Deformation of fluid elements

Phenomena bulk fluid movement and blending determine environment concentrations coarse scale turbulent exchange between the fresh feed and its surroundings inertial-convective disintegration of large eddies viscous-convective deformation of fluid elements followed by molecular diffusion... 

Figure 1.4 Model for deformation and break-up of fluid element due to pressing and shearing...

It is also interesting to briefly consider online measurements of variables different from temperature [5], Since pressure is defined as the normal force per unit area exerted by a fluid on a surface, the relevant measurements are usually based on the effects deriving from deformation of a proper device. The most common pressure sensors are piezoresistive sensors or strain gages, which exploit the change in electric resistance of a stressed material, and the capacitive sensors, which exploit the deformation of an element of a capacitor. Both these sensors can guarantee an accuracy better than 0.1 percent of the full scale, even if strain gages are temperature sensitive. 

Figure 9.2 Deformation and distribution of fluid elements in laminar flow for different types of flow...

The flow field in Eq. (Al-7) is really just a solid-body rotation which rotates, but does not deform, the fluid element. As a result, the rate-of-strain tensor D is the zero tensor, and the Finger strain tensor is the unit tensor. 

It is clear from Figure 11.12 that the slicing and replacing mechanism occurs in isolation and is not accompanied by either shearing or extension, both of which would deform the fluid elements. However, it is possible to create this distributive mixing action during laminar flow over appropriately designed... 

Mavridis, H., Hrymak, A.N., and Vlachopoulos, J. (1986) Deformation and orientation of fluid elements behind an advancing flow front./. Rheol,30,555-563. 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

A plausible assumption would be to suppose that the molecular coil starts to deform only if the fluid strain rate (s) is higher than the critical strain rate for the coil-to-stretch transition (ecs). From the strain rate distribution function (Fig. 59), it is possible to calculate the maximum strain (kmax) accumulated by the polymer coil in case of an affine deformation with the fluid element (efl = vsc/vcs v0/vcs). The values obtained at the onset of degradation at v0 35 m - s-1, actually go in a direction opposite to expectation. With the abrupt contraction configuration, kmax decreases from 19 with r0 = 0.0175 cm to 8.7 with r0 = 0.050 cm. Values of kmax are even lower with the conical nozzles (r0 = 0.025 cm), varying from 3.3 with the 14° inlet to a mere 1.6 with the 5° inlet. In any case, the values obtained are lower than the maximum stretch ratio for the 106 PS which is 40. It is then physically impossible for the chains to become fully stretched in this type of flow. 

Fig. 23. Schematic diagram of the 4-roll mill apparatus. Schematic representation of the flow field within the mill illustrating the deformation of a fluid element [35]...

Stress and Strain Rate The stress and strain-rate state of a fluid at a point are represented by tensors T and E. These tensors are composed of nine (six independent) quantities that depend on the velocity field. The strain rate describes how a fluid element deforms (i.e., dilates and shears) as a function of the local velocity field. The stress and strain-rate tensors are usually represented in some coordinate system, although the stress and strain-rate states are invariant to the coordinate-system representation. 

Fig. 2.6 Translation and deformation of a fluid element in the r-z plane. For the right-hand coordinate system (z, r, 9), note that the positive 6 direction is into the page. The displacements in the figure are grossly exaggerated to facilitate annotation. In the limit dr - 0, on which the analysis is based, the two elements approach being colocated.

Consider first the trivial case of a static fluid. Here there can only be normal forces on a fluid element and they must be in equilibrium. If this were not the case, then the fluid would move and deform. Certainly any valid relationship between stress and strain rate must accommodate the behavior of a static fluid. Hence, for a static fluid the strain-rate tensor must be exactly zero e(/- = 0 and the stress tensor must reduce to... 

Fig. 2.24 Translation and deformation of an initially rectangular fluid element in cartesian coordinates.

Fig. 2.4 The deformation of a fluid element in unidirectional shear flow.

At time t + At the rectangular fluid element is translated in the x direction and deformed into a parallelogram. We define the rate of shear as -dS/dt, where <5 is the angle shown in the figure. 

The temperature of an incompressible fluid element in a deforming medium is governed by the equation of thermal energy, Eq. 2.9-14. This excluding the reversible compression term, and in terms of specific heat, is... 

Still, sophisticated, exact, numerical, non-Newtonian and nonisothermal models are essential in order to reach the goal of accurately predicting final product properties from the total thermomechanical and deformation history of each fluid element passing through the extruder. A great deal more research remains to be done in order to accomplish this goal. 

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