Euler equations
- Fluid Mechanics Unclassified

05 Apr 2023 in Physics on Fluid Mechanics · 1 min

from 「Fluid Mechanics」: Landau, L.D. & Lifschitz, E.M.

Equations for adiabatic and inviscid flow.

• Momentum equation
  • Proof

Momentum equation

The momentum equation is a partial differential equation that describes the motion of an ideal fluid,

\[ \frac{D\b{u}}{Dt} = -\frac{1}{\rho}\nabla p \]

where $\frac{D}{Dt}$ is the material derivative. The momentum equation is derived from the conservation of momentum, which states that the total momentum of a system is constant.

Proof

Let us consider some volume $V_0$ in the fluid. The total force acting on this volume is equal to the surface integral of the pressure. It can be transformed to a volume integral by the divergence theorem:

\[ -\oint_{\partial V_0}p \dd{\b{f}} = -\int_{V_0} \nabla p \dd{V} \]