Introduction to Mathematical Physics/Continuous approximation/Momentum conservation

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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_idv+\int_{\partial D} \tau_{ij} n_jd\sigma=\int_D f_idv

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 p dp f(r,p,t)

Later on, fluid momentum is simply designated by p.