Viscoelasticity, linear Deborah number

In order to observe linear viscoelasticity, structural relaxation by diffusion must occur on a timescale comparable to our measurement time. The ratio of these times is the Deborah number. When this is of the order of unity our experiment will follow the relaxation processes in the material and the material will appear to be viscoelastic ... 

Figure 2.34 Schematic of Newtonian, elastic, linear, and non-linear viscoelastic regimes as a function of deformation and Deborah number during deformation of polymeric materials.

Nonlinear rheology vastly extends all the phenomena (elastic, viscous, and linear time dependent) discussed in Chapters 1-3. Elastic, viscous, and linear viscoelastic behaviors are but coastal zones on a continent of nonlinear rheology see Figure 4.1.2. The abscissa on Figure 4.1.2 is the Deborah number, which is generally defined as the ratio of the material s characteristic relaxation time k to the characteristic flow time t. 

The origin of Deborah s number is indicated in the frontispiece to this text. In Figure 4.1.2 we take the characteristic flow time to be the inverse of the typical deformation rate while in oscillatory flows we use the amplitude of the oscillatory strain times its frequency (yaO)). Die elastic, Newtonian, and linear viscoelastic limits illustrated in Figure 4.1.2 have already been discussed in Ch ters 1, 2, and 3, respectively. Second-order fluids, to be covered shortly, reside in a fringe of the regime of nonlinear viscoelasticity that lies just across the border from the Newtonian domain. 

The Weissenberg number is introduced in the following section in connection with flows with constant stretch history, i.e., flows in which the deformation rate and aU the stresses are constant with time. These are flows in which De is zero. And for deformations in which linear viscoelastic behavior is exhibited, Wi is zero. However, there are also flows of practical importance in which both Wi and De are nonzero and are sometimes even directly related to each other. This causes confusion, as authors often use the two groups interchangeably. This situation arises, for example, in the flow from a reservoir into a much smaller channel, either a slit or capillary. A Weissenberg number can readily be defined for this flow as the product of the characteristic time of the fluid and the shear rate at the wall of the flow channel. However, entrance flow is clearly not a flow with constant stretch history, and the Deborah number is thus non-zero as well and depends on the rate of convergence of the flow. 

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