Introduction to Mathematical Physics/Continuous approximation/Conservation laws

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Integral form of conservation laws[edit]

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_idv+\int_{\partial D} \alpha_{ij} n_jd\sigma=\int_D a_idv

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[edit]

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_idv=\int_D \frac{d}{dt} A_idv

It is also known that:


\frac{dA_i}{dt}=\frac{\partial A_i}{\partial t}+(A_iu_j)_{,j}

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


\int_{\partial D} \alpha_{ij}n_jd\sigma=\int_{D} \alpha_{ij,j}dv

Final equation is thus:


\frac{\partial A_i}{\partial t}+(A_iu_j+\alpha_{ij})_{,j}=a_i

Let us now introduce various conservation laws.