The Weissenberg Number

The Weissenberg number is introduced in the following section in connection with flows with constant stretch history, i.e., flows in which the deformation rate and aU the stresses are constant with time. These are flows in which De is zero. And for deformations in which linear viscoelastic behavior is exhibited, Wi is zero. However, there are also flows of practical importance in which both Wi and De are nonzero and are sometimes even directly related to each other. This causes confusion, as authors often use the two groups interchangeably. This situation arises, for example, in the flow from a reservoir into a much smaller channel, either a slit or capillary. A Weissenberg number can readily be defined for this flow as the product of the characteristic time of the fluid and the shear rate at the wall of the flow channel. However, entrance flow is clearly not a flow with constant stretch history, and the Deborah number is thus non-zero as well and depends on the rate of convergence of the flow. 

Just as the Deborah number is often used to plot data or predictions for transient tests at finit rates, the Weissenberg number (Wi) is used for a similar purpose for representing stresses that are independent of time. This is defined as the product of a time scale governing the onset of nonlinearity, let s call it /I, and the rate of strain of the experiment. We will see in the next section that a fluid that has a shear-rate dependent viscosity must have at least one material constant with units of time. For example, one might select the reciprocal of the shear rate at which the viscosity falls to 60% of its zero-shear value. This time then characterizes the nonlinearity of the behavior for this flow. Obviously, the degree to which a melt deviates from Newtonian behavior depends on how this time constant compares with the rate of the deformation. Thus, the Weissenberg number in simple shear flow is defined as  

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As an example of such applications, consider the dynamics of a hexible polymer under a shear how [88]. A shear how may be imposed by using Lees-Edwards boundary conditions to produce a steady shear how y = u/Ly, where Ly is the length of the system along v and u is the magnitude of the velocities of the boundary planes along the x-direction. An important parameter in these studies is the Weissenberg number, Wi = Ti y, the product of the longest... 

Elastic behavior of liquids is characterized mainly by the ratio of first differences in normal stress, Ai, to the shear stress, t. This ratio, the Weissenberg number Wi = Ai/r, is usually represented as a function of the rate of shear y. 

By keeping the Weissenberg number Wi (a pure material number) and the Hedstrom number He constant, measurements are performed and presented in a dimensionless frame ... 

In the case of non-Newtonian mixtures, which also exert pseudoplastic and viscoelastic behavior, the pi-space is widened by the Weissenberg number, Wi. In addition, it has to be decided which effective viscosity, peffi has to enter the Reynolds number ... 

These test data are correlated satisfactorily by the Weissenberg number (here ned as Wi = /.n) and the term ( x0, i/Mo.2) which accounts for the pumping irection, Fig 23. ji represents the starting viscosity of the lower, more viscous liqui ( )> an the upper, less viscous (2) liquid. Upflow z= 0.059, downflow z= 0.17. 

Examples of CFD applications involving non-Newtonian flow can be found, for example, in papers by Keunings and Crochet (1984), Van Kemenade and Deville (1994), and Mompean and Deville (1996). Van Kemenade and Deville used a spectral FEM and experienced severe numerical problems at high values of the Weissenberg number. In a later study Mompean and Deville (1996) could surmount these numerical difficulties by using a semi-implicit finite volume method. 

The parameters in these equations are the Reynolds number Re = pULfTf (U and L are a typical velocity and a t rpical length of the flow, and rj = r),+t]p is the total viscosity of the liquid), the Weissenberg number We = Xl/fL, and the retardation parameter e = fip/fi. Obviously, 0< <1 =1 corresponds to Maxwell-type fluids, and 0 < e < 1 corresponds to Jeffreys-type fluids. Observe the change of notation in equation (7), where /3(Vv,r) denotes now adl the nonlinear terms in Vv and t other than the term (v V)r. f denotes some given body forces. 

Many studies have been devoted to the Taylor-Couette problem (flow between two concentric cylinders with radii R and R2, Ri < R2, of infinite length, and rotating with angular velocities fij and 02 repectively). For instance Zielinska and Demay [74] consider the general Maxwell models with —1 <0 < 1. They show that the axisymmetric steady flow (the Couette flow) does not exist for 2dl values of parameters where the steady state exists moreover all models, except for a very close to —1, predict stabilization of the Couette flow in the spectral sense, for small enough values of the Weissenberg number. (See also [55].)... 

This method is highly stable and results have been obtained at large values of the Weissenberg number for various flows. Figure 2 from [28] show the evolution of velocity and T y stress... 

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]. 

If the ndgs values are plotted as a function of expression (3.26), in which Rew was formulated with and the Weissenberg number Wi with the relaxation time A , which correspond to the shear rate yielded a quite satisfactory correlation of mixing times. At the same time the results obtained with CMC converged further, as can be seen in Fig. 3.9. 

This parameter is known in the rheological literature as the Weissenberg number (also sometimes - mistakenly - called the Deborah number) 2... 

The quotient (r /r o), which indicates the magnitude of shear thinning effect with increasing y, can be estimated quite well as a function of the Weissenberg number (Nwg) and the polydispersity [7], N yg is the product of a characteristic time scale for the polymer with y, and can be approximated by Equation 13.27 where p is the polymer bulk density at the temperature of calculation. (The density must be expressed in g/m3 rather than g/cc in Equation 13.27 for consistency with the units used for the other quantities entering this equation.) Since the polydispersity Mw/Mn enters Equation 13.27, r depends on the polydispersity both indirectly via the dependence of Ny/g on Mw/Mn and directly. 

Calculate the Weissenberg number N yg at the shear rate of interest via Equation 13.27, with the value of the polymer bulk density estimated by using the equations given in Chapter 3. 

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 ... 

The asymptotic nature of the friction factor is clearly brought out in Fig. 10.24, which shows the measured fully established friction factors taken in three tubes of differing diameters as a function of the Weissenberg number based on the Powell-Eyring relaxation time for fixed values of the Reynolds number for aqueous solutions of polyacrylamide. The critical Weissenberg number for friction (Ws), is seen to be on the order of 5 to 10. When the Weissenberg number exceeds 10, it is clear that the fully developed friction factor is a function only of the Reynolds number. 

It is recommended that Eqs. 10.74 and 10.75, or equivalently Eqs. 10.76 and 10.77, be used to predict the heat transfer performance of viscoelastic aqueous polymer solutions for Reynolds numbers greater than 6000 and for values of the Weissenberg number above the critical value for heat transfer. This critical Weissenberg number for heat transfer based on the Powell-Eyring relaxation time is approximately 250 for aqueous polyacrylamide solutions. Appropriate care should be exercised in using this critical value for other viscoelastic fluids. 

The fully established friction factor for turbulent flow of a viscoelastic fluid in a rectangular channel is dependent on the aspect ratio, the Reynolds number, and the Weissenberg number. As in the case of the circular tube, at small values of Ws, the friction factor decreases from the newtonian value. It continues to decrease with increasing values of Ws, ultimately reaching a lower asymptotic limit. This limiting friction factor may be calculated from the following equation ... 

It should be noted that for some problems, more than one characteristic time for the flow can be identified [3]. Thus, a second dimensionless group, the Weissenberg number, Wi, is sometimes used in polymeric fluid dynamics. It can be defined as... 

The only way one may scale at constant Brinkman number is to maintain the characteristic velocity U constant as L increases, which would decrease the Weissenberg number. 

In most practical applications, polymer fluids do not behave like ideal Newtonian fluids. The occurrence of non-ideal viscoelastic behaviors of shear flow is often associated with a dimensionless number, called the Weissenberg number We (Weissenberg 1947 Dealy 2010),... 

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