Materials processing models often deal with the transport of mass, momentum, and energy. In this chapter we derive governing equations for the conservation of these quantities. We focus on differential forms, which are used in continuum models. We also include some integral forms, which help in the physical interpretation of the equations. The forms we derive are quite general, and you may wonder if the complexity this adds is worthwhile. Remember that we are seeking equations that have a very wide range of validity, so that, when we simplify them for a particular model, we can be sure that no term has been left out.

Although the focus of this chapter is on balance equations, we also introduce a few classical constitutive equations: the Newtonian fluid and Fourier's law of heat conduction. However, we also retain forms of the balance equations that can be used with other constitutive equations.

Insofar as possible, we avoid reference to particular coordinate systems, by using general vector and tensor notation. Readers who are unfamiliar with this style may want to review Appendix A. All of the governing equations derived in this chapter are written out in terms of their components for rectangular, cylindrical, and spherical coordinates in Appendix B. These component forms are the actual starting point for most models.

PRELIMINARIES

MATERIAL AND SPATIAL DESCRIPTIONS

In this chapter we want to develop equations for quantities that associate a value with each point in the material, quantities that include velocity, temperature, pressure, and stress. We also want to deal with materials that are moving and deforming.