Deborah numbers

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... 

A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For veiy large Deborah numbers, the behavior closely resembles that of an elastic solid. 

To do this, a Deborah Number Adeb. has been deflned as... 

The Deborah number Db has been defined by Metzner, White and Denn 201 as ... 

The ratio of a material s elastic time constant, te, to the time of an imposed process, tp, is the dimensionless Deborah number, De (= te/tp). When the imposed process time, tp, is rapid, the De number will be high, elastic stresses will dominate, the first (viscous) term in Eq. (2) will be negligible and the material will behave purely elastically. For rapid processing operations therefore, Eqs. (1) and (2) can be rewritten as ... 

Returning to Eq. (1) and considering the response of a biomaterials to an imposed stress in a relatively slow process (long process time, tp and low Deborah number, De), the elastic component, the second term, of Eq. (1) will be small and the wall material is likely to behave in a viscous manner. Thus Eq. (1) reduces to ... 

Vrentas, JS Duda, JL, Diffusion in Polymer-Solvent Systems, in. Construction of Deborah Number Diagrams, Journal of Polymer Science Polymer Physics Edition 15,441,1977. Wakao, N Smith, JM, Diffusion in Catalyst Pellets, Chemical Engineering Science 17, 825, 1962. 

Figure 4 A Deborah number diagram for the polystyrene-ethylbenzene system showing the diffusion behavior as a function of weight fraction and temperature. (From Ref. 33.)...

JS Vrentas, JL Duda. Diffusion in polymer-solvent systems. III. Construction of Deborah number diagrams. J Polym Sci, Polym Phys Ed 15 441-453, 1977. 

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

JS Vrentas, CM Jarzebski, JL Duda. Deborah number for diffusion in polymer-solvent systems. AIChE J 21 894-902, 1975. 

JC Wu, NA Peppas. Modeling of penetrant diffusion in glassy polymers with an integral sorption Deborah number. J Polym Sci Polym Phys Ed 31 1503-1518,... 

Then calculate the Deborah number from Eq. (6-99), using k = 0.088 and k2 = 0.0431 from Table 6-2 ... 

Diffusion-controlled drug delivery, 9 77 Diffusion Deborah number, 23 101 Diffusion flames... 

It is helpful here to introduce the Deborah number De defined by... 

The fluid s relaxation time A is the characteristic time of the fluid and, for oscillatory shearing, cu 1 can be taken as a measure of the characteristic time of the flow process, so De = A to. Thus, viscous behaviour occurs when the Deborah number is low, reflecting the fact that the fluid is able to relax. When the Deborah number is high, elastic behaviour is observed because the fluid is unable to relax sufficiently quickly. 

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 ... 

The similarity to the Peclet number is obvious but we should also bear in mind the relationship to the Deborah number. This becomes clear when we consider the fact that the mechanism of stress relaxation is due to the... 

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