Rheological model of the fine particles viscoplastic suspension

The analysis of the mechanical behaviour of some types of fine particles suspensions is carried out. A rheological model is proposed to describe the shear stress and shear rate dependence, taking into account the appropriate conditions imposed on the model parameters. It is shown that such model can take into account the “solidification” effect manifestation. A variation of generalization of the rheological model for two- and three-dimensional hydrodynamics problems of fine particles suspensions is proposed, and some methodological features for solving such problems are considered.


Introduction
As a rule, traditional fluids exhibit a monotonic viscosity change over a fairly wide range of shear rate changes.
In particular, the classical Newtonian fluid is characterized by the constant viscosity. Well known pseudoplastic fluids correspond to the monotonic viscosity decrease with increasing shear rate. Another group of so-called dilatant fluids, on the contrary, manifests itself in the monotonic viscosity increase as the shear rate increases.
In contrast to continua of such kind, some types of suspensions based on polymer fluids and fine particles demonstrate a more complex mechanical behavior [1][2][3][4].
Such behavior manifests itself, first of all, in the specific dependence of their viscosity on the shear rate. At different shear rate ranges, the suspension can exhibit a different viscosity change pattern. For example, a monotonic decrease in viscosity in a certain shear rate range can change to a monotonic increase, but in a different range. And then, in the next shear rate variation range, its character again changes to a monotonic decrease. In this case, inflection points appear on the flow curve -the dependence of the shear stress on the shear rate -corresponding to the boundaries of such ranges.
Such mechanical behavior anomalies are based on the material internal structure changes that occur in various deformation modes and are associated with the formation of the fine particles into associations or clusters of the "solid" structures type.
Suspensions of this kind, with the predetermined combination of the corresponding particles characteristics, taking into account their size and concentration, as well as the rheological parameters of the suspension fluid component, are the basis for a number of materials with rather specific mechanical behavior. This specific behavior is caused by viscosity anomalies. As the shear rate in the corresponding flow region zones approaches a certain critical value, such suspensions viscosity begins to increase sharply. Moreover, for some types of suspensions, this increase can be so significant that the viscosity value increases substantially. Then the suspension behavior in such flow region zones resembles the behavior of a solid.
Some examples of the use of suspensions of this kind in technical applications are given in [5][6][7].
Obviously, in order to manufacture products of "fixed" shape from such materials, they must demonstrate some plasticity. This can be achieved by using viscoplastic fluids with a sufficiently high shear stress as a liquid component.
Some examples of rheological models of similar fluids with viscous behavior anomalies were considered in [8][9].
Since these models do not fully describe all the features of the suspensions mechanical behavior, this article proposes a more complete rheological model of a nonlinear viscoplastic fluid, the flow curve of which has two inflection points corresponding to the changes in viscosity increase and decrease modes with increasing shear rate.

Rheological model
The review of the known experimental data, presented, for example, in [1][2][3][4], shows that fine particles suspensions in predetermined combination of their parameters (size, shape, concentration) and rheological characteristics of a viscous liquid base demonstrate a rather complex behavior.
As a rule, the dependence of the shear stress W and viscosity P on the shear rate J is qualitatively described as follows.
At relatively low shear rate values, the continuum behaves like a pseudoplastic fluid. Moreover, its viscosity decreases monotonically as the shear rate increases, starting from the level 0 P at 0 J . However, having reached the minimum level min , the viscosity begins to increase and the suspension mechanical behavior begins to correspond to the dilatant fluid behavior. In these two ranges of the shear rate change, the viscosity obviously changes, but in large it takes on values of approximately one order from the certain range s min P P P .
Subsequently, when the shear rate exceeds a certain threshold level s J J , the viscosity continues to increase, but this increase rate rises sharply. As shown by the known experimental data, for example, from [2], in this range, the viscosity can increase by several orders of magnitude.
In some cases, the viscosity increase becomes so significant that the suspension begins to behave like a solid. In such a situation, we can talk about the fluid "solidification" or the "solidification" effect manifestation.
Having finally reached the certain maximum level Note that in case of the shear stress and the shear rate linear dependence, which corresponds to the Newtonian fluid, the latter expression immediately leads to the traditional dynamic viscosity It is quite acceptable to describe the flow curve (the shear stress and the shear rate dependence) with one function, but it doesn't seem entirely rational. This is due to the fact that, as can be expected, such function will have a rather complex form. In turn, this will lead to difficulties in obtaining explicit analytical solutions, even for relatively simple hydrodynamics problems.
In this regard, taking into account the occurrence of characteristic sections of different mechanical behavior on the flow curve, it is proposed to take a rheological model of suspensions in the following form The rheological model type (4.2) predetermines the methodology for solving hydrodynamic problems for a viscoplastic suspension of fine particles.
The main feature of this method is the obligatory division of the flow region, in the most general case, into five separate zones. One of these zones must correspond to the plastic flow. In each of the remaining four zones, a shear flow corresponding to one of the ranges (4.2) of the strain rate tensor second invariant modulus variation must be realized.
In this case, the shear flow zones number is not fixed. It can vary from one to four zones.
In each specific case, the number of realized zones is determined by the flow mode corresponding parameters.
For example, for a flow in a channel, the pressure drop along its length can act as a flow mode parameter. In this case, in addition to the plastic flow zone, a shear flow zone will correspond to relatively small pressure drop values. If the pressure drop exceeds a certain threshold level, it will lead to the formation of the shear flow second zone. Further, as the pressure drop increases, the number of the realized shear flow zones will also increase. An example of this approach implementation, but for the case of a simpler rheological model, is given in [10].
Another feature of this method is that the separate spatial zones boundaries are not known in advance and must be determined in the course of solving the hydrodynamic problems under consideration.
Speaking about the formulation of the fine particles viscoplastic suspension hydrodynamic problems that satisfies the model of type (4.2), let us note the following. The possibility of the existence of separate zones with different mechanical behavior in the flow region presupposes, along with the traditional boundary conditions (at the flow region outer boundaries), the setting of additional boundary conditions at the common boundaries of these zones. Such boundary conditions should ensure the velocity and stress fields "stitching".

Conclusions
The proposed rheological model allows one to describe the mechanical behavior of the fine particles suspensions, the fluid component of which is a viscoplastic medium, over a wide range of the shear rate variation.
Such model takes into account the sequential alternation of the three sections as the shear rate increases: a decrease, an increase, and then again a decrease in viscosity.
The model configuration allows obtaining solutions of one-dimensional hydrodynamics problems in an analytical form. In this case, the model admits a special case of the "solidification" effect manifestation, when the viscosity increases so significantly (by several orders of magnitude) that at the model level it is permissible to assume f o max P . The proposed generalization of the considered rheological model to the flows can be used when considering hydrodynamics problems in a higher dimension.
The main features of solving hydrodynamic problems for viscoplastic suspensions of fine particles were shown.