The Equation of Motion

The equation of motion of the system is

\begin{equation} \label{dGM17}\rho\frac{dv_{\alpha}}{dt}=\frac{\partial}{\partial x_{\beta}}P^{\beta}_{\alpha}+\rho_{k}F^{k}_{\alpha}\ \ (\alpha=1,2,3),\ \ \ (k=1,\cdot n)\\ \end{equation}

where superscripts indicates rows, subscripts columns, and Einstein summation convention is used, i.e., for example:

\begin{equation} \rho_{k}F^{k}_{\alpha}:=\sum_{i=1}^{n}\rho_{k}F^{k}_{\alpha}\\ \end{equation}

where \(v_{\alpha}\) (\(\alpha=1,2,3\)) is a Cartesian component of \(\vec{v}\) and where \(x_{\alpha}\) (\(\alpha=1,2,3\)) are the Cartesian coordinates. The derivative \(dv_{\alpha}/dt\) is a component of the acceleration of the centre of gravity motion. The quantity \(P_{\alpha}^{\beta}\) (\(\alpha,\beta=1,2,3\)) are the Cartesian components of the pressure (or stress) tensor of the medium and \(F^{k}_{\alpha}\) and of the force for unit mass \(F^{k}\) exerted on the chemical component \(k\) respectively. We shall assume here\({}^{*}\) that the pressure P is symmetric, \(P_{\alpha}^{\beta}=P_{\beta}^{\alpha}\) (\(\alpha,\beta=1,2,3\))

In tensor notation, equation (\ref{dGM17}) are written as”

\begin{equation} \rho\frac{d\vec{v}}{dt}=-\nabla\cdot{\bf P}+\overset{\rightharpoonup}{\rho}\cdot\overset{\rightharpoonup}{\vec{F}}\\ \end{equation}

\(P_{\alpha}^{\beta}=P_{\beta}^{\alpha}\) (\(\alpha,\beta=1,2,3\)) In tensor notation, equation (\ref{dGM17}) are written as”

\begin{equation} \rho\frac{d\vec{v}}{dt}=-\nabla\cdot\vec{\bf P}+\overset{\rightharpoonup}{\rho}\cdot\overset{\rightharpoonup}{\vec{F}}\\ \end{equation}

From a microscopic point of view one can say that the pressure tensor \({\bf P}\) results from the short range interactions between the particle of the system , whereas \(\overset{\rightharpoonup}{\vec{F}}\) contains the external forces as well as a possible contribution from long-range interactions in the system. For the moment, we shall restrict the discussion to the consideration of conservative forces which can be derived from a potential \(\psi_{k}\) independent of time

\begin{equation} \label{dGM20}\vec{F}_{k}=-\vec{\nabla}\psi_{k},\ \ \ \frac{\partial\psi_{k}}{\partial t}=0\\ \end{equation}

Using relation \ref{dGM16}, the equation of motion (\ref{dGM19}) can be written as

\begin{equation} \frac{\partial\rho\vec{v}}{\partial t}=-\nabla\cdot(\rho\vec{v}\vec{v}+\vec{\bf P})+\overset{\rightharpoonup}{\rho}\cdot\overset{\rightharpoonup}{\vec{F}}\\ \end{equation}

Using relation (\ref{dGM16}), the equation of motion (\ref{dGM19}) can also be written as

\begin{equation} \label{dGM21}\frac{\partial\rho\vec{v}}{\partial t}=-\nabla\cdot(\rho\vec{v}\vec{v}+\vec{\bf P})+\rho_{k}\vec{F}^{k}\\ \end{equation}

wher \(\vec{v}\vec{v}\) is an ordered (dyadic) product (as explained in Appendix I). This equation has the form of a balance equation for momentum density \(\rho\vec{v}\). In fact, it is seen that one can interpret the quantity \(\rho\vec{v}\vec{v}+\vec{\bf P}\) as a momentum flow, with a convective part \(\rho\vec{v}\vec{v}\) and the quantity \(\rho_{k}\vec{F}^{k}\) as a source of momentum.

It is possible to derive from (\ref{dGM17}) a balance equation for the kinetic energy of the centre of gravity motion by multiplying both members with the component \(v_{\alpha}\) of the barycentric velocity and summing over \(\alpha\):