The Governing Equations for an Electrically Conducting Fluid

Abstract

Introduction In the first two chapters, the field of micro- and nanofluidics was introduced in a general framework, describing multiple applications as well as the scientific issues associated with fluid flow at small length scales. In Chapter 2, the fundamental transport properties involved in mathematical description of fluid flow, heat and mass transfer, and electrostatics were introduced. In this chapter, we use these concepts to derive, from first principles, the governing equations necessary for analyzing micro- and nanofluidic phenomena. For the purposes of this chapter, it will be assumed that the properties of the fluid medium are constant. Microfluidic devices are being used for rapid and continuous purification of proteins; a sketch of such a device is shown in Figure 3.1. The device addresses the need for high-throughput purification of very small amounts of proteins and enzymes from the carrier fluid. The term protein purification refers to a series of operations meant to isolate a single protein or enzyme in a complicated mixture. Here the microfluidic transport processes involve mass transport of a relatively large number of species with the target molecules present in as little as microgram per liter concentrations. This device can purify a sample in a short period of time and does not require a large amount of sample. The means of developing a model for such a device is discussed in this chapter. The governing equations of fluid motion on the macroscale are the (incompressible) Navier–Stokes equations. These equations, along with conservation of mass, or the continuity equation , enable the calculation of, in the general three-dimensional case, the three velocity components and the pressure.