Viscometric flows

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

Stress relaxation time, obtained from rheograms based on viscometric flows, is used to define a dimensionless parameter called the Deborah number , which quantifies the elastic character of a fluid... 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

MODELLING OF STEADY-STATE VISCOMETRIC FLOW 127 in Cartesian x, y) coordinate system... 

MODELLING OF STEADY-STATE VISCOMETRIC FLOW -WORKING EQUATIONS OF THE CONTINUOUS PENALTY SCHEME IN CARTESIAN COORDINATE SYSTEMS... 

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

The stress field corresponding to this regime is shown in Figure 6.18. As this figure shows the measuring surface of the cone is affected by these secondary stresses and hence not all of the measured torque is spent on generation of the primary (i.e, viscometric) flow in the circumferential direction. 

I liis simulation provides the quantitative measures required for evaluation of the extent of deviation from a perfect viscometric flow. Specifically, the finite element model results can be used to calculate the torque corresponding to a given set of experimentally determined material parameters as... 

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. 

Viscometric flows used for measurements include well known flows, such as flow in a narrow gap concentric cylinder device and between a small angle cone and a flat plate. In both of these cases the flows established in these devices approximate almost exactly simple shearing flow. There are other viscometric flows in which the shear rate is not constant throughout, these include the wide gap concentric cylinder flow and flow in a circular pipe, discussed above. 

Viscometric flow theories describe how to extract material properties from macroscopic measurements, which are integrated quantities such as the torque or volume flow rate. For example, in pipe flow, the standard measurements are the volume flow rate and the pressure drop. The fundamental difference with spatially resolved measurements is that the local characteristics of the flows are exploited. Here we focus on one such example, steady, pressure driven flow through a tube of circular cross section. The standard assumptions are made, namely, that the flow is uni-directional and axisymmetric, with the axial component of velocity depending on the radius only. The conservation of mass is satisfied exactly and the z component of the conservation of linear momentum reduces to... 

The second important feature of this technique is that it is independent of the constitutive relationship of the material. This is a direct reflection of its rigorous foundation in viscometric flow theory. 

B. D. Coleman, H. Markovitz, W. Noll 1966, Viscometric Flows of Non-Newtonian Fluids, Springer, Berlin. 

Figure 3.4 Vertical cross-sections of common measuring geometries used to provide simple viscometric flow...

Coleman, B.D., Markovitz.H., NoIl,W. Viscometric flows of non-newtonian fluids. Berlin-Heidelberg-New York Springer 1966. 

Pipkin,A.C. Small displacements superposed on viscometric flow. Trans. Soc. Rheology 12, 397-408 (1968). 

Markovitz,H. Small deformations superimposed-cm steady viscometric flows. In Onogi,S. (Ed.) Proc. 5th Intemat. Cong Rheology, Vol. I, pp. 499-510. Maryland University Park Press 1970. 

It is important to note that the rheological material functions obtained experimentally, using rheometers, are evaluated in simple flows, which are often called viscometric or rheometric. A viscometric flow is defined as one in which only one component of the velocity changes in only one spatial direction, vx (y). Yet these material functions are used to describe the more complex flow situations created by polymer processing equipment. We assume, therefore, that while evaluated in simple flows, the same rheological properties also apply to complex ones. 

Three kinds of viscometric flows are used by rheologists to obtain rheological polymer melt functions and to study the rheological phenomena that are characteristic of these materials steady simple shear flows, dynamic (sinusoidally varying) simple shear flows, and extensional, elongational, or shear-free flows. 

This section describes two common experimental methods for evaluating i], Fj, and IG as functions of shear rate. The experiments involved are the steady capillary and the cone-and-plate viscometric flows. As noted in the previous section, in the former, only the steady shear viscosity function can be determined for shear rates greater than unity, while in the latter, all three viscometric functions can be determined, but only at very low shear rates. Capillary shear viscosity measurements are much better developed and understood, and certainly much more widely used for the analysis of polymer processing flows, than normal stress difference measurements. It must be emphasized that the results obtained by both viscometric experiments are independent of any constitutive equation. In fact, one reason to conduct viscometric experiments is to test the validity of any given constitutive equation, and clearly the same constitutive equation parameters have to fit the experimental results obtained with all viscometric flows. 

In summary, and in terms of the viscometric flow notation, we conclude the following about the experimental capabilities of the cone-and-plate viscometric flow ... 

We wish to derive the steady state response of a linear viscoelastic body to an externally applied sinusoidal shear strain (dynamic testing) using the constitutive Eq. 3.3-8, which for this viscometric flow reduces to... 

We therefore observe that unlike in the Power Law model solution with a single shear stress component, xn, in the case of a CEF model, we obtain, in addition, two nonvanishing normal stress components. Adopting the sign convention for viscometric flow, where the direction of flow z is denoted as 1, the direction into which the velocity changes r, is denoted as 2, and the neutral direction 8 is denoted as direction 3, we get the expressions for the shear stress in terms of the shear rate, the primary, and secondary normal stress differences (see Eqs. 3.1-10 and 3.1-11) ... 

Stresses Generated by CEF Fluids in Various Viscometric Flows What stresses are necessary to maintain a CEF fluid flowing in the following flows (a) parallel-plate drag flow (b) Couette flow with the inner cylinder rotating and (c) parallel-plate pressure flow. Assume the same type of velocity fields that would be expected... 

Simplify the actual flow by assuming that it is a series of well-identified viscometric flows. 

By applying one or more mass balances, relate the volumetric flow rates in each of the viscometric flows. 

The same statement can be made about inelastic non-Newtonian fluids, such as the Power Law fluid, from a mathematical solution point of view. In reality, most non-Newtonian fluids are viscoelastic and exhibit normal stresses. For fluids such as those (i.e., fluids described by constitutive equations that predict normal stresses for viscometric flows), theoretical analyses have shown that secondary flows are created inside channels of nonuniform cross section (78,79). Specifically it can be shown that a zero second normal stress difference is a necessary (but not sufficient) condition to ensure the absence of secondary flow (79). Of course, the analyses of flows in noncircular channels in terms of constitutive equations—which, strictly speaking, hold only for viscometric flows—are expected to yield qualitative results only. Experimentally low Reynolds number flows in noncircular channels have not been investigated extensively. In particular, only a few studies have been conducted with fluids exhibiting normal stresses (80,81). Secondary flows, such as vortices in rectangular channels, have been observed using dyes in dilute aqueous solutions of polyacrylamide. Interestingly, these secondary flow vortices (if they exist) seem to have very little effect on the flow rate. 

Example 15.1 The Significance of Normal Stresses We consider the calender geometry of Fig. 6.22 (shown here) and make the same simplifying assumptions as in Section 6.4, but instead of a Newtonian or Power Law model fluid, we assume a CEF model that exhibits normal stresses in viscometric flows. By accepting the lubrication approximation, we assume that locally we have a fully developed viscometric flow because there is only one velocity component vx, which is a function of only one spatial variable y. 

The methods described below outline three dynamic adhesion/aggregation assays used to assess the in vitro and/or ex vivo efficacy of platelet antagonists (1) a viscometric-flow cytometric assay to measure shear-induced platelet-platelet aggregation in the bulk phase, (2) a perfusion chamber coupled with a computerized videomicroscopy system to visualize in real time and quantify (a) the adhesion and subsequent aggregation of platelets flowing over an immobilized substrate (e.g. extracellular matrix protein) and (b) free-flowing monocytic cell adhesion to immobilized platelets. 

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