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.
%0 Journal Article
%1 citeulike:5990493
%A Mason, Max
%A Weaver, Warren
%D 1924
%I American Physical Society
%J Physical Review Online Archive (Prola)
%K 76t20-suspensions
%N 3
%P 412--426
%R 10.1103/physrev.23.412
%T The Settling of Small Particles in a Fluid
%U http://dx.doi.org/10.1103/physrev.23.412
%V 23
%X 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.
@article{citeulike:5990493,
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.}},
added-at = {2017-06-29T07:13:07.000+0200},
author = {Mason, Max and Weaver, Warren},
biburl = {https://www.bibsonomy.org/bibtex/2b1aa9c1505038e178a90b777bd17c598/gdmcbain},
citeulike-article-id = {5990493},
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citeulike-linkout-0 = {http://dx.doi.org/10.1103/physrev.23.412},
citeulike-linkout-1 = {http://link.aps.org/abstract/PR/v23/i3/p412},
citeulike-linkout-2 = {http://link.aps.org/pdf/PR/v23/i3/p412},
comment = {(private-note)circulated by Mike Hudson, from Bob Cornell},
doi = {10.1103/physrev.23.412},
file = {mason_24_settling_33767.pdf},
interhash = {fd7d2445c5bc48c1f819c54f3497285f},
intrahash = {b1aa9c1505038e178a90b777bd17c598},
journal = {Physical Review Online Archive (Prola)},
keywords = {76t20-suspensions},
month = mar,
number = 3,
pages = {412--426},
posted-at = {2009-10-23 00:17:54},
priority = {2},
publisher = {American Physical Society},
timestamp = {2017-06-29T07:13:07.000+0200},
title = {{The Settling of Small Particles in a Fluid}},
url = {http://dx.doi.org/10.1103/physrev.23.412},
volume = 23,
year = 1924
}