Weissenberg equation

Equation (7-43) must also apply to Newtonian liquids. Since it is true here that y = 02 l n and d y/da2i = l/ry, this also gives dvjdR = a o2 lr]) + (1/4) (021/77) and, with dv/d/ = 021/77, it follows that a = 3/4. Equation (7-43) thus becomes the Weissenberg equation... 

The situation will be simpler to follow for set of reflections for which I = 0, i.e., for zero layer Weissenberg. Equation (8.1) will take a simpler form... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

The calculation of the shear rate at the capillary wall, 7 , is computed from the function slope of Fig 3.18 and the apparent shear rate using Eq. 3.36. The derivative of the function appears relatively constant over the shear stress range for Fig. 3.18. Many resin systems will have derivatives that vary from point to point. The corrected viscosity can then be obtained by dividing the shear stress at the wall by the shear rate i ,. Equation 3.36 is known as the Weissenberg-Rabinowitsch equation [9]. 

EitherEq. E3.1-9or Eq. E3.1-10, known as the Rabinowitsch or Weissenberg-Rabinowitsch equations, can be used to determine the shear rate at the wail yw by measuring Q and AP or r and Tw (21). Thus, in Eq. E3.1-4 both xw and yw can be experimentally measured for any fluid having a shear rate-dependent viscosity as long as it does not slip at the capillary wall. Therefore, the viscosity function can be obtained. 

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. 

Let V be a steady solution of the Navier-Stokes equations (with prescribed body forces and zero boundary velocity). Note that v is not assumed to be small . Then, there exists a steady solution (Vt,Tj,pj) of any Jeffreys model with a sufficiently small Weissenberg number We, and with a sufficiently small retardation parameter e, such that (ve,rj) is close to (v,0) and close 0. (See [26].)... 

The EVSS-G method introduced by Brown et al. uses the velocity gradient as an additional unknown ([7]). In order to come back to primitive variable, Guenette and Fortin ([20]) have introduced a (U, p, o, D) method where no explicit change of variable is performed in the constitutive equation. Hence this method is easier to implement. The elements used by these authors are continuous for velocity, discontinuous Pj for and pressure and continuous Qi for G and D. This method was tested on the 4 1 contraction and the stick-slip problem. This method seems robust and no limiting Weissenberg number was reached when using the PTT model for the stick-slip problem. 

As pointed out in Section II-3, numerical modelling of viscoelastic flows leads to numerical difficulties related to the mixed character (elliptic - hypeiholic) of the constitutive equation, to the propagation of "stress singularities" and to the so-called "High Weissenberg Number" problem. 

Equation 3B.18 is known as the Weissenberg-Rabinowitsch-Mooney (WRM) equation in honor of the three rheologists who have worked on this problem. An alternate equation can be derived for fluids obeying the power law model between shear stress and the pseudo shear rate ... 

Problem 3.8 Numerically solve Eq. (3-78), the differential approximation to the Doi-Edwards equation for entangled linear melts, in a steady-state shearing flow. Plot the dimensionless shear stress ayijG against Weissenberg number W/ = )>r for Wi between 0.1 and 100. 

Yin et al. (2006) qnalitatively showed this mechanism by solving relevant flow equations nnmericaUy. Xia et al. (2008) also developed a simplified pore scale model to describe polymer flow. The numerical solutions from Xia et al. have verified the proposed mechanism. Figure 6.22 shows the velocity contours of a Newtonian fluid with Weissenberg number (We) = 0 and a viscoelastic fluid with We = 0.35 in a flow channel with a dead end when the Reynolds number (Re) = 0.001. We can see that the velocity (m/day) of the viscoelastic fluid is higher than that of the Newtonian fluid at the same position of the dead end. This pulling mechanism also works in the case shown in Figure 6.20c, where the residual oil is trapped at the pore throats by capillary force. 

It was later possible to extract single crystals of the specimen, and from Weissenberg and oscillation photographs they were found to be PbCl2, which has in the equation for sin2 Q the constants A = 0.0073,... 

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. 

If we choose the coefficients p and s in (6.1.7) to be nonzero constants, then we arrive at the Reiner-Rivlin model, which additively combines the Newton model with a tensor-quadratic component. In this case the constants p and e are called, respectively, the shear and the dilatational (transverse) viscosity. Equation (6.1.7) permits one to give a qualitative description of specific features of the mechanical behavior of viscoelastic fluids, in particular, the Weissenberg effect (a fluid rises along a rotating shaft instead of flowing away under the action of the centrifugal force). 

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 ... 

Various forms of this equation are used, a common form (often termed the Weissenberg-Rabinowitsch or Rabinowitsch-Mooney equation) being,... 

---