Fluids and Porous Media as Continua

Abstract

Let us anticipate that we wish to treat fluids as continuous substances at a microscopic scale. To see the motivation for this, it is instructive to consider the possibility of describing fluid behavior at a molecular scale by making use of Lagrangian mechanics to track the behavior of each molecule, just as we would describe the ballistics of a moving, rigid body. Consider describing the state of a simple diatomic molecule at some instant; to do this, we must decide what minimum set of coordinates completely specifies the position and configuration of the molecule. For example, we must specify its position within an inertial reference frame, which requires the three Cartesian coordinates x, y, and z. We also must specify its velocity with respect to this coordinate system, which requires the three corresponding components of velocity um, vm, and wm. The molecule may be spinning; to describe this, we must assign to the molecule three local coordinate axes to specify three angular coordinates that give its orientation within the inertial reference frame. Because the axis of rotation may not coincide with one of the local axes, we also must specify two angular coordinates that give the orientation of the axis of rotation within the local coordinate system. Finally, we must specify the angular velocity about this axis of rotation. Thus, in addition to specifying the mass of a molecule, we need twelve variables or generalized coordinates to specify its state. Moreover, we must know initial values of these twelve coordinates, just as we need to know the initial position and velocity of a ballistic body to track its course. To track the behavior of N molecules, we therefore must know N masses plus 12N initial position, velocity, and orientation coordinates. It becomes clear that to adopt this approach to describe the behavior of a fluid constitutes a formidable task! One has no choice but to abandon a molecular scale treatment and adopt a view involving the microscopic scale, where behaviors of individual molecules are ignored, and instead, the collective behavior of a suitably defined ensemble of molecules is treated in a statistical (average) sense in terms of bulk properties such as fluid density, temperature, and viscosity.