Introduction to Mathematical Physics/Continuous approximation/Momentum conservation

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We assume here that external forces are described by and that internal strains are described by tensor .

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:

Using continuity equation yields to:

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

Later on, fluid momentum is simply designated by .