Second-order fluid simple shear

In a complex, polymeric liquid, normal stresses as well as the shear stress can be present, and these contributions will influence the shape of the structure factor. The simplest rheological constitutive model that can account for normal stresses is the second-order fluid model [64], where the first and second normal stress differences are quadratic functions of the shear rate. Calculations using this model [92,93,94,90,60], indicate that the appearance of normal stresses can rotate the structure factor towards the direction of flow in the case of simple shear flow and can induce a four-fold symmetry in the case of exten-sional flow. 

Second-Order Fluids in Simple Shearing Flow 515... 

SECOND-ORDER FLUIDS IN SIMPLE SHEARING FLOW... 

Three functions (q, P, and y) are necessary to characterize the flow of a second-order fluid. Let us determine these functions for a memory fluid under simple shearing flow (25, 26), as shown in Figure 13.1. In this case, the coordinates of a given point at time / will be X =Xi,X2 t) = [x2(/)/xi]xi = Xi tan5(/), and x = x. The velocity field is given by... 

For simple shear experiments, somewhat lengthy mathematical arguments indicate that the stress tensor for second-order fluids is given by (2)... 

Example 4.3.1 The Second-Order Fluid in Simple Shear... 

This example makes evident the usefulness of the equation of the second-order fluid. Once the three coefficients of the equation have been specified—and measurements in simple shear alone are enough to specify them—predictions can be made for the first deviations from Newtonian behavior for any other flow. This property has proved useful in analyzing slow but complex nonuniform flows, such as those observed in rod climbing (see Chapter 5) and flow over a pressure hole (see Chapter 6). 

Note that since m(s) and a( i, 2) are functions only of time y, then t]q, y3, and v are constants. A material that can be represented by the constitutive equation given in Eq. (3.76) is called a Coleman-Noll second-order fluid (Coleman and Markovitz 1964 Truesdell and Noll 1965). For steady-state simple shear flow, Eq. (3.76) yields... 

Measuring the surface tensions of solids poses a problem because it is almost impossible to extend a solid-fluid interface isothermally and reversibly. Stretching a solid-fluid interface is not performed against the interfacial tension, but against the interjacial stress r. The difference between r and y depends on the kinetics and history of the extension, so that generally the work w performed is not a sole characteristic of y or, for that matter, a function of state. Only when the extension can be ceirried out reversibly is it possible to relate r emd y. As r is a second order tensor (it has normal and shear components) and y is a scalar, this relation is complicated. For the very simple case that r does not depend on direction (a rather unrealistic situation for solids) and assuming reversibility the relation is ... 

Lee (1988) further connected the viscosities of liquid crystalline polymers (in the unit of f]) with concentration and molecular length as Equation (6.37), where A is a constant less than unity which is associated with the stability of simple shear flow, is a dimensionless parameter associated with the interaction in the fluid, R = (/A(a n) is the second order parameter where a is the molecular long axis and P4 is the fourth rank Legendre polynomial. 

Thus, the fluid is shear thinning for 0 < < 2, with a finite second normal stress difference. - V2/V1 is typically of order 0.1, so f will typically be of order 0.2. There is no simple analytical solution to the equations for transient stress growth in uniform uniaxial extension, and the coupled ordinary differential equations must be solved numerically. 

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