Scaling behaviours in settling process of fractal aggregates in water

We investigate the effect of permeant flow on the sedimentation of porous fractalaggregates in water. Our theoretical analysis gives explicit calculations on the scaling behaviours of settling velocities, taking into account the fractality through a proper permeability for fractal aggregates. The calculated results for the scaling behaviours of settling velocities fit remarkably well with experimental data. The analytic expression for the settling velocity provides a criterion for determining fractal dimensions Df of aggregates from sedimentation experiments. Copyright c © EPLA, 2007 Introduction. – The transport process of aggregates generated in fluids has been the subject of numerous studies ranging from basic to applied sciences [1], including biophysics [2]. The scaling relation between settling velocities ua and sizes a of aggregates such as ua ∼ a has been confirmed by settling experiments under the gravitational force [3–8]. In some cases, fractal dimensions Df obtained by using Stokes’ law ua ∼∆ρ(a)a2 are in agreement with those determined by direct observations [9]. We need, however, caution when applying Stokes’ law to permeable porous fractal-systems since it is in principle applicable to impermeable systems. Many theoretical attempts have been made to elucidate the interior flow of fractal aggregates [10–16]. Most of works have treated the hydraulic permeability as a function of the porosity P , ignoring the characteristics of fractal structures. In this letter, Brinkman equations are employed to obtain the analytic expression for the settling process of fractal aggregates in water. According to the close analogy between the electrical conductance and the fluid flow in porous fractal-media, we consider the effect of permeant flow through connected (backbone) channels in aggregates on the settling velocity. Our analysis gives theoretical foundation that the scaling relation between the settling (a)E-mail: [email protected] (b)E-mail: [email protected] velocities and the sizes of aggregates holds. In addition, we demonstrate that Stokes’ law is applicable under certain circumstances to derive fractal dimensions of aggregates generated in water. Theoretical analysis. – A feature of settling of aggregates is that streamlines traverse porous aggregates. The Brinkman equations are established to describe the permeant flow [17], where the porosity-dependent permeability k is introduced for describing the permeant flow in a spherical aggregate of radius a. The Brinkman equations are given by the following set of equations: ∇pi = μ∇ui − μ k (ui −ua) ; r a, ∇·ui = 0, (1) where pi denotes the pressure, ui the velocity of fluid inside an aggregate, ua the velocity of the aggregate, μ the kinetic viscosity of a fluid, respectively. The governing equation for the flow outside of an aggregate is given by Stokes equations, ∇po = μ∇2uo; r a, and ∇·uo = 0, where the subscripts o defines outside an aggregate. The vorticity is defined aswi =∇×ui,wo =∇×uo inside and outside of an aggregate, respectively. For convenience, we formulate the problem in spherical coordinates [r, θ, φ]. Since the flow is axial-symmetric, the fluid velocity in the φ direction is zero. Furthermore, we introduce the