Still, the complexity is somewhat mitigated by the fact that the components are stationary with respect to one another, that is, one component does not diffuse into another. It is possible to bypass the averaging process and to postulate balance, or conservation, equations directly.

The language even suggests the concept that there are many different events, or realizations, and that the expected value is an average over all of them. Only limited information can be obtained from these and similar calculations about higher concentrations and bubble deformation at higher Reynolds numbers. Let x be points in physical space. The key is the correct prescription of constitutive equations. Rather it is proper to identify the cause of the inadequacy of the model, obtain a better model, and reassess the constitutive equations and predictions.

The solutions of problems of hydromechanics of ordinary continua are used as motivation for the forms of certain constitutive equations in multicomponent materials. Thus it is extremely rare for us to revise its mathematical description. Except for simple systems, this is not practical. Indeed, it is actually at the point of the prescription of constitutive equations where the microscopic detail is lost. Many other materials, both manufactured and occurring in nature, are not homogeneous.

This allows computation of values for the undetermined parameters using the approximate solutions. With respect to notation, we use different notations for different purposes.

The assumed form for constitutive equations includes constants or functions to be determined. The problem with this approach is that if general exact solutions were known, they could be used directly and there would be no need for averaging. In the averaging approach, the undetermined parameters often have forms relating them to the exact or plausible approximate solutions of the microscopic equations of motion. We focus on the macroscopic viewpoint in this book.

In the former case, the ensemble consists of a large number of possible trajectories of each molecule. Even the concepts of point mass, rigid massless rod, and frictionless pivot are approximations to the description of the behavior seen by any observer. This suggests that each ensemble of realizations, corresponding to a given physical situation, has a probability measure on subsets of realizations. Oil, gas, and water coexist in rock. Moreover, such notation is intrinsically confusing to the eye, and becomes almost overwhelmingly so for multicomponent materials, where it is normal to denote the different components by indices.

Acknowledgments We owe gratitude to many for their support and encouragement. These are volume V and mass M.