New confinement effects on the viscosity of suspensions

We present rheological experiments of confined suspensions at moderate concentrations. The analysis is carried out in the framework of a previous study on particle suspensions [Davit and Peyla, EPL, 83, 64001 (2008)] where simulations revealed the presence of unusual effects attributed to confinement, i.e. when the gap size (h) becomes closer to the particle size (d). Deviations from the usual viscosity trends were found. The present work investigates these features further and confirms the important role of the confinement. Extensions are made from the classical approach to the case of confined suspensions where the importance of the reduced gap h d is taken into account. Introduction. – The viscosity of solid suspensions and pastes is important for industrial applications such as chemical, petrol engineering, food industry as well as in the field of environment [1] where more complex fluids are investigated. The rheology of suspensions is an evolving field where new measurement techniques are of interest [2,3] but where classical rheometrical tools are still proper devices for carrying out accurate measurements. Even when mastering the rheology of certain fluids, it can become quite difficult to study particular complex flows, such as the ones possessing free surfaces or those generated by complex geometries. An important recent topic is the one of confined flows [4–8] where interfaces or the presence of walls can give rise to intriguing new regimes. This study of confinement has been addressed previously [7] and proposed a possible way to determine viscosity or yield stress parameters using squeezing flows (implying smaller and smaller gaps), assuming that steady states were obtained. But this method raised the question of such steady states in particular since clusters of particles in complex fluids can lead to reorganization of substructures involving longer waiting times for a steady state [9]. Another drawback of the latter method [7] was that a known constitutive equation was needed to derive proper results (in fact a Herschel–Bulkley model). Most studies on confined suspensions deal usually with concentrated systems [7, 8, 10]. Another recent approach of confinement was based on the study of fluctuations and oscillations in concentrated colloidal suspensions leading to shear–thickening [6]. Finally, concentrated emulsions can also behave differently in confined flows [2] and exhibit a characteristic cooperation length which is concentration– dependent. In a recent study, the authors revealed the presence of anomalous effects due to the confinement of suspensions at moderate concentrations [11]. These effects were attributed to the presence of walls that change the nature of hydrodynamic interactions between the particles. The steady shear viscosity η was shown to behave differently from theories for unconfined suspensions. In general, previous authors [12–15] have proved that for moderate concentrations, the viscosity of a non-confined suspension of hard spherical particles behaves as η = η0(1 + a ∞ 1 φ+ a ∞ 2 φ ), (1) where η0 is the suspending fluid viscosity, φ is the volume concentration, and the symbol “∞“ denotes the limit for unbounded domain (i.e. not confined). Eq. (1) is a Taylor expansion where the linear term (a1 φ) holds for very dilute suspension and a1 = 2.5 has been calculated by Einstein [12, 13]. The quadratic term (a2 φ ) represents the contribution of the two-body hydrodynamic interaction. Batchelor and Green [14] gave the first good estimation of a2 = 5.2 ± 0.3 for homogeneous suspensions, this term being calculated more precisely later [15] and found close to a2 = 5. But the relationship (1) seems to fail as suspensions are confined, so that the usual con-