Introduction to Mathematical Physics/Continuous approximation/Momentum conservation

We assume here that external forces are described by $f$ and that internal strains are described by tensor $\tau _{ij}$.

${\frac {d}{dt}}\int _{D}\rho u_{i}dv+\int _{\partial D}\tau _{ij}n_{j}d\sigma =\int _{D}f_{i}dv$

This integral equation corresponds to the applying of Newton's law of motion\index{momentum} over the elementary fluid volume as shown by figure figconsp.

Momentum conservation law corresponds to the application of Newton's law of motion to an elementary fluid volume.}

figconsp

Partial differential equation associated to this integral equation is:

${\frac {\partial }{\partial t}}(\rho u_{i})+(\rho u_{i}u_{j})_{,j}+\tau _{ij,j}=f_{i}$

Using continuity equation yields to:

$\rho ({\frac {\partial }{\partial t}}u_{i}+u_{j}u_{i,j})+\tau _{ij,j}=f_{i}$

Remark: Momentum conservation equation can be proved taking the first moment of Vlasov equation. Fluid momentum ${\bar {p}}$ is then related to repartition function by the following equality:

${\bar {p}}=\int pdpf(r,p,t)$

Later on, fluid momentum is simply designated by $p$.