Liquid–solid relative motion during squeeze flow of pastes
Introduction
The deformation of a paste (a mixture of solid particles and liquid) can be accompanied by motion of the liquid component of the paste relative to the solid. We consider the effects of such relative motion during the squeezing of a paste between two circular, rigid, parallel plates which move towards each other. The geometry, depicted in Fig. 1, is axisymmetric.
The force F which must be applied to the plates is related both to the constitutive relation of the deforming paste and to the frictional forces between the paste and plates. The rate of deformation within the paste varies with position, so that to determine the rheology of the paste from the measured force F is not straightforward. This squeeze flow test is nevertheless a useful rheometer, as intimate contact is maintained between the paste and the plates of the device. Pastes and suspensions are known to slip at the walls of rheometers [1], [2], and the test may be considered as a method for investigating both the constitutive relation of the test material and the frictional interaction between the material and the walls of the device.
It is usual to assume that the test material is uniform and homogeneous. Covey and Stanmore [3] pointed out that liquid within the paste may move relative to the solid during the test, and such motion is now well documented [4], [5], [6], [7], [8]. If the test material is a filter cake its creation involved expression of liquid from an initial dilute slurry, and this process may continue during the squeeze flow test. Order-of-magnitude estimates suggest that, for bentonite filter cakes, relative motion will be very small over the timescale of a squeeze flow test because of the low filter cake permeability [9], but this is not necessarily so for more permeable pastes. Relative motion is known to occur in other rheometers (e.g. concentric cylinder rheometers), usually over long timescales, due either to shear-induced diffusion [10] or to gravitational settling [11]. Relative motion may perturb the homogeneity of a sample loaded into a parallel plate rheometer since the test material is squeezed when the upper plate is brought down into contact with the test material, and can also occur in extrusion or other processing of paste [12].
Any model which distinguishes liquid and solid phases is necessarily more complicated than one which treats the test material as a single homogeneous material. For small, elastic deformations of the test material we might assume the material to behave as a Biot poro-elastic material [13], [14]. The immediate, undrained response to an imposed stress σij is then an elastic deformation and a rise in the pore pressure by an amount −Bσkk/3, where B is Skempton’s parameter. If both the pore fluid and solid are incompressible, but the matrix of solid particles is compactible, then B=1 and the change in pore pressure is equal to the change in the isotropic imposed stress. Pore fluid subsequently moves due to gradients of pore pressure, and the final deformation of the poro-elastic material depends upon its drained constitutive relations.
However, in a constant velocity squeeze flow test the material is continuously deformed and strains are ultimately large. The material fails, and a small-strain poro-elastic model is inappropriate. Failure of a poro-elastic material is generally assumed to depend upon the difference between the imposed total stress and the pore pressure: this difference represents that part of the imposed load borne by the solid matrix.
Small oscillatory strains can be described in terms of a linearised two-fluid model with a no-slip boundary condition at the plates [6]. Such models require finite element computations when strains are large [7], [8].
We turn instead to an analysis for squeeze flow of a plastic material with slip at the walls [15]. This is combined with a constitutive relation which has proved useful [16], [17], [18] for describing the uniaxial compaction of suspensions in a filter press. The combination leads to governing equations for the liquid content of and stress within the paste as a function of radial position r and time t, independent of the axial coordinate z normal to the plane of the plates. These equations are set down in Section 2. Once the particular forms of the constitutive relations have been chosen (Section 2.4), the governing equations may be solved numerically and results are presented in 3 Results for the case of a partially filled gap,, 4 Numerical results for a fully filled gap,.
Section snippets
Pore pressure and Darcy flow of liquid within the paste
We assume that the total stress within the paste can be represented as the sum of the isotropic pressure p within the liquid and the matrix stress within the solid. Thus, where the matrix stress has been divided into an isotropic component and the deviatoric stresses . The matrix is assumed to be in the process of yielding, and the stress which it can support is a function of the solids volume fraction φ within the material, where
Governing equations
We scale all material coordinates by M, lengths in the r direction by R and lengths in the z direction by h0. Denoting non-dimensional quantities by a caret, the material coordinate (17) becomes If we now introduce the forms (29) for D, (30) for κ and (31) for f, and scale time by a timescale T=h0/U, we obtain, where plays the role of a Peclet number and the non-dimensional velocity of approach of
Numerical results for a fully filled gap, R=R0 constant
We keep the same constitutive relations as in Section 3, so that and we assume that the wall shear stress f=λψ.
We non-dimensionalise lengths in the radial direction by R0, lengths in the z direction by h0 and time by h0/U, and denote non-dimensional quantities by a tilde. Substituting , , into Eq. (24), we obtain where plays the role
Concluding remarks
We have combined earlier work on filtration and squeeze flow in order to show how squeeze flow of a paste with a particularly simple constitutive relation may be affected by motion of liquid relative to solid within the paste. Such computations are of course far removed from the problem of deriving a constitutive relation from a set of experimental measurements of the total force, which is difficult even when the material under test may be considered homogeneous. The constant volume experiment
Acknowledgements
I thank G.H. Meeten, J.R.A. Pearson, and H. Stone for helpful discussions.
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