its length scale is larger than the mean free path and at the same time is

Its length scale is larger than the mean free path

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, its length scale is larger than the mean free path), and at the same time is small compared with the typical length scales of the problem under consideration, then the resulting average will be useful for engineering analyses. Furthermore, we can imagine continuously sliding this averaging volume through the fluid, thus obtaining a continuous function defined at every point for the velocity in particular, and in general for any flow properties to be treated in more detail in the sequel. This permits us to state the following general idea that we will employ throughout these lectures, often without specifically noting it. Continuum Hypothesis . We can associate with any volume of fluid, no matter how small (but greater than zero), those macroscopic properties (e.g., velocity, temperature, etc.) that we associate with the bulk fluid. This allows us to identify with each point a “fluid particle,” or “fluid parcel,” or “fluid element,”
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2.2. DEFINITION OF A FLUID 13 (each with possibly its own set of properties, but which vary in a regular way, at least over short distances), and then consider the volume of fluid as a whole to be a continuous aggregation of these fluid particles. To make this discussion more concrete we provide some specific examples. First, we consider air at standard sea-level temperature and pressure. In this situation, the number of molecules in a cubic meter is O (10 25 ). If we consider a volume as small as a μ m 3 , i.e. , (10 6 m) 3 = 10 18 m 3 , there are still O (10 7 ) molecules in this region. Thus, meaningful averages can easily be computed. In fact, we can decrease the linear dimensions of our volume by nearly two orders of magnitude (nearly to nanoscales) before the number of molecules available for averaging would render the results unreliable. This suggests that sea-level air is a fluid satisfying the continuum hypothesis in the majority of common engineering situations—but not on nanoscales, and smaller. In contrast to this, consider a cube of volume O (10 6 ) cubic meters ( i.e. , a centimeter on a side) at an altitude of 300 km. At any given instant there is only one chance in 10 8 of finding even a single molecule in this volume. Averaging of properties would be completely impossible even though the linear length scale is macroscopic, viz. , 1 cm. Thus, in this case (low-Earth orbit altitudes) the continuum hypothesis is not valid. From the preceding discussions it is easy to see that while most analyses of fluid flow involve situations where the continuum hypothesis clearly is valid, there are cases for which it is not. Very high altitude (but still sub-orbital) flight at altitudes to be accessed by the next generation of military aircraft is an example. But a more surprising one is flow on length scales in the micro- and nano-scale ranges. These are becoming increasingly important, and because the length scales of the devices being studied are only a few mean free paths for the fluids being used, the continuum hypothesis is often not valid. As a consequence, in these situations the analytical methods we will study in the present lectures must be drastically modified, or completely replaced.
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