Rheological model parallel

Obtaining equivalent expressions for AP for non-Newtonian fluids is difficult and for most rheological models it is not possible to obtain analytical expressions. Good approximate solutions, however, can be obtained by replacing the narrow annulus geometry by a parallel plate model (108, 109), that is, annular flow is represented by slot flow. The... 

Figure 2 shows the basic physical idea of the microstructure of the continuum rheologicS model we proposed earlier (2). The layers can be idealized as separated by porous slabs, which are connected by elastic springs. Liquid crystals may flow parallel to the planes in the usual Newtonian manner, as if the slabs were not there. In the direction normal to the layers, liquid crystals encounter resistance through the porous medium, proportional to the normal pressure gradient, which is known as permeation. The permeation is characterized by a body force which in turn causes elastic compression and splay of the layers. Applied strain from the compression causes dislocations to move into the sample from the side in order to relax the net force on the layers. When the compression stops and the applied stress is relaxed the permeation characteristic has no influence on stress strain field. 

From eqs. (5) to (10) it is possible to obtain the pressure drop for the laminar regime of the suspensions that follows the Robertson and Stiff of this model flowing on a given annular geometry. The development parallels that one made by Hanks J for the Herschel and Bulkley rheological model. 

The values of v in the above equation are the equilibrium values after an infinite pressing time. There are kinetics involved here too in that the deformation of the cake due to a mechanical stress is not instantaneous but dampened by the necessary permeation of the liquid out of the cake. Baluais et al. considered a rheological model for this in analogy with the action of a shock absorber in parallel with one spring and in series with another. 

Rheological Models One-dimensional constitutive models for viscoelasticity based on spring, dashpot, and spring-pot elements. The elements maybe cormected in series or in parallel. In models where the elements are cormected in series the strain is additive while fire stress is equal in each element. In parallel cormections, the stress is additive while the strain is equal in each element. 

Figure 2.11 Electrical equivalent circuits for arranging rheological model elements, (a) Capacities in series (b) Capacities in parallel.

Another way to introduce fractional derivatives is through rheological models of fractional order. In particular, the fractional Maxwell element corresponds to a spring in series with a fractional damper. The one-dimensional linear stress, <7, versus strain, e, relation of a spring in parallel with the fractional Maxwell element can expressed in terms of fractional derivatives [171], e.g.,... 

For the shear of the lubricant the rheological model by Evans and Johnson [26] is adopted to take account of the the non-Newtonian behaviour of the lubricant that takes place throughout the cycle. Hence the shear stress between the two parallel surfaces is given by ... 

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

Example 6.14 Squeezing Flow between Two Parallel Disks This flow characterizes compression molding it is used in certain hydrodynamic lubricating systems and in rheological testing of asphalt, rubber, and other very viscous liquids.14 We solve the flow problem for a Power Law model fluid as suggested by Scott (48) and presented by Leider and Bird (49). We assume a quasi-steady-state slow flow15 and invoke the lubrication approximation. We use a cylindrical coordinate system placed at the center and midway between the plates as shown in Fig. E6.14a. 

It is convenient to describe these properties in terms of the following mechanical models [396] the Hooke body (an elastic spring), the Saint-Venant body modeling dry friction (a bar on a solid surface), and the Newton body (a piston in a vessel filled with a viscous fluid). By using various combinations of these elementary models (connected in parallel and/or in series), one can describe situations which are rather complex from the rheological viewpoint. 

The key point in the rheological classification of substances is the question as to whether the substance has a preferred shape or a natural state or not [19]. If the answer is yes, then this substance is said to be solid-shaped otherwise it is referred to as fluid-shaped [508]. The simplest model of a viscoelastic solid-shaped substance is the Kelvin body [396] or the Voigt body [508], which consists of a Hooke and a Newton body connected in parallel. This model describes deformations with time-lag and elastic aftereffects. A classical model of viscoplastic fluid-shaped substance is the Maxwell body [396], which consists of a Hooke and a Newton body connected in series and describes stress relaxation. 

The DNF model incorporates the experimentally observed characteristics by using a micromechanism-inspired approach in which the material behavior is decomposed into a viscoplastic response, corresponding to irreversible molecular chain sliding due to the lack of chemical crosslinks in the material, and atime-dependent viscoelastic response. The viscoelastic response is further decomposed into the response of two molecular networks acting in parallel the first network (A) captures the equilibrium response and the second network (B) the time-dependent deviation from the viscoelastic equilibrium state. A onedimensional rheological representation of the model framework and a schematic illustrating the kinematics of deformation are shown in Fig. 11.6. 

The cone-and-plate and parallel-plate rheometers are rotational devices used to characterize the viscosity of molten polymers. The type of information obtained from these two types of rheometers is very similar. Both types of rheometers can be used to evaluate the shear rate-viscosity behavior at relatively low vales of shear rate therefore, allowing the experimental determination of the first region of the curve shown in Figure 22.6 and thus the determination of the zero-shear-rate viscosity. The rheological behavior observed in this region of the shear rate-viscosity curve cannot be described by the power-law model. On the other hand, besides describing the polymer viscosity at low shear rates, the cone-and-plate and parallel-plate rheometers are also useful as dynamic rheometers and they can yield more information about the stmcture/flow behavior of liquid polymeric materials, especially molten polymers. 

A polymer solution (of density 1000 kg/m ) is flowing parallel to a plate (300 mm x 300 mm) the free stream velocity is 2 m/s. In the narrow shear rate range, the rheology of the polymer solution can be adequately approximated by both the power-law (m = 0.3 Pa-s" and n =0.5) and the Bingham plastic model (tq = 2.28 Pa s and /xg = 7.22 mPa-s). Using each of these models, estimate and compare the values of the shear stress and the boundary layer thickness 150 mm away from the leading edge, and the total frictional force on each side of the plate. 

In spite of the complexity of the situation, there have been attempts to create theories of rheological behavior of PLCs. Wissbrun [79] represents a PLC material as a space-filling system of domains. At rest, the minimum energy arrangement is achieved when the directors in the planes of contact are parallel. Under shear, the domains slide over each other. The model predicts shear sensitiveness, a phenomenon observed experimentally the curves of viscosity as a function of the shear rate are horizontal for low shear rates and then go down. In fact, for instance the results for PCarb - - PET/ 0.6PHB blends—if we employ such coordinates rather than those in Fig. 41.12—exhibit shear sensitiveness [76]. 

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