# The Equation of Motion

The equation of motion of the system is

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

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

$$\rho_{k}F^{k}_{\alpha}:=\sum_{i=1}^{n}\rho_{k}F^{k}_{\alpha}\\$$

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”

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

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

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

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

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

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

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

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

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

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 $$\alpha$$: