Cauchy momentum equation

The local balance law for linear momentum in a continuum, relating mass density times acceleration to stress divergence and body-force density.

Cauchy stress tensor: A tensor describing internal force per unit area across oriented surfaces in a continuum. Its spatial divergence supplies the internal-force density in the equation.
Continuum mechanics: The study of material bodies modeled as continuously distributed matter. The equation is a foundational local law within this framework.
Navier–Stokes equations: The equations governing motion of viscous fluids under mass and momentum conservation. A Newtonian-fluid stress law reduces the momentum balance to these equations.
Integral momentum equation: A control-volume statement relating momentum change to surface and body forces. It expresses the same balance over a finite region rather than locally.
Material derivative: The rate of change of a field following a moving material point. It gives the acceleration when applied to the velocity field.
Linear momentum: The product of a body's mass and velocity, conserved in isolated systems. The equation expresses how momentum density changes under force.
Euler equations: The equations of motion for an inviscid fluid, derived from momentum conservation. Neglecting viscous stress gives this fluid-flow form of the balance.
Conservative form: A formulation that writes evolution equations as flux balances plus source terms. It makes momentum flux explicit and remains useful across discontinuities.
Body force: A force distributed through a material's volume, such as gravity or electromagnetic force. Its force per unit volume appears alongside stress divergence.
Density: Mass per unit volume of a material. It converts acceleration into force per unit volume.

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