Transient flow of viscoelastic, thixotropic fluid in a vane rheometer or infinite slot
Introduction
The rheology of concentrated suspensions is usually complex. For engineering calculations such fluids are often considered to have a yield stress; their behaviour prior to yield, if considered at all, might be assumed to be elastic, viscous (with a high viscosity) or viscoelastic. Measurements of the yield stress can be affected by slip at the walls of conventional rheometers, and one way to reduce wall slip is to use a vane rheometer [1], [2], [3], [4], [5], [6], [7], [8]. The vane rotates within a large sample of gelled fluid. Fluid between the blades of the vane tends to rotate, undeformed, with the blades, so that the stress measured is that in the highly sheared region between the rotating cylinder of test fluid and the outer, gelled fluid.
When a yield stress fluid is investigated by means of a vane rheometer rotating at constant velocity, there is usually an initial period during which the stress increases to a maximum, and subsequently decreases to a plateau value. If initially the stress is approximately proportional to strain, a shear modulus for the gelled fluid can be estimated [9]. However, it is not clear whether it is more appropriate to use the maximum stress or the subsequent (lower) plateau value to estimate a yield stress, nor is it obvious to what extent the results will be modified by viscoelasticity or breakdown of the structure of the fluid. Our aim here is to investigate how viscoelasticity is likely to modify the stresses during the early stages of rotation of the vane.
As the number of blades of the vane increases, motion of fluid between the blades of the vane decreases in the frame rotating with the vane [9], [10]. In the limit of an infinite number of blades, the vane becomes equivalent to a rotating cylinder. We take advantage of this, and consider the simplified geometry of a wide-gap Searle viscometer, infinitely long so that end effects can be ignored. The geometry is shown in Fig. 1, and is described in terms of cylindrical polar coordinates . The inner radius of the annular rheometer is and the outer radius is , with for the numerical results presented in Section 4. The components of velocity are . The inner cylinder rotates at an angular velocity , so that in the absence of slip . Fluid velocity and stresses are independent of , and we look for plane solutions independent of z. We assume that inertia is sufficiently small to be neglected.
Heymann and Aksel [11] recently performed experiments to investigate the changeover between elastic behaviour at small strain and viscous behaviour at large strains. This change in behaviour depends upon the rate of shear, and the rate at which any structure within the fluid is broken down. Here we compute the stresses which will be observed during start-up flow of a viscoelastic shear-thinning fluid. We assume that the constitutive relation obeys the MBM model [12], which is a modified form of the Bautista–Manero model [13], a thixotropic Maxwell model originally developed for viscoelastic surfactant solutions. At short timescales the MBM model is viscoelastic. In steady shear flow the stress is proportional to shear rate at low and high shear rates. At intermediate shear rates the fluid structure breaks down as the shear rate increases, and the shear stress stays close to a plateau value that might be interpreted as a yield stress if nothing were known about lower shear rates. In the version of the model used here, the stress is monotonic increasing with shear rate (albeit increasing only slowly at intermediate shear rates). However, a flow curve with a maximum can be introduced by suitable modification of the model [14].
The MBM model, although not originally intended to represent the rheology of a concentrated suspension, is used here because it conveniently includes viscoelasticity and structure breakdown. Other thixotropic models are available and might instead have been used to predict the stress overshoot typically observed when shear is imposed starting from rest. Many such models are reviewed by Mujumdar et al. [15], with more recent examples discussed in [16], [17]. Some models include a yield stress, either explicitly or by allowing the viscosity to become infinite in the rest state, and of these we draw attention to another thixotropic Maxwell model, that of Coussot et al. [18]. Although the stress plateau of the MBM model can extend to low shear rates, in the form used here the model does not predict a yield stress, and the low shear-rate Newtonian response allows creep to occur. It is not our intention here to enter the debate concerning the existence of yield stresses [19], but it is important to know how creep may manifest itself in a rheological test.
Recent experiments have determined flow profiles within wide-gap rheometers. Both Nouar et al. [20] and Raynaud et al. [21] discuss experiments on fluids thought to have a true yield stress over the timescales involved. Raynaud et al. did not report the evolution of stress in their experiments, but their observed velocity profiles will be discussed in Section 4. Nouar et al. worked at constant applied stress, and so their results cannot be compared to the predictions of Section 4.
In Section 2 we present the rheological model, and in Section 3 we obtain non-dimensional equations governing flow in the wide-gap rheometer. Numerical results obtained for a particular set of rheological parameters are presented in Section 4. The equations governing time-dependent flow in an infinitely long two-dimensional channel are very similar to those for flow in the wide-gap rheometer, and are discussed in Section 5, with results in Section 6.
Section snippets
Constitutive relation: the modified Bautista–Manero model
Boek et al. [12] modified the Bautista–Manero constitutive relation by splitting the stress into a Newtonian solvent stress:(where is the solvent viscosity) and a non-Newtonian stress which in a viscoelastic surfactant solution is due to the micelles, but which more generally is a particle stress. The particle stress satisfies an upper-convected Maxwell equation:
The fluidity is governed by an evolution equation:
Non-dimensional equations
We non-dimensionalize stresses by and lengths by the inner radius . We scale time by the timescale for recovery of the fluidity , rather than by the elastic relaxation timescale (which varies with ) or by a timescale based on a combination of velocity and . Consequently, we use non-dimensional quantities:and the non-dimensional constitutive coefficients are
The governing Eq. (4) become, after non-dimensionalization:
Material coefficients
The MBM model has in the past been used mainly to characterize viscoelastic surfactant solutions [12], [25], rather than suspensions of particles. For illustrative purposes we present numerical results corresponding to a 3% (by mass) solution of surfactant concentrate (a mixture of the cationic surfactant erucyl bis(hydroxyethyl)methylammonium chloride (EHAC) and 2-propanol in a 3:2 ratio (by mass), in a 4% aqueous solution of KCl at 60 C. Thus we take Pa, Pa−2 s−1, Pa
Non-dimensional equations
The equations governing time-dependent flow along an infinite straight two-dimensional channel (Fig. 7) are almost the same as those in a circular annulus (which may in some ways be considered to be a channel of finite length wrapped around the axis and joined up on itself). We can therefore compute motion in the channel by means of a numerical scheme very similar to that used in Sections 3 Flow in an annular rheometer, 4 Numerical results for an annular rheometer for the Searle viscometer. The
Material coefficients
For numerical computations we use the material coefficients of Section 4.1. If the flow profile is parabolic (40), the non-dimensional wall shear-rate and the shear rate at which non-linearities appear is, by (7), corresponding to a non-dimensional flow rate . The limiting particle contribution (8) to the shear stress corresponds (in the absence of any solvent viscosity) to a non-dimensional pressure gradient . This will
Concluding remarks
The results for a wide-gap Searle rheometer give a clear indication of what might be expected if a thixotropic, highly shear thinning viscoelastic fluid is investigated by means of a vane rheometer [8]. The stress maximum cannot be considered to be a static yield stress unless it is independent of the speed at which the vane rotates. This problem is well recognised in the published literature on vane rheometry. On the other hand, the final steady stress can be related directly to that which
References (32)
- et al.
Vane rheometry of bentonite gels
J. Non-Newton. Fluid Mech.
(1991) - et al.
Measuring the yield behaviour of structured fluids
J. Non-Newton. Fluid Mech.
(2004) - et al.
The use of the vane to measure the shear modulus of linear elastic solids
J. Non-Newton. Fluid Mech.
(1991) - et al.
Constitutive equations for extensional flow of wormlike micelles: stability analysis of the Bautista–Manero model
J. Non-Newton. Fluid Mech.
(2005) - et al.
Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model
J. Non-Newton. Fluid Mech.
(1999) - et al.
On the shear banding flow of elongated micellar solutions
J. Non-Newton. Fluid Mech.
(2000) - et al.
Transient phenomena in thixotropic systems
J. Non-Newton. Fluid Mech.
(2002) - et al.
A structural kinetics model for thixotropy
J. Non-Newton. Fluid Mech.
(2006) - et al.
Rheology of concentrated dispersed systems in a low molecular weight matrix
J. Non-Newton. Fluid Mech.
(1993) The yield stress—a review or ‘’—everything flows?
J. Non-Newton. Fluid Mech.
(1999)
Propagation of the interface in a fluid suspension after the onset of shear flow
J. Non-Newton. Fluid Mech.
Laminar, unidirectional flow of a thixotropic fluid in a circular pipe
J. Non-Newton. Fluid Mech.
Stability analysis of shear banding flow with the BMP model
J. Non-Newton. Fluid Mech.
Thixotropy modelling at local and macroscopic scales
J. Non-Newton. Fluid Mech.
The vane borer—an apparatus for determining the shear strength of clay soils directly in the ground
The measurement of the yield stress of liquids
Rheol. Acta
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