Chapter 4 - Conservation Laws

Abstract

This chapter focuses on conservation laws. The governing principles in fluid mechanics are the conservation laws for mass, momentum, and energy. These laws are presented in this order in this chapter and are stated in integral form, applicable to an extended region, or in differential form, applicable at a point or to a fluid particle. Both forms are equally valid and may be derived from each other. The integral forms of the equations of motion are stated in terms of the evolution of a control volume and the fluxes of mass, momentum, and energy that cross its control surface. The integral forms are typically useful when the spatial extent of potentially complicated flow details are small enough for them to be neglected and an average or integral flow property, such as a mass flux, a surface pressure force, or an overall velocity or acceleration, is sought. Setting aside nuclear reactions and relativistic effects, mass is neither created nor destroyed. Thus, individual mass elements—molecules, grains, fluid particles, etc.—may be tracked within a flow field because they will not disappear and new elements will not spontaneously appear. The equations representing conservation of mass in a flowing fluid are based on the principle that the mass of a specific collection of neighboring fluid particles is constant. The momentum-conservation equivalent is developed from Newton's second law, the fundamental principle governing fluid momentum. The integral energy-conservation equivalent is developed from a mathematical statement of conservation of energy for a fluid particle in an inertial frame of reference. The general equations of motion for a fluid may be put into a variety of special forms when certain symmetries or approximations are valid. Several special forms are presented in this chapter.