Classical frames, primitive quantities and SI units

Use the nonrelativistic Newtonian spacetime $I\times E$, where $I\subseteq \mathbb{R}$ is an open interval with absolute time $t$ in seconds. Here $E$ is a three-dimensional oriented Euclidean affine space: its translation vectors form an oriented Euclidean vector space, but no origin in $E$ is distinguished. A Cartesian inertial chart is an event-coordinate map $(t,X)\mapsto (t,x_t(X))$, where $b:E\to {\mathbb{R}}^3$ is an orientation-preserving affine isometry, $w\in {\mathbb{R}}^3$ is constant, and $x_t(X)=b(X)+tw$ for $t\in I$. The affine map $b$ is auxiliary data defined on $E$, so this formula requires no evaluation at time $0$ when $0\mathrm{\notin }I$. Each fixed-time spatial map is an orientation-preserving affine isometry with fixed spatial orientation; the spatial origin has constant velocity. Spatial coordinates are measured in metres and $w$ in metres per second. A chart may also use a translated absolute-time coordinate; at corresponding events the inertial-chart transformations are the ones displayed below. Write $\mathrm{S}\mathrm{O}(3)=\{Q\in {\mathbb{R}}^{3\times 3}:Q^{T}Q=I_3,\det Q=1\}$. Inertial charts are related by

$t^{\ast }=t+t_0,x^{\ast }=Qx+vt+a,$

where $t_0$ is a constant time translation in seconds, $Q\in \mathrm{S}\mathrm{O}(3)$ is constant, $v\in {\mathbb{R}}^3$ is a constant relative velocity in m s${}^{-1}$, and $a\in {\mathbb{R}}^3$ is a constant spatial translation in metres. A worldline is a curve $t\mapsto (t,x(t))$ with $x\in C^2(I,{\mathbb{R}}^3)$, parameterized by absolute time; this model imposes no relativistic speed or causal condition. A rigid observer change is the more general coordinate change $t^{\ast }=t,x^{\ast }=Q(t)x+c(t)$, with $Q\in C^2(I,\mathrm{S}\mathrm{O}(3))$ and $c\in C^2(I,{\mathbb{R}}^3)$ in metres; it is not called inertial unless $Q$ is constant and $c$ is affine in time. A frame here means an adapted Cartesian chart, not an observer congruence or a basis field.

For the initial-chart material-motion convention below, require $0\in I$. Let $B\subseteq {\mathbb{R}}^3$ be a bounded open reference body whose boundary is locally the graph of a $C^1$ function, with $B$ lying on one side of each such graph. Identify the reference coordinates with the chosen initial Cartesian chart, and let $U$ be an open neighborhood of $\bar{B}$. A material motion is a map $\chi \in C^2(I\times U,{\mathbb{R}}^3)$ such that each ${\chi }_t=\chi (t,\cdot )$ is a bijection from $U$ onto an open image, has a $C^2$ inverse, and has positive Jacobian determinant; in the initial chart set $\chi (0,a)=a$. The label $a\in U$ and position $x=\chi (t,a)$ are measured in metres. Write

$F(t,a)=D_a\chi (t,a),J(t,a)=\det F(t,a)>0,A(t,x)={\chi }_t^{-1}(x),{\mathrm{\Omega }}_t={\chi }_t(B)\mathrm{.}$

Here $F$ and $J$ are dimensionless and $A$ has units metres. The material map $\chi$ describes motion of the medium; it is not a change of observer or frame. The Eulerian velocity is the vector field defined by $u(t,\chi (t,a))={\mathrm{\partial }}_t\chi (t,a)$, measured in m s${}^{-1}$. For a differentiable scalar or component field $f$, its material derivative is $D_{t}f={\mathrm{\partial }}_{t}f+u\cdot \mathrm{\nabla }f$. Set $L_{ij}={\mathrm{\partial }}_j{u}_i$, $D=(L+L^T)\mathrm{/}2$, $W=(L-L^T)\mathrm{/}2$, $\theta =\mathrm{tr}D=\mathrm{div}u$, and $D_0=D-(\theta \mathrm{/}3)I_3$; $L,D,W,\theta ,D_0$ have units s${}^{-1}$. Define vorticity by

$\mathrm{curl}u=({\mathrm{\partial }}_y{u}_z-{\mathrm{\partial }}_z{u}_y,{\mathrm{\partial }}_z{u}_x-{\mathrm{\partial }}_x{u}_z,{\mathrm{\partial }}_x{u}_y-{\mathrm{\partial }}_y{u}_x),$

also in s${}^{-1}$. In this library, a motion is material-incompressible on a region when $\mathrm{div}u=0$ there; spatially homogeneous density means $\mathrm{\nabla }\rho =0$ there. These are distinct conditions. A pathline is a $C^1$ curve $X$ on an interval where the field is defined and the solution exists, satisfying $dX\mathrm{/}dt=u(t,X(t))$. A streamline at time $t_0$ is a $C^1$ integral curve satisfying $dX\mathrm{/}ds=u(t_0,X(s))$ for a frozen $t_0$, with curve parameter $s$ measured in seconds. These are different constructions for an unsteady field.

A spatial vector is an element of the Euclidean translation space; its components rotate by $Q$ under a rigid observer change. A spatial second-order tensor is a linear map from that vector space to itself, represented by a $3\times 3$ matrix in a Cartesian orthonormal basis; it transforms as $S^{\ast }=QS{Q}^T$. A tensor-valued constitutive response is objective when its values at corresponding events satisfy this transformation rule for every $C^2$ rigid observer change. For a stress tensor $\sigma$, use the tension-positive convention in which the traction (force per unit current area) on a surface with outward unit normal $n$ is $\sigma n$. Set $(\mathrm{div}\sigma {)}_i={\mathrm{\partial }}_j{\sigma }_{ij}$; the tensor contraction is $S:T={\sum }_{i,j}{S}_{ij}{T}_{ij}$. Stress and traction have units Pa $=\mathrm{N}{\mathrm{m}}^{-2}=\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$.

Within a specified model, a primitive quantity is a quantity adopted as input rather than constructed from the other data of that model; a derived quantity is assigned by an explicit construction from stated inputs and results. This status is framework-local and is independent of whether its SI unit is base or derived. In the continuum model used on this page, ${\rho }_0(a)$ is a reference mass-density field in kg m${}^{-3}$, $b(t,x)$ is an adopted body-acceleration field in m s${}^{-2}$, $e(t,x)$ is specific internal energy (energy per unit mass) in J kg${}^{-1}={\mathrm{m}}^2{\mathrm{s}}^{-2}$, $T(t,x)>0$ is temperature in K, $s(t,x)$ is specific entropy in J kg${}^{-1}$ K${}^{-1}$, $q(t,x)$ is heat flux (heat-transfer rate per unit area) in W m${}^{-2}$, and $r(t,x)$ is specific heat supply (power per unit mass) in W kg${}^{-1}$. Their adoption is a physical-model choice, not a consequence of the SI. The current mass density $\rho (t,x)$ means mass per unit current volume and has units kg m${}^{-3}$; for a mass-conserving motion, its construction from ${\rho }_0$ and $\chi$ is established from the stated balance hypotheses in the theorem on this page. A pressure $p$ is a scalar in Pa whose role depends on the selected constitutive branch. The shear and bulk dynamic-viscosity parameters $\mu ,\zeta$, when that branch is adopted, have units Pa s $=\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$; thermal conductivity ${\kappa }_T$ has units W m${}^{-1}$ K${}^{-1}$. The gradient $\mathrm{\nabla }$ has units m${}^{-1}$. Thus the metre, kilogram, second, and kelvin are SI base units in this dictionary, while density, velocity, acceleration, pressure, energy, power, viscosity, and heat-flux units are derived according to the BIPM SI conventions.

MIT OpenCourseWare 8.01SC, Chapter 11, §§11.1–11.3 (Galilean frames)

MIT OpenCourseWare 8.09, full lecture notes, §1.1 (inertial and rotating frames)

David Tong, Fluid Mechanics, Part I, §§1.1.1–1.1.3 (Eulerian fields, paths, and material derivative)

BIPM, SI Brochure 9th edition, version 4.01, §§2.2–2.3 and Tables 4–6

NIST SP 330-2019, §§2–3 and Tables 4–6

The batch-2 supplier Smooth material configurations and Eulerian fields is now present: its inspected definition gives the $C^2$ family of orientation-preserving diffeomorphisms, positive Jacobian, inverse label, Eulerian velocity, material derivative, and strain-rate interface used here. This item states its own initial-chart normalization and explicit SI dictionary as additional conventions.