Deformation and Stress A fluid is a substance which undergoes continuous deformation when subjected to a shear stress. Figure 1 illustrates this concept. A fluid is bounded by two large parallel plates, of area A, separated by a small distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U.
The fluid continues to deform as long as the force is applied, unlike a solid, which would undergo only a finite deformation.
Fig 1.
The force is directly proportional to the area of the plate; the shear stress is t = F/A. Within the fluid, a linear velocity profile u = Uy/H is established; due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient g˙ = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds.
The flow in Fig. 1 is a simple shear flow. Viscosity The ratio of shear stress to shear rate is the viscosity, m. m = (1)
The SI units of viscosity are kg/(m × s) or Pa × s (pascal second). The cgs unit for viscosity is the poise; 1 Pa × s equals 10 poise or 1000 centipoise (cP) or 0.672 lbm/(ft × s). The terms absolute viscosity and shear viscosity are synonymous with the viscosity as used in Eq. (1).
Kinematic viscosity n;m/r is the ratio of viscosity to density. The SI units of kinematic viscosity are m2/s. The cgs stoke is 1 cm2/s. Rheology In general,
fluid flow patterns are more complex than the one shown in Fig. 1, as is the relationship between fluid deformation and stress. Rheology is the discipline of fluid mechanics which studies this relationship. One goal of rheology is to obtain constitutive equations by which stresses may be computed from deformation rates. For simplicity, fluids may be classified into rheological types in reference to the simple shear flow of Fig. 1.
Fig 2.
Fluids without any solidlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of deformation (strain).
Those which exhibit both viscous and elastic properties are called visco-elastic fluids.
Purely viscous fluids are further classified into time-independent and time-dependent fluids. For time-independent fluids, the shear stress depends only on the instantaneous shear rate. The shear stress for time-dependent fluids depends on the past history of the rate of deformation, as a result of structure or orientation buildup or breakdown during deformation.
A rheogram is a plot of shear stress versus shear rate for a fluid in simple shear flow, such as that in Fig. 6-1. Rheograms for several types of time-independent fluids are shown in Fig. 2. The Newtonian fluid rheogram is a straight line passing through the origin. The slope of the line is the viscosity. For a Newtonian fluid, the viscosity is independent of shear rate, and may depend only on temperature and perhaps pressure. By far, the Newtonian fluid is the largest class of fluid of engineering importance. Gases and low molecular weight liquids are generally Newtonian. Newton’s law of viscosity is a rearrangement of Eq. (6-1) in which the viscosity is a constant:
t = mg˙ = m (6-2)
All fluids for which the viscosity varies with shear rate are non- Newtonian fluids. For non-Newtonian fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distinction from Newtonian behavior.
Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids.
