22 Definition of a Fluid We are now in a position to examine the physical

22 definition of a fluid we are now in a position to

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study in the present lectures must be drastically modified, or completely replaced. 2.2 Definition of a Fluid We are now in a position to examine the physical behavior of fluids, and by so doing see how they differ from other forms of matter. This will lead us to a precise definition that does not suffer the deficiency alluded to earlier, i.e. , being unable to determine whether a substance is, or is not, a fluid. 2.2.1 Shear stress induced deformations The characteristic that distinguishes a fluid from a solid is its inability to resist deformation under an applied shear stress (a tangential force per unit area). For example, if one were to impose a shear stress on a solid block of steel as depicted in Fig. 2.2(a), the block would not begin to change shape until an extreme amount of stress has been applied. But if we apply a shear stress to a fluid element, for example of water, we observe that no matter how small the stress, the fluid element deforms, as shown in Fig. 2.2(b) where the dashed lines indicate the vertical boundaries of the fluid element before application of the shear stress. Furthermore, the more stress that is applied, the more the fluid element will deform. This provides us with a characterizing feature of liquids (and gases—fluids, in general) that distinguishes them from other forms of matter, and we can thus give a formal definition. Definition 2.1 A fluid is any substance that deforms continuously when subjected to a shear stress, no matter how small.
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14 CHAPTER 2. SOME BACKGROUND: BASIC PHYSICS OF FLUIDS (b) (a) Water Steel Figure 2.2: Comparison of deformation of solids and liquids under application of a shear stress; (a) solid, and (b) liquid. We remark that continuous deformation under arbitrarily small shear stresses is not seen in a number of common substances that appear to “flow,” for example, various household granular cooking ingredients such as sugar, salt, flour, many spices, etc . Clearly, any of these can be “poured,” but their response to shear stress is very different from that of a fluid. To see this, consider pouring a cup of sugar onto a table. If we do this carefully we will produce a pile of sugar having a nearly conical shape, as indicated in Fig. 2.3(a). If we were to analyze the outer surface pile of sugar (a) pool of coffee (b) Figure 2.3: Behavior of things that “flow”; (a) granular sugar, and (b) coffee. of this cone it would be seen that the grains of sugar on this surface must be supporting some shear stress induced by gravitational forces. But the shape of the pile is not changing—there is no deformation ; hence, sugar is not a fluid, even though it flows. By way of contrast, pour a cup of coffee onto the table. The coffee will spread across the surface of the table, as shown in Fig. 2.3(b), stopping only for somewhat detailed reasons we will be able to understand later. But in any case, it is clear that coffee cannot “pile up.” The shear stresses induced by gravitational forces cannot be supported, and deformation occurs: coffee is a fluid.
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