(t,a):=D_a\chi (t,a)$ and $J(t,a):=\det F(t,a)$. The joint map $H(t,a):=(t,\chi (t,a))$ is injective and has block-triangular derivative with determinant $J(t,a)>0$. The Euclidean inverse function theorem therefore gives local $C^1$ inverses; they agree on overlaps because the label is unique, so $A$ is globally $C^1$ on $\mathrm{\Omega }$. Define the Eulerian velocity field $u(t,x):={\mathrm{\partial }}_t\chi (t,A(t,x))$; an Eulerian field is a function of time and current position. Thus $u$ is $C^1$.

For any $C^1$ scalar or vector field $f$ on $\mathrm{\Omega }$, its material derivative is $D_{t}f:={\mathrm{\partial }}_{t}f+(D_{x}f)u$, componentwise; it is the derivative along the curves $t\mapsto \chi (t,a)$. Here $D_{x}f$ and $D_a\chi$ are the matrices of coordinate partial derivatives. Define the spatial velocity gradient $L:=D_{x}u$, its symmetric and skew-symmetric parts $D:=(L+L^{\mathsf{T}})\mathrm{/}2$ and $W:=(L-L^{\mathsf{T}})\mathrm{/}2$, the scalar $\theta :=\mathrm{tr}L={\sum }_{i=1}^3{\mathrm{\partial }}_{x_i}{u}_i$, and the trace-free part $D_0:=D-\theta I_3\mathrm{/}3$. The trace is the sum of diagonal entries, and $I_3$ is the $3\times 3$ identity matrix.

For any interval $K\subseteq \mathbb{R}$ containing at least two points, a map from $K$ is $C^1$ if it is continuous on $K$, is continuously differentiable on its interior, and has at every included finite endpoint the corresponding one-sided derivative, with the derivative continuous there. A pathline is a $C^1$ map $x:\mathcal{J}\to {\mathbb{R}}^3$, where $\mathcal{J}\subseteq I$ is an interval containing at least two points, such that $(t,x(t))\in \mathrm{\Omega }$ and $x^{\mathrm{\prime }}(t)=u(t,x(t))$ for every $t\in \mathcal{J}$; derivatives at included finite endpoints are one-sided. Each curve $t\mapsto \chi (t,a)$ on $I$ is a pathline. For a fixed $t_0\in I$, a streamline at frozen time $t_0$ is a $C^1$ map $X:\mathcal{S}\to {\chi }_{t_0}(U)$, where $\mathcal{S}\subseteq \mathbb{R}$ is an interval containing at least two points, satisfying $X^{\mathrm{\prime }}(s)=u(t_0,X(s))$ for every $s\in \mathcal{S}$; derivatives at included finite endpoints are one-sided. No equality between pathlines and frozen-time streamlines is part of this definition. These are mathematical kinematic definitions; no units or physical balance postulates are imposed.

Euclidean inverse function theorem; local inverse supplier

R. Abeyaratne, Continuum Mechanics, Volume II, Chapter 3 §§3.1–3.3 and 3.8 (printed pp. 64–68, 89–91)

James F. Price, Lagrangian and Eulerian Representations of Fluid Flow, Part 1, §§1.2.1–1.2.2 (MIT OpenCourseWare, Fall 2024)