Mathematical functions that define the fluid state

Mathematical functions:-

Following the continuous assumption, the mathematical description of the state of a moving fluid can be characterized by functions of the coordinates x, y, z and of the time t. These functions of (x, y, z, t) are related to the quantities defined for the fluid at a given point (x, y, z) in space and at a given time t, which refers to fixed points in space and not to fixed particles of the fluid. For example, we can consider the mean local velocity v(x, y, z, t) of fluid particles or fluid points, called the fluid velocity, and two thermodynamic quantities that characterize the fluid state, the pressure p(x, y, z, t) and the mass density (x, y, z, t), the mass per unit volume of fluid. Following the discussion of §2, two remarks can be done at this stage:

 i). The fluid is assumed to be a continuum. This implies that all space-time derivatives of all dependent variables exist in some reasonable sense. In other words, local properties such as density pressure and velocity are defined as averages over elements large compared with the microscopic structure of the fluid but small enough in comparison with the scale of the macroscopic. This allows the use of differential calculus to describe such a system.

 ii). All the thermodynamic quantities are determined by the values of any two of them, together with the equation of state. Therefore, if we are given five quantities, namely the three components of the velocity v, the pressure p and the mass density , the state of the moving fluid is completely determined. We recall that only if the fluid is close to thermodynamic equilibrium, its thermodynamic properties, such as pressure, density, temperature are well-defined. This requires (as a very former hypothesis) that local relaxation times towards thermal equilibrium are much shorter than any macroscopic dynamical time scale. In particular, microscopic collision time scale (between elementary constituents of the fluid) needs to be much shorter than any macroscopic evolution time scales. This hypothesis is almost a tautology for standard fluids build up by molecules at reasonable density, but becomes not trivial in the case of some hot dense matter state created in high energetic hadronic collisions.

In the following, we prove that these five unknown quantities describe completely the case of what we define as ideal fluids, in which we take no account of processes of energy dissipation. Energy dissipation may occur in a moving fluid as a consequence of internal friction (or viscosity) within the fluid and heat exchange between different parts of it. Neglecting this phenomenon, we can find a set of five equations that are sufficient obtain a closed system: 5 equations for 5 unknown quantities.

 Interestingly, we can gain some intuition about the behavior of the ideal flow by expressing in more details its pressure field. An ideal fluid, in particular, is characterized by the assumption that each particle pushes its neighbors equally in every direction. This is why a single scalar quantity, the pressure, is sufficient to describe the force per unit area that a particle exerts on all its neighbors at a given time. Also, we know that a fluid particle is not accelerated if its neighbors push back with equal force, which means that the acceleration of the fluid particle results from the pressure differences. In short, the pressure force can be seen as a global interaction of all fluid particles. 

When the energy dissipation inside the fluid is not neglected, we need to consider also the internal energy density e(x, y, z, t) and heat flux density q(x, y, z, t) as four additional unknown functions to be determined by a proper set of closed equations: nine equations are needed in such cases. This concerns what we define as real fluids. We discuss the case of real fluids in more details later in the document. However, a few intuitive arguments can be made with no mathematical formalism. When the energy dissipation is not neglected, this means that we take into account frictional forces inside the fluid. Their main effect is that they enhance the local coherence of the flow. They counteract at each point the deviation of the velocity field from its local average. This means that if a fluid particle moves faster than the average of its neighbors, then friction