1.2 Velocity, Acceleration, and the Material Derivative

A fluid is defined as a material that will undergo sustained motion when shearing forces are applied, the motion continuing as long as the shearing forces are maintained. The general study of fluid mechanics considers a fluid to be a continuum. That is, the fact that the fluid is made up of molecules is ignored but rather the fluid is taken to be a continuous media.

In solid and rigid body mechanics, it is convenient to start the geometric discussion of motion and deformation by considering the continuum to be made up of a collection of particles and consider their subsequent displacement. This is called a Lagrangian, or material, description, named after Joseph Louis Lagrange (1736 1836). To illustrate its usage, let ( X( X ₀, Y ₀, Z ₀, t), Y( X ₀, Y ₀, Z ₀, t), Z( X ₀, Y ₀, Z ₀, t)) be the position at time t of a particle initially at the point ( X ₀, Y ₀, Z ₀). Then the velocity and acceleration of that particle is given by

The partial derivatives signify that differentiation is performed holding X ₀, Y ₀ and Z ₀ fixed.

This description works well for particle dynamics, but since fluids consist of an infinite number of flowing particles in the...