Fluid mechanical basic equations

Energy systems are, by their very nature, the seat of transfers, and for the most part, mass and heat transfers. This is particularly true when these systems include flowing fluids, which is very often the case. The energetic quality of heat transfer is obvious. In the case of mass transfers, energy lies beneath the surface, in the form of internal energy (characterized essentially by the temperature level), enthalpy (the internal energy associated with pressure potential energy), kinetic energy, gravitational potential energy, chemical energy, etc., etc., etc.

Knowledge of fluid flow transfer is therefore fundamental to solving a large number of energy problems. Two different analyses are generally applied for this purpose: one is of the local type, the other of the global type. Both can be approached through the same study, that of the balances we present in this article.

Whatever the quantity under examination, it is possible to write that its variation over time, for a given domain, is due to a flux of this quantity across the domain boundary, to a diffusion of the quantity with respect to the mean flux (superposition of a microscopic movement on the observed macroscopic movement) and to sources or production (positive or negative) of this quantity. The translation of this (common sense) principle into "mathematical" form is the general balance equation. Its application to the mass of a fluid leads to the so-called conservation of mass equation (no source, no diffusion). If the magnitude is momentum, Newton's principle is translated by this balance equation, in which diffusion is due to the fluid's viscosity, while the source is due to the various field and pressure forces. Finally, the first principle of thermodynamics is reflected in the energy balance equation, which leads to an enthalpy balance, with the diffusive term corresponding to the conduction (linear – Fourier law) of heat in the fluid, and the source term to heat input by radiation, for example, or by chemical reactions. The kinetic energy balance presented in this article is not independent of other balances. It is merely a "mechanistic" presentation of the energy balance, and corresponds in fact to the integration, in one direction of space, of the momentum balance. Other classical balances could have been presented in this article, notably the species balance (for flows of reactive or non-reactive mixtures) and the entropy balance. This has not been done, so as not to overload this "introduction" to the study of balances.

By mathematical transformation of integrals, all balance equations can be given a local form from which an integration, almost always numerical, can be performed, taking into account boundary and initial conditions, to determine vector (velocity) and...