De Deborah number

Cl screw center-to-center distance D deformability parameter d droplet diameter d equilibrium droplet diameter De Deborah number do initial diameter of droplet Dp particle diffusion coefficient dPidz pressure gradient E energy of mixing e relative flight width = e/(7iDsSincp)... 

The appropriate apparent viscosity is estimated at the effective shear rate, = 5,000 Nakanoh and Yoshida found it necessary to introduce another correction in case of viscoelastic liquids [61]. The right hand side of Equation 35 must be divided by (1 +0.13 De ) where De, Deborah number, is defined as XWJ d j, with Vg the bubble swarm velocity and d is the sauter mean diameters K is the fluid relaxation time, which was arbitrarily defined as the reciprocal of the shear rate at which the apparent viscosity of the solution had dropped to (2/3) of its zero shear viscosity. 

De Deborah number Re Reynolds number We Weber number Wi Weissenberg number... 

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 ... 

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... 

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. 

The dimensionless number in rheology that compares relative importance of the time scale of the deformation process tD over the observation time tQ is called the Deborah number (De) ... 

A useful parameter often used to estimate the elastic effects during flow is the Deborah2 number, De. The Deborah number is defined by... 

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

The plots in Figure 10.17 show the axial velocity profiles for various positions along the channel for a Deborah number, De = 1.87. The vertical, dashed lines indicate the locations of each scan of the channel. The initial profile, which is furthest upstream of the cylinder, shows an expected, parabolic shape. As the cylinder is approached, the velocity field becomes distorted, with a larger portion of the flow being diverted toward the lower region where the gap between the cylinder and the boundary of the channel is the largest. 

Figure 10.17 Axial velocity profiles for a Deborah number of De = 1.87. The vertical, dashed lines indicate the axial positions at which the channel was translated.

Note that since in this case, the Deborah number, De = Aw, the moduli and the loss tangent, G. G", tan S, are functions of the Deborah number. 

---