Conservation of Energy

Abstract

We briefly considered in Chapter 6 a geologically important class of flows—buoyancy driven flows—in which thermal effects hold an essential part in creating the forces that induce flow. We also should recall that fluid properties such as viscosity can vary with temperature. As fluids are heat-conducting media, taking into account the thermal energy conditions and heat flow within a fluid therefore is often an essential part of describing a flow field. Thermal energy, however, is not transported merely by conduction; in a moving fluid, thermal energy also is advected from position to position. In addition, recall that the thermal energy of a fluid, according to the first law of thermodynamics, is inextricably coupled with energy in the form of work performed between a fluid element and its surroundings. It is therefore important to consider the mechanical energy of a fluid when describing its thermal conditions. To this end, the developments below concern conservation of energy in both thermal and mechanical forms. Analogous to our treatment of conservation of mass, we will derive equations that describe a condition—conservation of energy—which must be satisfied at each coordinate position in a fluid. An important outcome of our development of expressions for conservation of energy is a set of dynamical equations for the special case of an ideal fluid. In component form, these are referred to as Euler’s equations, and arise from conservation of purely mechanical energy, neglecting thermal forms. (We will derive Euler’s equations again in Chapter 10 using an explicit treatment of the forces involved in fluid motion.) Conservation of energy also applies to flow in porous media; the relevant expressions are similar to those for purely fluid flow, but with several important differences that arise from the two-phase character (solid and fluid) of flow in porous media. In relation to this topic, we also will develop the idea of Hubbert’s potential, and the relation of this potential to piezometric head. This is a cornerstone of the theory of flow in porous media. Our objective is to illustrate how Hubbert’s potential, and head, are obtained from applying the idea of conservation of mechanical energy to a fluid.