The Settling of Small Particles in a Fluid

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 $n_0=f\left(x\right)\mathrm{}$. 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 $x=0\mathrm{}$ to $x=\infty$, the boundary conditions are satisfied by assuming a suitable fictitious initial distribution over the range from $x=-\infty$ to $x=0\mathrm{}$. 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.