Viscoelasticity Deborah number

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. 

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 ... 

We can get a first approximation of the physical nature of a material from its response time. For a Maxwell element, the relaxation time is the time required for the stress in a stress-strain experiment to decay to 1/e or 0.37 of its initial value. A material with a low relaxation time flows easily so it shows relatively rapid stress decay. Thus, whether a viscoelastic material behaves as a solid or fluid is indicated by its response time and the experimental timescale or observation time. This observation was first made by Marcus Reiner who defined the ratio of the material response time to the experimental timescale as the Deborah Number, Dn-Presumably the name was derived by Reiner from the Biblical quote in Judges 5, Song of Deborah, where it says The mountains flowed before the Lord. ... 

The difference between solids and liquids is found in the magnitude of D. Liquids, which relax in small fractions of a second, have small D. Solids have a large D. A sufficient lime span can reduce the Deborah number of a solid to unity, and impact loading can increase D of a liquid. Viscoelastic materials are best characterized under conditions in which D lies within a few decades of unity. 

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.

Deborah number De kca fluid relaxation time flow characteristic time Viscoelastic flow... 

Whether we have a liquid or a solid, is governed by the Deborah number (Figure C4-6). This is a ratio of the relaxation time to the time used to shear the fluid over one radian. If the Deborah number is very small, we are dealing with a Newtonian liquid. If it has a value around unity, we have a viscoelastic fluid if the value is very large, a creeping solid. 

In their ki a measurements Yagi and Yoshida [596] also tested two non-Newtonian fluids CMC and PANa = sodium salt of polyacrylic acid. They found that equation (4,31) had to be expanded by an extra term, which contains the Weiflenberg (or Deborah) number Wi = De = An, to correlate kiu values for viscoelastic fluids see equations (1.53) and (1.45) ... 

Basically, Eq. 6.17 results from combining Eqs. 6.12, 6.13, and 6.15 with Cei = 1. According to Marshall and Mentzner (1964), the onset of viscoelastic behavior occurs at a Deborah number around 0.1. From the work of Durst et al. (1982), the Deborah number is 0.5. The smaller the Deborah number, the more the material appears like a fluid. 

The flow velocity is in the order of 10 m/s. The radius of an oil thread is about 10 m. The relaxation time of polymer solution used in the oil displacement process is about 10 to 10 s. Under these conditions, the range of Deborah number, Noe, is between 0.1 and 10. Figure 6.26 shows the normal stress of the viscoelastic fluid with different Deborah numbers. The stress acts on the undulated oil/water interface. When the representation in Figure 6.26 was constructed, the fluid velocity of 3.47 x 10 m/s and the relaxation time of 0.247 s were used. In the figure, negative stress represents that the stress direction is opposite to the external normal line of the acting surface. We can... 

If De -4 1 we call the material a liquid and at De 1 the material appears solid. It is only if De is of order unity that we observe viscoelastic behavior. Variation of the frequency (to) in dynamic tests thus amounts to varying the Deborah number. 

If a given deformation is applied to a viscoelastic material, the stress slowly relaxes the characteristic time for this is called the relaxation time. The Deborah number (De) is defined as the ratio of this relaxation time over the observation time. For a solid De is very large, for a liquid very small, and for a viscoelastic material of order unity. It thus depends on the time scale of observation whether we call a material solid or liquid. Several foods appear to be solid at casual observation, but show flow during longer observation. 

A. B. Metzner, J. L. White, and M. M. Denn, Constitutive Equation for Viscoelastic Fluids for Short Deformation Periods and for Rapidly Changing Flows Significance of the Deborah Number, AIChEJ. (12) 863,1966. 

It should be noted here that in polymer rheology, for viscoelastic fluids the commonly used dimensionless parameter to characterize the ratio of elastic force to viscous force is the Deborah number denoted by the symbol De. This parameter is essentially just the Peclet number. In terms of characteristic times, it is equal to the ratio of the largest time constant of the molecular motions or other appropriate relaxation time of the fluid compared to the characteristic flow time. 

A physical insight into the viscoelastic character of a material can be obtained by examining the material response time. This can be illustrated by defining a characteristic time for the material — for example, the relaxation time for a Maxwell element, which is the time required for the stress in a stress relaxation experiment to decay to e (0.368) of its initial value. Materials that have low relaxation times flow easily and as such show relatively rapid stress decay. This, of course, is indicative of liquidlike behavior. On the other hand, those materials with long relaxation times can sustain relatively higher stress values. This indicates solidlike behavior. Thus, whether a viscoelastic material behaves as an elastic solid or a viscous liquid depends on the material response time and its relation to the time scale of the experiment or observation. This was first proposed by Marcus Reiner, who defined the ratio of the material response time to the experimental time scale as the Deborah number, D . That is. 

A new parameter, the Deborah number, De, is used to define viscoelastic behavior. It is derived by dividing the relaxation (or retardation) time by the duration of the process, as follows ... 

It is known that a viscoelastic fluid, e.g., a solution with a trace amount of highly deformable polymers, can lead to elastic flow instability at Reynolds number well below the transition number (Re 2,000) for turbulence flow. Such chaotic flow behavior has been referred to as elastic turbulence by Tordella [2]. Indeed, the proper characterization of viscoelastic flows requires an additional nondimensional parameter, namely, the Deborah number, De, which is the ratio of elastic to viscous forces. Viscoelastic fluids, which are non-Newtonian fluids, have a complex internal microstructure which can lead to counterintuitive flow and stress responses. The properties of these complex fluids can be varied through the length scales and timescales of the associated flows [3]. Typically the elastic stress, by shear and/or elongational strains, experienced by these fluids will not immediately become zero with the cessation of fluid motion and driving forces, but will decay with a characteristic time due to its elasticity. 

For a given geometry, the viscoelastic effects of a fluid flow can be characterized by the Deborah number, De. The Deborah number is a dimensionless parameter which typifies the relative importance of the elastic stresses of the fluid with the... 

They investigated the expansion of viscoelastic and Newtonian fluids in an identical channel. They found that at a high Reynolds number, Newtonian fluids generated large circulation zones or comer vortices downstream of the contraction, which are a feature of the expansion flow behavior for Newtonian flow [15]. However, for a viscoelastic fluid with the same Reynolds number and a low Deborah number, the exit vortex behavior was completely suppressed. 

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