Statistical theory of sedimentation of disordered suspensions.

An analytical treatment for the sedimentation rate of disordered suspensions is presented in the context of a resistance problem. From the calculation it is confirmed that the lubrication effect is important in contrast to the suggestion by Brady and Durlofsky (Phys.Fluids 31, 717 (1988)). The calculated sedimentation rate agrees well with the experimental results in all range of the volume fraction. 05.60.+w,05.40.+j,47.55.Kf,83.70.Hg Typeset using REVTEX e-mail: [email protected] e-mail: [email protected] Present address 1 The sedimentation of disordered suspensions is important in both technology and laboratory [1]. The role of the sedimentation is relevant to the current topics of statistical mechanics such as fluidized beds of gas-solid or liquid-solid mixtures [3–5], and the density waves in the granular flows in vertical tubes [6]. We believe the subject is a fundamental one in fluid mechanics [7]. The rate of sedimentation for disordered suspensions under gravity has yet to be determined theoretically except for a problem for dilute spheres with hard core interactions at a small Reynolds number [1,8]. Our present understanding of theoretical studies of sedimentation of monodisperse random suspensions can be summarized as follows. Batchelor [8] has calculated the sedimentation rate in the dilute limit of hard core particles with the radius a based on the following assumptions: (i) The rate can be obtained from the combination of the mobility matrix of two particles and the two-body correlation function geq(r) where r is the relative distance of particles, and (ii) the correlation function is assumed to be geq(r) = θ(r − 2a) , where θ(x) is the step function θ(x) = 1 for ≥ 0 and θ(x) = 0 otherwise. His result at the volume fraction φ can be written as U(φ)/U0 = 1 − 6.55φ + O(φ ) for φ → 0, where U(φ) the sedimentation velocity at φ and U0 is the equilibrium sedimentation velocity of one particle. The result of Batchelor consists of two parts: one is 1 − 5φ from the Rotne-Prager tensor which represents the effects of long-range hydrodynamic interaction, and another −1.55φ from the lubrication, the hydrodynamic repulsive, force. Extensions of this dilute theory to concentrated suspensions require the account of many body hydrodynamic interactions. A generalization [9], based on the method of O’Brien [10] predicts negative sedimentation rate for φ > 0.27. Brady and Durlofsky [11] have also obtained a negative sedimentation rate for φ > 0.23 when they adopt well accepted correlation function geq(r) for concentrated suspensions. As a result, they claim that the Rotne-Prager approximation actually captures the correct features of sedimentation and ignored all of the contributions from the lubrication force. We feel, however, the statement by Brady and Durlofsky [11] unacceptable, because there is no reason to ignore lubrication effects in the dilute limit [8]. On the other hand, Beenakker and Mazur [12,13] also calculated the sedimentation rate based on an effective 2 medium approximation and multipole expansions. Although they did not present an explicit expression of the sedimentation rate, Ladd [14] indicated that their result is better than the result by Brady and Durlofsky [11] for concentrated suspensions. In this Rapid Communication, we wish to demonstrate the relevance of the lubrication force and improve the theory by Brady and Durlofsky [11]. We also clarify the relationship between our theory and that by Beenakker and Mazur [11,12]. The problem of sedimentation of N particles with the radius a at low Reynolds numbers is equivalent to obtaining the resistance matrix R or the mobility matrix M in U = 1 6πμa M · F, M = R, (1) where U and F denote the sets of the velocity field of N particles and the force exerted on N particles, respectively, and μ is the shear viscosity. These mobility and resistance problems are not easy to solve even numerically. One of the most successful numerical methods, the Stokesian dynamics, has been developed by Brady and his coworkers [15,16]. The extension by Ladd [14] also follows a similar algorithm to the Stokesian dynamics. They decouple the resistance matrix into the far-field part (M) and the lubrication part R as R = (M) + R, (2) where R is calculated by the pairwise additive expression of the two-body lubrication matrix R 2B = R2B−(M ∞ 2B) . The resistance matrix is calculated as a function of the particle configuration at each numerical step. Then the force exerted on spheres and consequently the equation of motion are obtained. The success in the Stokesian dynamics suggests that the problem for sedimentations should be considered based on a resistance picture. In fact, some unphysical results of simulations based on a mobility picture supports this statement. We may understand the relevance of a resistance picture as follows. Since the contribution of the lubrication is proportional to the number of particles, as will be shown, the direct addition of the lubrication for the mobility cannot avoid a negative sedimentation rate. In other words, the linear contribution of the lubrication to the drag is reasonable, while the 3 linear addition of the lubrication to the mobility cannot produce any nonlinear complicated motion of particles in experiments. Thus, we are not surprized by the failure of direct generalizations of Batchelor’s theory which is described as a mobility problem. We must calculate the sedimentation rate in the context of a resistance problem. The problem is, thus, reduced to obtaining < (M∞) > + < R >, where the bracket is the average over the particle configurations. Note that < (M) > and < R > are the scalar quantities. The far-field part can be calculated from < (M) >≃< M >= M̃(k = 0), where M̃(k) is defined by