Open VOGEL/Fluids

Introduction
When we talk about fluids we talk about matter. At nanometric scale, fluids are naturally made of atoms and molecules. But what gives fluids their unique property, and sets them apart from solids, is that each one of these particles is free, and not rigidly linked to its neighbors. Despite the atomic nature of fluids, we don't study them by treating molecules one by one. Instead we assume the whole fluid is a sort of continuous ether, with mass and other properties distributed over space. The difference between this ether and a real fluid, is that we could infinitely divide it in small pieces and, unlike in reality, each piece will always retain the same intensive properties. This is the reason why we can use the term point particle. A point particle is not an atom or molecule, not even a quark. It only refers to the description of specific properties at a given point.

So although we now all agree that matter is not continuous, the idea of continuity is necessary to describe the flow in a mathematical way, this is, by using functions in the form of scalar fields, vector fields and tensor. Throughout this chapter we will use several concepts provided by calculus to describe the behavior of fluids. We will start by introducing the most basic concept, which is the velocity field.

Kinematics of fluids
As we said in the introduction, the difference between solids and fluids is that the particles of the former ones are rigidly held together. The particles of a fluid are not hardly bound to their neighbors, so that they are able to undergo continuous deformations. A fluid particle has temporary neighbors. When we study solids in elasticity, we analyse the displacement between the point particles, and based on that, we deduce the deformation, stress and internal energy. In fluids dynamics we do it differently. We study the velocity field, which tells us how fast particles are moving, and based on the variations of this field we can deduce the stress around that particle.

The successive positions of a given particle over time can be written as an unsteady vector field. Imagine we take a picture of the flow at a given moment. This instant we can take as reference to name each particle, and from then on, the successive positions of a particle $$\mathbf{P}_O$$ will be given by a vector function like this:


 * $$\mathbf{P}=[x,y,z,t]=\mathbf{P}(\mathbf{P}_O,t)$$

The points associated to this function form lines in space that are called trajectory lines.

In general, we could have written this for any property $$A$$ of a certain particle, like for instance its density, as follows:


 * $$A=F(\mathbf{P}_O,t)$$

Because this representation describes the evolution of a property of a unique particle, we call it a material representation. If, instead of following an specific particle, we would have opted for sensing the value of a property at a fixed point in space, we would have come out with a different tratment: a local representation.


 * $$A=F(\mathbf{P},t)$$

A local representation senses the value of the property carried by each particle that occupies the control point during the observation. So here we clearly distinguish two different kinds of representations.