Ideal Liquids

The ideal liquid is often referred to as a Newtonian liquid, after Isaac Newton, the scientist who first described its behavior (Whorlow 1992, Macosko 1994, Rao 1995). When a shear stress is applied to an ideal liquid, it continues to flow as long as the stress is applied. Once the stress is removed, there is no elastic recovery of the material (i.e., it does not return to its original shape).

The viscosity of a liquid is a measure of its resistance to flow: the higher the viscosity, the greater the resistance (Macosko 1994). The concept of viscosity can be understood by considering a liquid which is contained between two parallel plates (Figure 8.3). The bottom plate is at rest, while the top plate moves in the ^-direction with a constant velocity (v). It is assumed that the liquid between the plates consists of a series of infinitesimally thin layers. The liquid layers in direct contact with the bottom and top plates are assumed to "stick" to them, so that they have velocities of 0 and v, respectively. The intervening liquid layers slide over each other with velocities that range between 0 and v, the actual value being given by dy(dv/dy), where dy is the distance from the bottom plate and dv/dy is the velocity gradient between the plates. The shear stress applied to the fluid is equal to the shear force divided by the area over which it acts (t = FA). The rate of strain is given by the change in displacement of the layers per unit time: dy/dt (or y) = dv/dy. For an ideal liquid, the shear stress is proportional to the rate of strain (Figure 8.4):

FIGURE 8.3 The viscosity of a liquid is related to the friction between the liquid layers as they slide across each other: the greater the friction, the higher the viscosity.

Rate of Strain Strain

FIGURE 8.4 Stress is proportional to rate of strain for an ideal liquid.

Rate of Strain Strain

FIGURE 8.4 Stress is proportional to rate of strain for an ideal liquid.

where the constant of proportionality (n) is called the viscosity. The viscosity arises from the friction between the liquid layers as they slide past one another (Macosko 1994). The lower the viscosity of a liquid, the less resistance between the liquid layers, and therefore the smaller the force required to cause the top plate to move with a given velocity, or the faster the top plate moves when a given force is applied. The ideal viscous fluid differs from the ideal elastic solid because the shear stress is proportional to the rate of strain (Figure 8.3), rather than the strain (Figure 8.1).

The units of shear stress (t) are N m-2 (or Pa), and those of shear rate (y) are s-1; thus the viscosity (n) has units of N s m-2 (or Pa s) in the SI system. Viscosity can also be expressed in the older cgs units of poise, where 1 Pa s = 10 P. Thus the viscosity of water can be quoted as 1 mPa s, 0.001 Pa s, 0.01 P, or 1 cP, depending on the units used.

Ideally, a Newtonian liquid should be incompressible (its volume does not change when a force is applied to it), isotropic (its properties are the same in all directions), and structureless (it is homogeneous). Although many liquid foods do not strictly meet these criteria, their rheological behavior can still be described excellently by Equation 8.2 (e.g., milk). Nevertheless, there are many others that exhibit nonideal liquid behavior and so their properties cannot be described by Equation 8.2.

The type of flow depicted in Figure 8.3 occurs at low shear rates and is known as laminar flow, because the liquid travels in a well-defined laminar pattern. At higher shear rates, eddies form in the liquid and the flow pattern is much more complex. This type of flow is referred to as turbulent, and it is much more difficult to mathematically relate the shear stress to the rate of strain under these conditions. For this reason, instruments that measure the viscosity of liquids are designed to avoid turbulent flow.

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