Liquid–solid relative motion during squeeze flow of pastes

doi:10.1016/S0377-0257(02)00011-3

Journal of Non-Newtonian Fluid Mechanics

Volume 104, Issue 1, 20 April 2002, Pages 1-32

https://doi.org/10.1016/S0377-0257(02)00011-3 Get rights and content

Abstract

The force required to squeeze a thin cylinder of paste between two approaching parallel circular plates depends upon the rheology of the paste and upon friction at the paste–plate interface. Pressure developed at the centre of the paste can move liquid relative to solids within the paste. We study the case in which the approach velocity of the plates is sufficiently slow that solid–liquid relative motion cannot be neglected during the timescale of the squeeze test. It is assumed that slip occurs at the paste–plate interface, and that the total stress within the paste may be predicted by a lubrication analysis. The deformation of the paste is close to a uniform straining motion. Gradients of liquid volume fraction across the narrow gap between the plates therefore develop slowly, and can be balanced by diffusion of liquid. The pressures and liquid volume fraction are therefore, to a first approximation, independent of the coordinate z normal to the plates. We consider two cases, both accessible to experiment. In the first, the plates are sufficiently large that the paste does not extend to their periphery. The radius of the region occupied by the constant volume of paste increases as the plates approach each other. In the second, the gap between plates of finite radius is always filled with paste which extrudes at the periphery of the plates as they approach each other. In both cases the liquid volume fraction is reduced in the central region of the paste near the axis, and is sometimes reduced to such a small value that the force required to push the plates together becomes exceedingly large. Numerical solution of the lubrication equations then requires very fine resolution near the axis, where the paste yield stress is high and pressure gradients are large. The contribution to the total force from the region near the axis becomes important and the lubrication approximation, which is poor in this region, becomes inadequate.

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 σ_ij is then an elastic deformation and a rise in the pore pressure by an amount −Bσ_kk/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 $σ$ 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, $σ =−p I −ψ I + S,$ where the matrix stress has been divided into an isotropic component $−ψ I$ and the deviatoric stresses $S$ . 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 $φ= volume of solid volume of solid + pore fluid = 1 1+e,$

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 $m ̂ = 2 φ_{0} ∫_{0}^{r̂} r ̂ ′φ(r ̂ ′) d r′.$ 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, $U ̂ ∂e ∂ t ̂_{m} =− ∂ ∂ m ̂ λα R_{0} h_{0} (1+e)e^{(1−β)+1/α} r ̂ h ̂^{1/2} −e^{−β} r ̂^{2} h ̂ φ_{0} ∂e ∂ m ̂,$ where $U ̂ = UR_{0}^{2} φ_{0} 4h_{0} D_{0},$ 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 $D= κφ^{3} μ d ψ d φ =D_{0} e^{−β} = D_{0} φ^{β} (1−φ)^{β},$ $ψ=p_{1} e^{−1/α} = p_{1} φ^{1/α} (1−φ)^{1/α},$ $κ= D_{0} μα(1−φ)^{(1−β)+1/α} p_{1} φ^{2−β+1/α},$ 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 $U ̃ ∂ ∂ t ̃ (h ̃ φ)=− 1 r ̃ ∂ ∂ r ̃ r ̃^{2} φ U ̃ −2α r ̃ λR_{0} h_{0} 1−φ φ^{1−β} − r ̃ φ^{2−β} (1−φ)^{β} ∂(h ̃ φ) ∂ r ̃,$ where $U ̃ = UR_{0}^{2} D_{0} h_{0}$ 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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Liquid–solid relative motion during squeeze flow of pastes

Abstract

Introduction

Section snippets

Pore pressure and Darcy flow of liquid within the paste

Governing equations

Numerical results for a fully filled gap, R=R0 constant

Concluding remarks

Acknowledgements

J. Non Newtonian Fluid Mech.

J. Non Newtonian Fluid Mech.

J. Non Newtonian Fluid Mech.

J. Non Newtonian Fluid Mech.

J. Non Newtonian Fluid Mech.

Adv. Colloid Interf. Sci.

J. Non Newtonian Fluid Mech.

J. Non Newtonian Fluid Mech.

Colloids Surf.

J. Colloid Interf. Sci.

J. Non Newtonian Fluid Mech.

Rheological behavior of a concentrated suspension: a solid rocket fuel simulant

J. Rheol.

Role of permeation in the linear viscoelastic response of concentrated emulsions

Langmuir

Numerical results for a fully filled gap, R=R₀ constant