19.1 : Euler's Equations of Motion

In fluid mechanics, shear stresses arise from viscosity, which represents a fluid's internal resistance to deformation. For low-viscosity fluids, like water, these stresses are minimal, simplifying flow analysis by allowing the fluid to be treated as inviscid, or frictionless. In an inviscid fluid, shear stresses are absent, leaving only normal stresses, which act perpendicularly to fluid elements. Notably, pressure — defined as the negative of the normal stress — remains uniform across different directions and is unaffected by the orientation of stress.

The motion of inviscid fluids is governed by Euler's equations, derived from Newton's second law as applied to fluid dynamics. These equations describe momentum conservation by relating the rate of change in velocity to the forces acting on a fluid element. For an inviscid fluid, the relevant forces include pressure gradients and body forces, such as gravity. Euler's equations express the balance of these forces, excluding viscous terms, as:

where ρ is the fluid density, v is the velocity vector, p is the pressure, and g represents gravitational acceleration.

While Euler's equations ignore viscosity, they remain complex due to their nonlinear terms, which account for the convective acceleration of fluid particles. When integrated along a streamline, these equations yield Bernoulli's equation, which describes a fundamental relationship in fluid mechanics, linking pressure, velocity, and elevation for inviscid flows. Bernoulli's equation is a powerful tool for analyzing energy distribution within fluid flows, supporting applications in fluid transport, aerodynamics, and hydrodynamics.