Dimensionless groups Deborah Number

The dimensionless flow exponent m and the rheological time constant are additional influencing variables that turn the one-dimensional problem for Newtonian fluids into a three-dimensional problem. The rheological time constant , when multiplied by the revolution speed n, forms an independent dimensionless group (Deborah number). 

Although a mechanism for stress relaxation was described in Section 1.3.2, the Deborah number is purely based on experimental measurements, i.e. an observation of a bulk material behaviour. The Peclet number, however, is determined by the diffusivity of the microstructural elements, and is the dimensionless group given by the timescale for diffusive motion relative to that for convective or flow. The diffusion coefficient, D, is given by the Stokes-Einstein equation ... 

While the Deborah number is often used to compare the time for deformation with the time of observation in experiments, it also inspires us to identify and formulate other dimensionless groups that compare the various characteristic times and forces relevant in colloidal phenomena. We discuss some of the important ones. 

Other dimensionless groups similar to the Deborah number are sometimes used for special cases. For example, in a steady shearing flow of a polymeric fluid at a shear rate y, the Weissenberg number is defined as Wi = yr. This group takes its name from the discoverer of some unusual effects produced by normal stress differences that exist in polymeric fluids when Wi 1, as discussed in Section 1.4.3. Use of the term Weissenberg number is usually restricted to steady flows, especially shear flows. For suspensions, the Peclet number is defined as the shear rate times a characteristic diffusion time to [see Eq. (6-12) and Section 6.2.2]. 

In dealing with viscoelastic fluids, especially under turbulent flow conditions, it is necessary to introduce a dimensionless number to take account of the fluid elasticity [29-33], Either the Deborah or the Weissenberg number, both of which have been used in fluid mechanical studies, satisfies this requirement. These dimensionless groups are defined as follows ... 

It is useful to describe non-Newtonian flows in terms of the Deborah number, a dimensionless group describing the ratio of the viscoelastic relaxation time, X, and a characteristic timescale for the flow, tpow. 

Lii is a characteristic length in the direction of flow. The Deborah number is equivalent to a Weissenberg number with the characteristic length in the direction of flow. Our dimensionless groups thus reduce to... 

Rheologists resort to dimensionless groups such as the Weissenburg and Deborah numbers and the stress ratio Sjt to characterize the elasticity of the flow. High IVs corresponds to high fluid elasticity. 

For viscoelastic fiuids with a relaxation time k, several dimensionless groups can be defined, but these can be seen as being equivalent [21]. For example, the Deborah number (De) is defined as... 

Representative data are presented in Figure 3.7 for a PPS melt. We note that Zy and Ni overshoot their equilibrium values and that the maximum in Ni usually occurs later than that in Xyx at the same value of )>o. Because many processes take place in short time intervals, it may be that the transient properties are more important than the steady shear ones. More wiU be discussed about the importance of the transient behavior in Section 3.2, when we define a dimensionless group called the Deborah number. 

A few additional comments about when and under what conditions one must use a nonlinear viscoelastic constitutive equation are discussed here. At this time it seems that whenever the flow is unsteady in either a Lagrangian DvIDt 0) or a Eulerian (9v/9r 0) sense, then viscoelastic effects become important. In the former case one finds flows of this nature whenever inhomogeneous shear-free flows arise (e.g., flow through a contraction) and in the latter case in the startup of flows. However, even in simple flows, such as in capillaries or slit dies, viscoelastic effects can be important, especially if the residence time of the fluid in the die is less than the longest relaxation time of the fluid. Then factors such as stress overshoot could lead to an apparent viscosity that is higher than the steady-state viscosity. In line with these ideas one defines a dimensionless group referred to as the Deborah number ... 

The ratio of the fluid relaxation time to the timescale for flow If defines a dimensionless group termed the Deborah number, De = /tf. This group has been used in the literature to characterize deviations from Newtonian flow behavior in polymers [6]. Specifically, in flows such as simple steady shear flow where a single flow time tf = y can be defined, it has been observed that for De 1, a Newtonian fluid behavior is observed, whereas for De 1, a non-Newtonian fluid response is observed. However, in flows where multiple timescales can be identified, for example, shear flow between eccentric cylinders, the Deborah number is clearly not unique. In this case, it is generally more useful to discuss the effect of flow on polymer liquids in terms of the relative rates of deformation of material lines and material relaxation. In a steady flow, this effect can be captured by a second dimensionless group termed the Weissenberg number, Wi = k A, where k is a characteristic deformation rate and A is a characteristic fluid relaxation time. For polymer liquids, A is typically taken to be the longest relaxation time Ap, and for steady shear flow, k = y, which leads to Wi = y Ap. 

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