We should note further that the structure of granular substances such as sugar

We should note further that the structure of granular

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We should note further that the “structure” of granular substances such as sugar is very different from that of fluids, as a little further investigation will show. Again, using sugar as a common
2.3. FLUID PROPERTIES 15 example, it can be seen that there is a very different kind of interaction between pairs of small but macroscopic solid grains of sugar than occurs at microscopic levels for atoms and/or molecules (or even fluid parcels) of gases and liquids. In particular, among other differences is the very important one of sliding friction generated by the movement of grains of sugar past one another. This is the main reason such granular materials are able to support some shear stress without deforming. But we should remark that despite these fundamental physical differences between fluids and other somewhat similar substances that flow, the equations of fluid motion to be developed and studied in these lectures often provide at least a reasonable approximation in the latter cases, and they are often used for practical calculations in these contexts. 2.2.2 More on shear stress We will later need to be able to study forces acting on fluid elements, and shear stress will be very important in deriving the equations of fluid motion, and later in the calculation of drag due to flow over submerged objects and flow resistance in pipes and ducts. Since shear stress is a (tangential) force per unit area, we can express this for a finite area A as ¯ τ = F A , where F is the tangential force applied over the area A . This is the average shear stress acting on the finite area A . But in the derivation and analysis of the differential equations describing fluid motion it is often necessary to consider shear stress at a point . The natural way to define this is τ = lim A 0 F A . We note that this limit must, of course, be viewed in the context of the continuum hypothesis since, as we have previously stressed, there may not be any molecules of the fluid at the selected point at an arbitrary particular time. Because of this the formal limit shown above must be replaced with ∆ A ǫ , ǫ > 0, where ǫ is sufficiently small to be negligible in comparison with macrosopic length scales squared, but still sufficiently large to contain enough molecules to permit calculation of averaged properties and “construction” of fluid parcels. 2.3 Fluid Properties Our next task is to consider various properties of fluids which to some extent permit us to distinguish one fluid from another, and they allow us to make estimates of physical behavior of any specific fluid. There are two main classes of properties to consider: transport properties and (other, general) physical properties. We will begin by considering three basic transport properties, namely viscosity, thermal conductivity and mass diffusivity; but we will not in these lectures place much emphasis on the latter two of these because they will not be needed for the single-phase, single-component, incompressible flows to be treated herein. Following this, we will study basic physical properties

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