Retarded motion expansion

As our second major topic, we present the simplest equations from each of the three important classes of constitutive equations, namely the differential equations from the retarded-motion expansion, the Maxwell-type differential equations, and the integral equations. Third and finally, we summarize the more accurate constitutive equations that we feel are the most promising for simply and realistically describing viscoelastic fluids and for modeling viscoelastic flows. More complete treatments of nonlinear constitutive equations are available elsewhere (Tanner, 1985 Bird et al., 1987 Larson, 1988 Joseph, 1990). Throughout this chapter, our examples are drawn from the literature on polymeric... 

The equation of the second-order fluid is so named because it contains all terms up to second order in the velocity gradient in a perturbation expansion about the rest state. The Newtonian term 2jjoD is the first-order term, and pi is the zeroth-order term. There are also equations of the third-order fluid, and so on (Bird et al., 1987). These form a series of equations in the retarded motion expansion so called because it assumes that the flow is a small perturbation from the state of test. The equations that are higher in order than second are of less practical importance because of their complexity and the restricted conditions under which they are accurate. 

Most flows of polymeric fluids are not slow enough for the second-order-fluid equation or any of the equations of the retarded-motion expansion to apply to them. Of course eq 4.3.1 can be made accurate for any steady shearing flow merely by replacing the constants rjo, 1,0, and 2,0 by the shear rate dependent coefficients l(y). and V 2(i )- Although the resulting equation, called... 

The retarded motion expansion (ref. 5, chapter 6) written through terms of third order, is ... 

This model has been found to be useful in studying the slow motion of viscoelastic fluids around particles, droplets and bubbles, and in predicting the directions of secondary flows in rotating systems. Furthermore it is very helpful in providing a framework for the presentation of kinetic theory results obtained by perturbation theories. The retarded motion expansion is not, however, useful for most industrial flow problems, in which large velocity gradients or rapid time responses are generally encountered. 

If the retarded motion expansion is written for steady shear flows, there is a considerable simplification inasmuch as Y( ) = 0 for n> 3. As a result the series collapses to the form... 

Here y[o] is shorthand for YiojitXl- This can be regarded as an expansion about the Lodge rubberlike liquid, which in turn includes the general linear viscoelastic model. By expanding the strain tensors in equation (49) about time t, the retarded-motion expansion of equation (38) is obtained, with the... 

The reason heat is absorbed on expansion of a surface is that the molecules must be transferred from the interior against the inward attractive force to form the new surface. In this process, the motion of the molecules is retarded by this inward attraction so that the temperature of the surface layers is lower than that of the interior unless heat is supplied from outside. 

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