Tenseur de densité du flux d'impulsion

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Le tenseur de densité du flux d'impulsion est donné par :

\Pi _{ik}=p\delta _{ik}+\rho v_{i}v_{k}

Considérons un fluide ayant la vitesse ${\vec {v}}$ : l'impulsion de l'unité de volume du fluide est égale à $\rho {\vec {v}}$ avec $\rho$ la masse volumique du fluide. Le taux de sa variation est donc:

{\frac {\partial \rho {\vec {v}}}{\partial t}}

En notations tensorielles, nous avons alors: ${\frac {\partial \rho v_{i}}{\partial t}}=\rho {\frac {\partial v_{i}}{\partial t}}+v_{i}{\frac {\partial \rho }{\partial t}}$

Utilisons l'équation de continuité: ${\frac {\partial \rho }{\partial t}}=-{\frac {\partial \rho v_{k}}{\partial x_{k}}}$ ; et l'équation d'Euler: ${\frac {\partial v_{i}}{\partial t}}=-v_{k}{\frac {\partial v_{i}}{\partial x_{k}}}-{\frac {1}{\rho }}{\frac {\partial p}{\partial x_{i}}}$

Grâce à ces deux équations, nous déduisons:

{\frac {\partial \rho v_{i}}{\partial t}}=-\rho v_{k}{\frac {\partial v_{i}}{\partial x_{k}}}-{\frac {\partial p}{\partial x_{i}}}-v_{i}{\frac {\partial \rho v_{k}}{\partial x_{k}}}=-{\frac {\partial p}{\partial x_{i}}}-{\frac {\partial \rho v_{i}v_{k}}{\partial x_{k}}}

Nous pouvons très bien écrire que: ${\frac {\partial p}{\partial x_{i}}}=\delta _{ik}{\frac {\partial p}{\partial x_{k}}}$

d'où, ${\frac {\partial \rho v_{i}}{\partial t}}=-{\frac {\partial (\delta _{ik}p+\rho v_{i}v_{k})}{\partial x_{k}}}=-{\frac {\partial \Pi _{ik}}{\partial x_{k}}}$

Afin de mettre en évidence la signification du tenseur $\Pi _{ik}$ , intégrons l'équation précédente dans un certain volume:

{\frac {\partial }{\partial t}}\int \rho v_{i}dV=-\int {\frac {\partial \Pi _{ik}}{\partial x_{k}}}dV

D'après le théorème d'Ostrogradsky-Gauss, nous pouvons écrire:

{\frac {\partial }{\partial t}}\int \rho v_{i}dV=-\oint \Pi _{ik}dS_{k}

Dans le premier membre nous avons la variation par unité de temps de la i-ème composante de l'impulsion dans le volume considéré. Ce qui signifie que le second membre représente la quantité de cette impulsion s'écoulant par unité de temps à travers la surface délimitant le volume considéré.

En écrivant $dS_{k}=n_{k}dS$ , nous pouvons dire que: $\Pi _{ik}n_{k}=pn_{i}+\rho v_{i}v_{k}$ ce qui correspond à l'expression vectorielle $p{\vec {n}}+\rho {\vec {v}}({\vec {v}}\cdot {\vec {n}})$ . Nous concluons que $\Pi _{ik}$ est la i-ème composante de la quantité d'impulsion traversant par unité de temps l'unité de surface normale à l'axe $x_{k}$ .

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