Equations of Motion

Fundamental differential Euler and Navier-Stokes equations governing fluid motion.

General idea of the topic

The Navier-Stokes equations are Newton's second law applied to a fluid element: the acceleration of a fluid particle results from pressure, viscosity and body forces. Together with the continuity equation they form the complete model of a Newtonian fluid.

The topic presents first the Euler equation, which is the case without viscosity and from which Bernoulli is obtained, and then an introduction to Navier-Stokes: what each term represents and why, in general, it is solved only with numerical methods.

What you will truly understand

By the end you will be able to identify in the equation the inertia, pressure, viscous and external force terms, and recognize the limits in which it simplifies to already known cases.

The aim is not to solve it analytically but to understand its meaning.

Recommended learning route

1. Euler Equation for Fluids: force balance per unit volume without viscosity, from which Bernoulli is derived.
2. Introduction to the Navier-Stokes Equation: term-by-term reading of the equation with viscosity and its relation to the second law.

Physical criterion you must acquire

The essential criterion is to understand that the equation expresses a force balance per unit volume. Without viscosity it reduces to Euler's, from which Bernoulli is obtained along a streamline. The Reynolds number measures the relative importance of the inertia and viscous terms.

Connection with real problems

• numerical flow simulation.
• weather forecasting.
• design of aircraft and vehicles.
• hemodynamics.

Final goal of the topic

The final goal is to understand the structure of the equation that governs real fluids and place the simplified models within it.