Basics of governing equations

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The thermal-fluids systems can either possess one phase or multiple phases: the former is referred to as single-phase system while the latter is referred to as multiphase system. A multiphase system, which is distinguished from a single-phase system by the presence of one or more interfaces separating the phases, can be considered as a field that is divided into single-phase regions by those interfaces, or moving boundaries, between phases. A single-phase system can be described using the standard local instance governing equations with appropriate boundary conditions.

Two common techniques are used for describing fluid flow: Lagrangian and Eulerian. The Lagrangian approach requires that the properties of a particular element of the fluid particles be tracked as it traverses the flow. This approach is similar to what we used in particle and rigid-body dynamics. The location of this fluid element is described by its coordinates (x, y, z), which are functions of time. The fluid element can be identified by tracking it from its initial location (x0, y0, z0) at time t = 0, and the velocity of this element at an arbitrary time t is expressed as . In order to describe a fluid flow using the Lagrangian approach, the sensors that monitor fluid properties would have to move at the same velocity as the fluid element; this is an impractical and often impossible requirement to meet, especially for such complex cases as three-dimensional transient flow. Therefore, the Lagrangian approach is rarely used in description of fluid flow.

The Eulerian approach, on the other hand, observes the flow properties from a fixed location relative to a reference frame, which can be stationary or more generally moves at its own velocity. The Eulerian approach gives the values of the fluid variable at a given point (x,y,z) at a given time t. For example, the velocity can be expressed as , where x, y, and z are independent of t. The Eulerian approach requires that the fluid properties be measured at spatial locations that are fixed relative to the reference frame in the fluid field, so the sensors are not required to move with the individual particles. Since the Eulerian approach is consistent with conventional experimental observation techniques, it is widely used in fluid mechanics and will also be adopted in this textbook.

The conservation laws for flow and heat transfer can be expressed for a fixed-mass or for a control volume. While the former is a fixed collection of particles with constant mass (fixed-mass), the latter is a defined region relative to the reference frame in space (fixed-volume). From a thermodynamic point of view, the fixed-mass and the control volume can be considered as closed and open systems, respectively. Writing the governing equations for a fixed-mass requires tracking the motion of the particles, i.e., the Lagrangian approach. As we have already mentioned, although the Lagrangian description is applicable to some fluid mechanics problems, it is not a very practical way to describe multiphase systems, with a few exceptions such as liquid droplet tracking in a thermal plasma. The governing equations written for a control volume, on the other hand, express the relationship between the change of properties inside the control volume and the property of the flow into or out of the control volume. The governing equations obtained by writing the conservation laws for a control volume are consistent with the Eulerian approach. However, all of the fundamental laws of mechanics, including conservation of mass, momentum, and energy, are formulated for a collection of particles with fixed identity; that is to say, they are Lagrangian in nature. Therefore, it is necessary to apply the fundamental laws to the fixed-mass first, and then to convert them into expressions for the control volume.

Please refer to the following articles for different topics about governing equations:

References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Further Reading

External Links