In the case of gases it is easiest to analyze a fixed volume closed system as

# In the case of gases it is easiest to analyze a fixed

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In the case of gases it is easiest to analyze a fixed-volume closed system as shown in Fig. 2.7. Figure 2.7(a) depicts the low-temperature situation, and from this it is easy to imagine that molecular collisions and hence, also intermolecular exchanges of momentum, are relatively infrequent. But increasing the temperature results in higher energy and momentum; moreover, because of the corresponding higher velocities any given molecule covers a greater distance, on average, in a unit of time, thus enhancing the probability of colliding with another molecule. Thus, both the number of collisions and the momentum exchanged per collision increase with temperature in a gas, and it is known from the kinetic theory of gases that both of these factors result in increased viscosity. Diffusion of Momentum We know from basic physics that diffusion corresponds to “mixing” of two or more substances at the molecular level . For example, we can consider the (mass) diffusion of salt into fresh water and quantify the degree of mixing with the concentration of salt. But we can also analyze diffusion of energy and momentum in a similar way. We will later see, after we have derived the equations of fluid motion, that diffusion of momentum is one of the key physical processes taking place in fluid flow; moreover, it will be clear, mathematically, that viscosity is the transport property that mediates this process. Here we will provide a brief physical description of how this occurs. We first note that by diffusion of momentum we simply mean mixing on molecular scales of a high-momentum portion of flow (and thus one of higher speed in the case of a single constant- temperature fluid) with a lower-momentum portion. The end result is a general “smoothing” of the velocity profiles such as were first shown in Fig. 2.4. The physical description of this process is best understood by considering the initial transient leading to the velocity profile of that figure. In particular, let both plates in Fig. 2.4 have zero speed until time t = 0 + . Then at time t = 0 the fluid is motionless throughout the region between the plates. An instant later the top plate is impulsively set into motion with speed U , due to a tangential force F . At this instant the velocity profile will appear as in Fig. 2.8(a). The fluid velocity immediately adjacent to the top plate is essentially
2.3. FLUID PROPERTIES 21 (a) (b) Figure 2.7: Effects of temperature on molecular motion of gases; (a) low temperature, (b) higher temperature. (c) (b) U U U (a) Figure 2.8: Diffusion of momentum—initial transient of flow between parallel plates; (a) very early transient, (b) intermediate time showing significant diffusion, (c) nearly steady-state profile. that of the plate (by the no-slip condition), but it is nearly zero at all other locations. (Notice that the velocity profile is very nonsmooth.) This does not, however, imply that the molecules making up the fluid have zero velocity, but only that when their velocities are averaged over a finite, but arbitrarily small, volume (continuum hypothesis, again!) this average is everywhere zero.

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