Introduction to Mathematical Physics/Continuous approximation/Conservation laws

Integral form of conservation laws

A conservation law\index{conservation law} is a balance that can be applied to every connex domain strictly interior to the considered system and that is followed in its movement. such a law can be written:

eqcon

${\frac {d}{dt}}\int _{D}A_{i}dv+\int _{\partial D}\alpha _{ij}n_{j}d\sigma =\int _{D}a_{i}dv$

Symbol ${\frac {d}{dt}}$ represents the particular derivative (see appendix chapretour). $A_{i}$ is a scalar or tensorial\footnote{ $A_{i}$ is the volumic density of quantity ${\mathcal {A}}$ (mass, momentum, energy ...). The subscript $i$ symbolically designs all the subscripts of the considered tensor. } function of eulerian variables $x$ and $t$. $a_{i}$ is volumic density rate provided by the exterior to the system. $\alpha _{ij}$ is the surfacic density rate of what is lost by the system through surface bording $D$.

Local form of conservation laws

Equation eqcon represents the integral form of a conservation law. To this integral form is associated a local form that is presented now. As recalled in appendix chapretour, we have the following relation:

${\frac {d}{dt}}\int _{D}A_{i}dv=\int _{D}{\frac {d}{dt}}A_{i}dv$

It is also known that:

${\frac {dA_{i}}{dt}}={\frac {\partial A_{i}}{\partial t}}+(A_{i}u_{j})_{,j}$

Green formula allows to go from the surface integral to the volume integral:

$\int _{\partial D}\alpha _{ij}n_{j}d\sigma =\int _{D}\alpha _{ij,j}dv$

Final equation is thus:

${\frac {\partial A_{i}}{\partial t}}+(A_{i}u_{j}+\alpha _{ij})_{,j}=a_{i}$

Let us now introduce various conservation laws.