Jeffreys model

Shearing flows of the convected Jeffreys model. The convected Jeffreys model [6] or Oldroyd s B-fluid [54] is given by,... 

Indicating that the convected Jeffreys model gives a constant viscosity and first normal stress coefficient, while the second normal stress coefficient is zero. 

Develop expressions for the elongational viscosities fji and fj2 for steady shearfree flows of a convected Jeffreys model. Comment how this expression compares with experiments. 

Including a velocity gradient in the time derivative, we obtain the Jeffreys model (31)... 

This equation for the dielectric constant is the analogue of the compliance of a mechanical model, the so-called Jeffreys model, consisting of a Voigt-Kelvin element characterised by Gi and rp and t =t /Glr in series with a spring characterised by Gz- The creep of this model under the action of a constant stress aQ is (Bland, 1960)... 

Another peculiar property of LCPs is shown in Fig. 15.47, where the transient behaviour of the shear stress after start up of steady shear flow is shown for Vectra A900 at 290 °C at two shear rates. We will come back to this behaviour in Chap. 16 for lyotropic systems where this behaviour is quite common and in contradistinction to the transient behaviour of conventional polymers, as presented in Fig. 15.9. This damped oscillatory behaviour is also found for simple rheological models as the Jeffreys model (Te Nijenhuis 2005) and according to Burghardt and Fuller, it is explicable by the classic Leslie-Ericksen theory for the flow of liquid crystals, which tumble, rather than align, in shear flow. Moreover, it is extra complicated due to the interaction between the tumbling of the molecules and the evolving defect density (polynomial structure) of the LCP, which become finer, at start up, or coarser, after cessation of flow. 

Transient behaviour of lyotropic MCLCPs is similar to that of thermotropic MCLCPs as discussed in Sect. 15.7 (Fig. 15.47). An example is given in Fig. 16.35 for PpPTA in sulphuric acid (Doppert and Picken, 1987). It shows damped oscillating behaviour, which is in contradistinction to conventional polymers, but which is also found as a result of simple rheological models, like the Jeffreys model (see Chap. 15). This behaviour turns out to be a general feature of lyotropic MCLCPs. The oscillatory behaviour is easier to measure for lyotropic than for thermotropic systems, where it is less pronounced. 

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].)... 

Concerning global (in time) existence of solutions, the first general result is the following [23], which is established for Jeffreys models with a sufficiently large Newtonian contribution to the extra-stress. 

Remark 4.6 A number of problems are still open. For example the existence of (small) periodic solutions for Mcixwell models is unknown, as is that of arbitrary (not small) periodic solutions for Jeffreys models. For example too, nothing is known concerning the global existence of unsteady solutions for differential models in two or three space dimensions (say, weak solutions, singularities in finite time,. ..). See below, for some examples in one space dimension. 

The only parameter in (11) having dimensions of time is C. Although this is not a relaxation time per se, it can be associated with a "Maxwell-type relaxation time, as follows. Although the linear Maxwell model predicts a constant (Newtonian) viscosity, it may be generalized by utilizing a co-rotational reference frame which follows the local rotation and translation of each fluid element [9]. When a term is added to account for the high shear limiting behavior, the result is the co-rotational form of the Jeffreys model ... 

Strain relation of a Jeffrey model. In the dilute solution theory, it is assumed that the viscoelastic fluid consists of a suspension of dumbbells in a Newtonian solvent. A dumbbell has two identical beads connected by a spring. An illustration is shown in Fig. 1.27. The Oldroyd-B fluid has viscoelastic properties with constant viscosity. To characterize the viscoelastic flow, the Weissenberg number... 

It can be seen that Eq. (3.15) is a special case of Eq. (3.18). Note that Eq. (3.18) predicts shear-rate dependent viscosity and also nonzero negative Nj, giving rise to —Nj/Nj = 0.5. It should be mentioned that there is no a priori reason for preferring any single generalization of the classical Maxwell model, Eq. (3.3), or the classical Jeffreys model, Eq. (3.8). Only a careful comparison of the model predictions with experimental results will distinguish between the possible alternatives. 

An example of (a) is the convected Jeffreys model or Oldroyd B model " ... 

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