Moreover we will see that in this context dudy is a part of the strain rate

Moreover we will see that in this context dudy is a

This preview shows page 24 - 27 out of 164 pages.

3. Moreover, we will see that in this context du/dy is a part of the “strain rate,” from which it follows that stresses in fluids are proportional to strain rate , rather than to strain, itself, as in solid mechanics. Units of Viscosity At this point it is useful to consider the dimensions and units of viscosity. To do this we solve Eq. (2.1) for μ to obtain μ = Fh AU .
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2.3. FLUID PROPERTIES 19 We can now easily deduce the units of μ in terms of very common ones associated with force, distance, area and velocity. We will usually employ “generalized” dimensioins in these lectures, leaving to the reader the task of translating these into a specific system, e.g. , the SI system, of units. We will typically use the notation T time L length F force M mass . Now recall that velocity has dimension L/T in this formalism, so it follows that the dimension of viscosity is given by μ F · L L 2 ( L/T ) F · T L 2 . In SI units this would be n · s/m 2 ; i.e. , Newton · seconds/square meter. In many applications it is convenient to employ the combination viscosity/density, denoted ν : ν = μ ρ . This is called the kinematic viscosity, while μ is termed the dynamic viscosity. Since the (general- ized) dimension for density is M/L 3 , the dimension for kinematic viscosity is ν F · T/L 2 M/L 3 F · T · L M . But by Newton’s second law of motion F/M acceleration L/T 2 . Thus, dimension of viscosity is simply ν L 2 T , or, again in SI units, m 2 /s. Physical Origins of Viscosity Viscosity arises on molecular scales due to two main physical effects: intermolecular cohesion and transfer of molecular momentum. It should be expected that the former would be important (often dominant) in most liquids for which molecules are relatively densely packed, and the latter would be more important in gases in which the molecules are fairly far apart, but moving at high speed. These observations are useful in explaining the facts that the viscosity of a liquid decreases as temperature increases, while that of a gas increases with increasing temperature. First consider the liquid case, using water (H 2 O) as an example. We know that the water molecule has a structure similar to that depicted in Fig. 2.6(a), i.e. , a polar molecule with weak intermolecular bonding due to the indicated charges. We also know that the kinetic energy and momentum increase with increasing temperature. Thus, at higher temperatures the forces available to break the polar bonds are much greater than at lower temperatures, and the local order shown in Fig. 2.6(a) is reduced (as indicated in Fig. 2.6(b)); so also is the “internal friction” reduced, and the liquid is then less viscous than at lower temperatures.
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20 CHAPTER 2. SOME BACKGROUND: BASIC PHYSICS OF FLUIDS electron cloud disorder due to higher molecular energy (b) (a) local short-range structure H H H H H H O O H H H O O H H O O H H H H O O H O H H H H O O H H H H + + + + + + + + + + + + + + + + + + + + + + - - - - - - - - - - - Figure 2.6: Structure of water molecule and effect of heating on short-range order in liquids; (a) low temperature, (b) higher temperature.
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