Hydrodynamics of suspensions

Several experiments have been reported that indicate a significant difference between the hydrodynamic properties of suspensions and conventional viscous fluids. According to measurements made with different types of viscometers, the viscosity of the suspension in many cases exhibits anomalous behavior that is not compatible with descriptions of the suspension as a conventional Newtonian or non-Newtonian fluid [1-3]. Stratification into solid and liquid phases also plays an important role in the flow of suspensions. In particular, in the significant practical case of Poiseuille flow of an equidense suspension in a circular pipe, wall and core effects were observed. The wall effect, i. e., the migration of suspended particles toward the pipe centerline and the corresponding reduction of the solid phase concentration near the walls, was observed in suspensions of different nature (particles of different nature and form in a viscous liquid [4] and in blood, i.e., suspensions of blood corpuscles in blood plasma [3, 5]) over a wide range of values of average concentration. In contrast with the wall effect, the core effect, which involves an increase of the solid phase concentration in an annular region at a distance of 0.5-0.7 R from the pipe centerline (R is the pipe radius), was observed only for small values of the average concentration [6, 7]. An experiment [8] showed that the presence of the core effect is associated with rotation of the solid particles. When the center of gravity of each of the particles was shifted, their rotation was hindered. In the absence of rotation the particles always migrated toward the tube centerline. Many studies have been devoted to explanation of the wall effect. The magnitude of the transverse force acting on a sphere rotating in a translational viscous liquid flow was calculated in [9] (the limits of applicability of the resulting formula do not permit examination of those values of the parameters for which the core effect is observed in practice). If we substitute the expression for force into the particle equation of motion and integrate, we find that the particle trajectory approaches the tube axis asymptotically [10], regardless of the initial position of the particle. Another approach is the averaged description of the suspension behavior. The Navier-Stokes equation, with a coefficient of viscosity that depends in known fashion on the local solid phase concentration, was written as the equation of motion for the steady flow of a suspension in a circular tube. The variational principle (the principle of minimum energy dissipation) was formulated to find the concentration distribution. The wall effect was also obtained using this approach [11, 12]. An analogous study using the Kessonmodel yielded practically no results [13]. The complete system of equations of two-fluid hydrodynamics was recently constructed in which the fluid and the dispersed phase are considered as two interpenetrating, interacting continua, with rotation and deformation of the dispersed phase particles being taken into consideration (Nguen Van D'ep, Candidate's dissertation: Some questions on the Hydrodynamics of Structured Fluids [in Russian], Voronezh State University, Voronezh, 1968). However, the practical use of this system for solving, for example, the Poiseuille problem appears to be difficult, since it is necessary to know in detail the interaction forces between the phases. In the present study we use the single-fluid approximation, i.e., the quantities introduced (velocity, density, and so on) characterize the motion of the suspensions as a whole, and not any single phase. Only two quantities characterize the solid phase proper: the bulk concentration and the local angular velocity of the solid particles. This approach does not require knowledge of the interaction forces between the phases, and the final equations are considerably simpler than those for the two-fluid description. The thermodynamics of irreversible processes is used to construct a closed system of equations that includes the diffusion equations and the generalized moment of momentum equation, which make it possible to find the three-dimensional distribution of the concentration and the angular velocity of the solid particles. The generalized Fick law, which contains three additional diffusion coefficients, is obtained. In contrast with classical models, the viscous stress tensor is asymmetric, and its antisymmetric part is proportional to the difference in the angular rates of rotation of the solid particles and the suspension as a whole. From these equations follow under certain particular assumptions the equations of the theory of a fluid with internal rotation [14]. As an example of the application of the theory, the Poiseuille problem of flow in a flat channel is solved. The concentration distributions obtained agree well with the experimental data described above.