Article,
The Settling of Small Particles in a Fluid
Physical Review, 23 (3): 412--426 (Mar 1, 1924)
DOI: 10.1103/physrev.23.412
Abstract
Settling of small particles in a fluid; mathematical theory .—Small particles immersed in a liquid experience a motion which is the combination of a steady gravitational drift and a Brownian movement. If there are space variations in the density of distribution of particles; the Brownian movement produces a diffusion which tends to equalize the density. In the steady state the density n of particles is an exponential function of x ; the distance below the surface of the liquid. This paper investigates the manner in which the steady state is established . A consideration of the combined effect of fall and diffusion leads to a partial differential equation for the number density of particles as a function of depth and time. A set of special solutions is obtained in terms of which a solution satisfying initial and boundary conditions can be expressed. (1) Liquid of finite depth . The solution is obtained for a liquid of finite depth with an arbitrary initial distribution n 0 = f ( x ). For the case of uniform initial distribution a reduced form of the solution is obtained which contains a single parameter. This one parameter family of curves is plotted; and from these curves; either directly or by interpolation; may be obtained the density distribution at any time for a solution of any depth; density; and viscosity; and for particles of any size and density. For small values of t ; since the solution obtained converges slowly; an image method is used to obtain an integral formula for the density. (2) Liquid of semi-infinite or infinite depth . In the case of a liquid of infinite depth the solution for an arbitrary initial distribution is expressed by the Fourier integral identity. The case of zero initial density for negative x ; and constant initial density for positive x is calculated; as is also the case of particles initially uniformly distributed over a layer of depth h . In the case of a liquid extending from x =0 to x =∞; the boundary conditions are satisfied by assuming a suitable fictitious initial distribution over the range from x =-∞ to x =0. The cases of uniform initial distribution; and initial distribution over a layer; are calculated. The latter case; while derived for a liquid of semi-infinite depth; gives approximately the distribution of density during the settling of a layer of particles initially distributed uniformly over a depth h at the upper end of a very long column of liquid.
