Finally we comment that the preceding discussion applies equally for all

Finally we comment that the preceding discussion

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Finally, we comment that the preceding discussion applies equally for all fluids—both liquids and gases, provided the continuum hypothesis has been satisfied. But at the same time, details of the behavior of μ will in fact differ, especially with respect to temperature, as we noted earlier. Non-Newtonian Fluids Before going on to a study of other properties of fluids it is worthwhile to note that not all fluids satisfy Newton’s law of viscosity given in Eq. (2.2). In particular, in some fluids this simple
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2.3. FLUID PROPERTIES 23 linear relation must be replaced by a more complicated description. Some common examples are ketchup, various paints and polymers, blood and numerous others. It is beyond the intended scope of these lectures to treat such fluids in any detail, but the reader should be aware of their existence and the form of their representation. It is common for the shear stress of non-Newtonian fluids to be expressed in terms of an empirical relation of the form τ = K parenleftbigg du dy parenrightbigg n , where the exponent n is called the flow behavior index , and K is termed the consistency index . This representation is called a “power law,” and fluids whose shear stress can be accurately represented in this way are often called power-law fluids . For more information on such fluids the reader is referred to more advanced texts and monographs on fluid dynamics. 2.3.2 Thermal conductivity Thermal conductivity is the transport property that mediates diffusion of heat through a substance in a manner analogous to that already discussed in considerable detail with respect to viscosity and momentum. We can associate heat with thermal energy , so thermal conductivity provides an indication of how quickly thermal energy diffuses through a medium. The basic formula representing this process is Fourier’s law of heat conduction which is typically written in the form q = k dT dy , (2.3) with q representing the heat flux (amount of heat per unit area per unit time), k is the thermal conductivity, and dT/dy is the component of the temperature gradient in the y direction—thus chosen to coincide with the velocity gradient component appearing in Newton’s law of viscosity. It is evident that this formula is quite analogous to Newton’s law of viscosity except for the minus sign; this sign convention is not necessary, but is widely used—sometimes also in the context of Newton’s law of viscosity. We remark that the behavior of k with respect to changes in temperature is very similar, at least qualitatively, to that of μ in the case of fluids, especially for gases; indeed, the underlying physics is the same for both properties. On the other hand, we do not consider viscosity of solids (until they become molten and then are no longer solid), but thermal conductivity in solids is an important property with rather different physical origins. We will not pursue this further in these lectures.
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