Conservation of Energy

According to the principle of conservation of energy, the total energy content within an arbitrary volume V in the system can only change if energy flows into (or out of) the volume through its boundary \(\Omega\)

\begin{equation} \label{dGM30}\frac{d}{dt}\int^{V}\rho edV=\int^{V}\frac{\partial\rho e}{\partial t}dV=-\int^{\Omega}\vec{J}_{e}\cdot d\Omega\\ \end{equation}

Here \(e\) is the energy per unit mass, and \(\vec{J}_{e}\) the energy flux per unit surface and unit time. We shall refer to \(e\) as the total specific energy, because it includes all forms of energy in the system. Similarly we shall call \(\vec{J}_{e}\) the total energy flux. With the help of Gauss’ theorem, we obtain the differential or local form of the law of conservation of energy

\begin{equation} \label{dGM31}\frac{\partial\rho e}{\partial t}+\nabla\cdot\vec{J}_{e}=0\\ \end{equation}

In order to relate this equation to previously obtained results (\ref{dGM29}) for the kinetic and potential energy, we must specify which are the various contributions to energy \(e\) and the flux \(\vec{J}_{e}\). The total specific energy \(e\) includes the specific kinetic energy \(\frac{1}{2}\vec{v}^{2}\), the specific potential energy \(\psi\) and the specific internal energy \(u\):

\begin{equation} \label{dGM32}e=\frac{1}{2}\vec{v}^{2}+\psi+u\\ \end{equation}

From a macroscopic point of view, this relation can be considered as the definition of internal energy \(u\). With these definitions, we actually introduce thermodynamics into the game. From a microscopic point of view, \(u\) represents the energy and thermal agitation, as well as the energy due to the short-range molecular interactions. Similarly the total energy flux includes a convective term \(\rho e\vec{v}\), and energy flux \(\vec{\bf P}\cdot\vec{v}\) due to the mechanical work performed on the system, a potential energy flux \(\sum_{k}\psi_{k}\vec{J}_{k}\) due to the diffusion of the various components in the field of force, and finally a “heat flow” (transfer of thermal energy) \(\vec{J}_{q}\):

\begin{equation} \label{dGM33}\vec{J}_{e}=\rho e\vec{v}+\vec{\bf P}\cdot\vec{v}+\sum_{k}\psi_{k}\vec{J}_{k}+\vec{J}_{q}\\ \end{equation}

This relation maybe considered as defining the heat flow \(\vec{J}_{q}\). If we subtract equation (\ref{dGM29}) from equation (\ref{dGM31}), we obtain, using also (\ref{dGM32}) and (\ref{dGM33}), the balance equation for the internal energy \(u\):

\begin{equation} \label{dGM34}\frac{\partial\rho u}{\partial t}=-\nabla\cdot(\rho u\vec{v}+\vec{J}_{q})-\vec{\bf P}:\vec{\nabla}\vec{v}+\sum_{k}\vec{J}_{k}\vec{F}_{k}\\ \end{equation}

It is apparent from this equation that internal energy \(u\) is not conserved. In fact a source term appear, which is equal but of opposite sign to the source term of the balance equation (\ref{dGM29}) for kinetic and potential energy.

The equation (\ref{dGM34}) may be written in an alternate form. If we restrict our analysis on non-elastic fluids, we can split the total pressure tensor into a scalar hydrostatic part \(p\) and a tensor \(\Pi\):

\begin{equation} \label{dGM35}\vec{\bf P}:=p\vec{\bf I}+\vec{\bf\Pi}\\ \end{equation}

where \(\vec{\bf I}\) is the unit tensor matrix with elements \(\delta_{\alpha\beta}\) (\(\delta_{\alpha\beta}=1\) if \(\alpha=\beta\), \(\delta_{\alpha\beta}=0\) if \(\alpha\neq\beta\). With this relation (\ref{dGM16}), equation (\ref{dGM34}) becomes

\begin{equation} \label{dGM36}\rho\frac{du}{dt}=-\nabla\cdot\vec{J}_{q}-p\nabla\cdot\vec{v}-\vec{\bf P}:\vec{\nabla}\vec{v}+\sum_{k}\vec{J}_{k}\vec{F}_{k}=\rho\frac{dq}{dt}--p\nabla\cdot\vec{v}-\vec{\bf P}:\vec{\nabla}\vec{v}+\sum_{k}\vec{J}_{k}\vec{F}_{k}\\ \end{equation}

where use has been made of the equality:

\begin{equation} \label{dGM37}\vec{\bf I}:\vec{\nabla}\vec{v}=\sum_{\alpha,\beta=1}^{3}\delta_{\alpha\beta}\frac{\partial}{\partial x_{\beta}}v_{\alpha}=\nabla\cdot\vec{v}\\ \end{equation}

and where:

\begin{equation} \label{dGM38}\rho\frac{dq}{dt}+\nabla\cdot\vec{J}_{q}=0\\ \end{equation}

defines \(dq\), the “heat” (thermal energy) added per unit mass. With (\ref{dGM14}) equation (\ref{dGM36}), the first law of thermodynamics can be written in the form:

\begin{equation} \label{dGM39}\frac{du}{dt}=\frac{dq}{dt}-p\frac{dv}{dt}-v\vec{\bf\Pi}:\vec{\nabla}\vec{v}+v\sum_{k}\vec{J}_{k}\vec{F}_{k}\\ \end{equation}

where \(v:=\rho^{-1}\) is called the specific volume.

AS stated in the preceding section, we have restricted in this chapter to the consideration of conservative forces \(\vec{F}_{k}\) of the type (\ref{dGM20}). The more general case, which arises for instance when electromagnetic forces are considered, will be treated elsewhere.

For elastic fluids: ****

How heat can be split