# 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 $$\Omega$$

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

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

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

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 $$u$$:

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

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) $$\vec{J}_{q}$$:

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

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 $$u$$:

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

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 $$\Pi$$:

$$\label{dGM35} \label{dGM35}\vec{\bf P}:=p\vec{\bf I}+\vec{\bf\Pi}\\$$

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

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

where use has been made of the equality:

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

and where:

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

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:

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

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