Continuous shear strain, motion

Fluids flowing in a capillary rheometer see a shearing mode of deformation. Since fluids are always in motion and cannot hold a shape, a static measure of the strain is not useful a continuous deformation rate must be determined. Dilferentiating the shear strain with respect to time gives the shear strain rate or shear rate ... 

When a fluid system is studied by the application of a stress, motion is produced until the stress is removed. Consider two surfaces separated by a small gap containing a liquid, as illustrated in Figure 1.4. A constant shear stress must be maintained on the upper surface for it to move at a constant velocity, u. If we can assume that there is no slip between the surface and the liquid, there is a continuous change in velocity across the small gap to zero at the lower surface. Now in each second the displacement produced is x and the strain is... 

A fluid packet, like a solid, can experience motion in the form of translation and rotation, and strain in the form of dilatation and shear. Unlike a solid, which achieves a certain finite strain for a given stress, a fluid continues to deform. Therefore we will work in terms of a strain rate rather than a strain. We will soon derive the relationships between how forces act to move and strain a fluid. First, however, we must establish some definitions and kinematic relationships. 

The constitutive equation for a dry powder is a governing equation for the stress tensor, t, in terms of the time derivative of the displacement in the material, e (= v == dK/dt). This displacement often changes the density of the material, as can be followed by the continuity equation. The constitutive equation is different for each packing density of the dry ceramic powder. As a result this complex relation between the stress tensor and density complicates substantially the equation of motion. In addition, little is known in detail about the nature of the constitutive equation for the three-dimensional case for dry powders. The normal stress-strain relationship and the shear stress-strain relationship are often experimentally measured for dry ceramic powders because there are no known equations for their prediction. All this does not mean that the area is without fundamentals. In this chapter, we will not use the approach which solves the equation of motion but we will use the friction between particles to determine the force acting on a mass of dry powder. With this analysis, we can determine the force required to keep the powder in motion. 

There are other classes of fluids, such as Herschel-Bulkley fluids and Bingham plastics, that follow different stress-strain relationships, which are sometimes useful in different drilling and cementing applications. For a discussion on three-dimensional effects and a rigorous analysis of the stress tensor, the reader should refer to Computational Rheology. For now, we will continue our discussion of mudcake shear stress, but turn our attention to power law fluids. The governing partial differential equations of motion, even for simple relationships of the form given in Equation 17-57, are nonlinear and therefore rarely amenable to simple mathematical solution. For example, the axial velocity v (r) in our cylindrical radial flow satisfies... 

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