Note on the rheology of a dilute suspension of dipolar spheres with weak Brownian couples

1972 ◽  
Vol 56 (4) ◽  
pp. 803-813 ◽  
Author(s):  
E. J. Hinch ◽  
L. G. Leal
Keyword(s):  
Rheological Properties ◽  
Applied Field ◽  
Asymptotic Form ◽  
Present Note ◽  
Orientation Distribution ◽  
Dipole Axis ◽  
Brownian Rotation ◽  
Bulk Stress ◽  
Dilute Suspension ◽  
Dipolar Particles

A problem of theoretical interest in suspension rheology is the calculation of bulk rheological properties for a dilute suspension of spherical dipolar particles in the presence of weak Brownian rotation, when the applied field is perpendicular to the local vorticity of the bulk flow. In the present note, we determine the asymptotic form for the orientation distribution of the dipole axis in the limit of weak Brownian motion and use this distribution to determine the corresponding rheological properties of the suspension. The bulk stress is then discussed in terms of an effective viscosity for shear flow.

The stress generated in a non-dilute suspension of elongated particles by pure straining motion

1971 ◽  
Vol 46 (4) ◽  
pp. 813-829 ◽  
Author(s):  
G. K. Batchelor
Keyword(s):  
Theoretical Formula ◽  
Rigid Sphere ◽  
Cross Sectional ◽  
Outer Flow ◽  
Rigid Particles ◽  
Bulk Stress ◽  
Order Of Magnitude ◽  
Sectional Plane ◽  
Dilute Suspension ◽  
A Minor

In a pure straining motion, elongated rigid particles in suspension are aligned parallel to the direction of the greatest principal rate of extension, provided the effect of Brownian motion is weak. If the suspension is dilute, in the sense that the particles are hydrodynamically independent, each particle of length 2l makes a contribution to the bulk deviatoric stress which is of roughly the same order of magnitude as that due to a rigid sphere of radius l. The fractional increase in the bulk stress due to the presence of the particles is thus equal to the concentration by volume multiplied by a factor of order l2/b2, where 2b is a measure of the linear dimensions of the particle cross-section. This suggests that the stress due to the particles might be relatively large, for volume fractions which are still small, with interesting implications for the behaviour of polymer solutions. However, dilute-suspension theory is not applicable in these circumstances, and so an investigation is made of the effect of interactions between particles. It is assumed that, when the average lateral spacing of particles (h) satisfies the conditions b [Lt ] h [Lt ] l, the disturbance velocity vector is parallel to the particles and varies only in the cross-sectional plane. The velocity near a particle is found to have the same functional form as for an isolated particle, and the modification to the outer flow field for one particle is determined by replacing the randomly placed neighbouring particles by an equivalent cylindrical boundary. The resulting expression for the contribution to the bulk stress due to the particles differs from that for a dilute suspension only in a minor way, viz. by the replacement of log 2l/b by log h/b, and the above suggestion is confirmed. The relative error in the expression for the stress is expected to be of order (log h/b)−1. Some recent observations by Weinberger of the stress in a suspension of glass-fibre particles for which 2l/h = 7·4 and h/2b = 7·8 do show a particle stress which is much larger than the ambient-fluid stress, although the theoretical formula is not accurate under these conditions.

An averaged-equation approach to particle interactions in a fluid suspension

1977 ◽  
Vol 83 (4) ◽  
pp. 695-720 ◽  
Author(s):  
E. J. Hinch
Keyword(s):  
Constitutive Relations ◽  
Equations Of Motion ◽  
Fixed Bed ◽  
Hydrodynamic Interactions ◽  
Particle Interactions ◽  
Consistent Field ◽  
Averaged Equations ◽  
Bulk Stress ◽  
Self Consistent Field ◽  
Dilute Suspension

Earlier ideas are combined to produce a systematic approach both to forming the bulk equations of motion of a dilute suspension and to calculating the overall hydrodynamic interactions between the suspended particles. Equations governing averaged field quantities are derived by taking ensemble averages of the conservation laws and constitutive relations. The bulk equations thus produced contain a term in which the averaging is performed holding one particle fixed. If now the same prescription is applied to fields averaged with one particle fixed, equations are produced containing a term averaged with two particles fixed, and so on up an infinite hierarchy. The hierarchy can be truncated in an asymptotic analysis for small particle concentrations.This approach to the mechanics of suspensions is illustrated by applying it to three problems which have already been well studied by different methods. The problems concern the first effects of hydrodynamic interactions on the bulk stress and sedimentation velocity of a free suspension, and on the permeability of a fixed bed. Earlier results are recovered in a new light. Multiparticle effects, which before have occurred as divergent sums, are seen to arise because the suspension described by the averaged equations assumes a viscosity and density different from the solvent, or in the case of the fixed bed because the suspension starts behaving as a porous medium instead of as a Newtonian solvent. A close connexion is thus revealed between the averaged-equation description of the interactions and a self-consistent-field model.

The effect of Brownian motion on the bulk stress in a suspension of spherical particles

1977 ◽  
Vol 83 (1) ◽  
pp. 97-117 ◽  
Author(s):  
G. K. Batchelor
Keyword(s):  
Brownian Motion ◽  
Probability Density ◽  
Volume Fraction ◽  
Joint Probability ◽  
Small Degree ◽  
Spherical Particles ◽  
Stress System ◽  
Bulk Stress ◽  
Dilute Suspension ◽  
Thermodynamic Forces

The effect of Brownian motion of particles in a statistically homogeneous suspension is to tend to make uniform the joint probability density functions for the relative positions of particles, in opposition to the tendency of a deforming motion of the suspension to make some particle configurations more common. This smoothing process of Brownian motion can be represented by the action of coupled or interactive steady ‘thermodynamic’ forces on the particles, which have two effects relevant to the bulk stress in the suspension. Firstly, the system of thermodynamic forces on particles makes a direct contribution to the bulk stress; and, secondly, thermodynamic forces change the statistical properties of the relative positions of particles and so affect the bulk stress indirectly. These two effects are analysed for a suspension of rigid spherical particles. In the case of a dilute suspension both the direct and indirect contributions to the bulk stress due to Brownian motion are of order ø2, where ø([Lt ] 1) is the volume fraction of the particles, and an explicit expression for this leading approximation is constructed in terms of hydrodynamic interactions between pairs of particles. The differential equation representing the effects of the bulk deforming motion and the Brownian motion on the probability density of the separation vector of particle pairs in a dilute suspension is also investigated, and is solved numerically for the case of relatively strong Brownian motion. The suspension has approximately isotropic structure in this case, regardless of the nature of the bulk flow, and the effective viscosity representing the stress system to order ϕ2 is found to be \[ \mu^{*} = \mu(1+2.5\phi + 6.2\phi^2). \] The value of the coefficient of ø2 for steady pure straining motion in the case of weak Brownian motion is known to be 7[sdot ]6, which indicates a small degree of ‘strain thickening’ in the ø2-term.

The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles

1972 ◽  
Vol 52 (4) ◽  
pp. 683-712 ◽  
Author(s):  
E. J. Hinch ◽  
L. G. Leal
Keyword(s):  
Brownian Motion ◽  
Shear Flow ◽  
Spherical Particles ◽  
Aspect Ratios ◽  
Bulk Stress ◽  
Strong Shear ◽  
Steady Shear Flow ◽  
Dilute Suspension ◽  
Limiting Case ◽  
Particle Aspect Ratio

The effect of rotary Brownian motion on the rheology of a dilute suspension of rigid spheroids in shear flow is considered for various limiting cases of the particle aspect ratio r and dimensionless shear rate γ/D. As a preliminary the probability distribution function is calculated for the orientation of a single, axisymmetric particle in steady shear flow, assuming small particle Reynolds number. The result for the case of weak-shear flows, γ/D [Lt ] 1, has been known for many years. After briefly reviewing this limiting case, we present expressions for the case of strong shear where (r3 + r−3) [Lt ] γ/D, and for an intermediate regime relevant for extreme aspect ratios where 1 [Lt ] γ/D [Lt ] (r3 + r−3). The bulk stress is then calculated for these cases, as well as the case of nearly spherical particles r ∼ 1, which has not hitherto been discussed in detail. Various non-Newtonian features of the suspension rheology are discussed in terms of prior continuum mechanical and experimental results.

On the creeping motion of two arbitrary-sized touching spheres in a linear shear field

1973 ◽  
Vol 59 (2) ◽  
pp. 209-223 ◽  
Author(s):  
Avinoam Nir ◽  
Andreas Acrivos
Keyword(s):  
Entire Range ◽  
Stokes Equations ◽  
Material Point ◽  
Shear Field ◽  
Bulk Stress ◽  
Linear Shear ◽  
Dilute Suspension ◽  
Freely Suspended ◽  
The Individual ◽  
Creeping Motion

The Stokes equations describing the creeping motion of two arbitrary-sized touching spheres are solved exactly through the use of tangent-sphere coordinates. For the case of a linear shear field at infinity, explicit results covering the entire range of size ratios are presented for: (a) the forces and torques on the aggregate; (b) the hydrodynamic forces on the individual spheres comprising a freely suspended aggregate, which are in general non-zero; (c) the contribution of the pair to the bulk stress of a dilute suspension; and (d) under free suspension conditions, the velocity of any material point relative to that of the undisturbed flow.

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