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Euler’s Equations of Motion

A set of partial differential equations describing the motion of inviscid (no viscosity) fluid flow. These are fundamental to aerodynamics.
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The statement of the theorem

Euler's Equations of Motion constitute a system of partial differential equations governing the flow of a compressible, inviscid fluid (u\mathbf{u} is the velocity vector, ρ\rho is the density, pp is the pressure, and f\mathbf{f} represents external body forces per unit mass). The system is derived from the conservation laws (mass, momentum, and energy) and is expressed in the material derivative form DDt=∂∂t+u⋅∇\frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf{u} \cdot \boldsymbol{\nabla}.\n\n**1. Continuity Equation (Conservation of Mass):**\n∂ρ∂t+∇⋅(ρu)=0\frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \cdot (\rho \mathbf{u}) = 0\n\n**2. Momentum Equation (Euler Equation):**\nρ(∂u∂t+(u⋅∇)u)=−∇p+f\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u} \right) = -\boldsymbol{\nabla} p + \mathbf{f} \n\n**3. Energy Equation (Conservation of Energy):**\nAssuming the fluid is adiabatic and no external work is done by non-pressure forces, the total energy E=e+12u⋅uE = e + \frac{1}{2} \mathbf{u} \cdot \mathbf{u} (where ee is specific internal energy) satisfies:\nρ(∂E∂t+(u⋅∇)E)=−p(∂ρ∂t+u⋅∇ρ)+u⋅fnon−pressure\rho \left( \frac{\partial E}{\partial t} + (\mathbf{u} \cdot \boldsymbol{\nabla}) E \right) = -p \left( \frac{\partial \rho}{\partial t} + \mathbf{u} \cdot \boldsymbol{\nabla} \rho \right) + \mathbf{u} \cdot \mathbf{f}_{non-pressure} \n\n*Note: For a perfect gas, the equation of state p=p(ρ,T)p = p(\rho, T) and the specific enthalpy hh are often used to close the system, relating the three equations.*
Source: Wikipedia