Deborah number definition

Dealy DM (2010) Weissenberg and Deborah numbers— their definition and use. Rheol Bull 79 14-18... 

The Weissenberg number compares the elastic forces to the viscous effects. It is usually used in steady flows. One can have a flow with a small Wi number and a large De number, and vice versa. Sometimes the characteristic time of the flow in the deflnition of the Deborah number has been taken to be the reciprocal of a characteristic shear rate of the flow in these cases, the Deborah number and the Weissenberg number have the same definition. Pipkin s diagram (see Fig. 3.9 in Tanner 2000) classifies shearing flow behavior in terms of De and Wi, and provides a useful guide for the choice of constitutive equations. 

SO that for Marshall and Mentzner s definition a" = 0.5. Marshall and Men tzner (1964), using the contra variant convected Maxwell model to obtain a relaxation time for their polymer solutions, noticed the onset of viscoelastic behaviour at a Deborah number between 0.1 and 1.0 for flow through sintered bronze. Some of their results for various flexible coil polymers are shown in Figure 6.10. 

A further perspective into such classifications may be gained by considering the element of time. In the case of the familiar silly putty which bounces like a rubber ball, but flows when left alone, the crucial factor that decides between its solid or liquid-like response is the time scale of the experiment. This leads naturally to the definition of a dimensionless parameter called the Deborah number, viz. 

The second investigation concerns the influence of the viscosity ratio p. In the study, we change the values of solvent and polymer viscosity but keep the total viscosity and other parameter unchanged. The Deborah number is this study is De = 717.05. From the definition of the viscosity ratio, we know the larger the ratio, for a fixed total viscosity, the smaller portion of polymers in the fluid. 

To simulate the viscoelastic flow, the Oldroyd-B model has been implemented in the VOF-code. Stabilization approaches, such as the Positive Definiteness Preserving Scheme and the Log-Conformation Representation approach have been adapted and implemented in the code to stabilize the simulations at high Weissenberg numbers. The collision of viscoelastic droplets behaves as an oscillation process. The amplitude of the oscillation increases and the oscillation frequency decreases when the Deborah number becomes larger. The phenomenon can be explained with the dilute solution theory with Hookean dumbbell models. An increase of the fluid relaxation time yields a decrease of the stiffness of the spring in the dumbbell and restrains the deformation of the droplets. In addition, with larger the viscosity ratio the collision process is more similar to the Newtonian one since the fluid has less portion of polymers. 

The first is the weak viscoelasticity approach employed in the theory. For common polymers with long flexible chains, the weak viscoelasticity simply means the smallness of the Deborah number, De = Qy < 1 where 0 is the relaxation time averaged over the relaxation spectrum. In case of anisotropic LCPs where at least two relaxation times exist, the definition of Deborah number is not clear. 

It is diflicult to write a concise definition of a time constant that governs the rate at which stored elastic energy changes in a given deformation without reference to a specific flow, and we therefore give a definition of the Deborah number in general terms... 

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