Settling Velocities of Circular Cylinders at Low Reynolds Numbers

Experiments have been conducted on the settling rates of cylindrical-shaped grains in a fluid, having application to the settling of certain heavy minerals and fecal pellets in water. The settling objects were fabricated from circular glass rods with diameters 3-8 mm, giving length to diameter ratios ranging 2-12. Glycerine was used as the fluid in the experiments, yielding Reynolds numbers equivalent to quartz-density silt and fine sand grains of similar shape settling in water. Analysis of the data shows that the settling velocity \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{wasysym} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$${\mathrm{w_{s}}}$$\end{document} can be calculated with the modified Stokes relationship \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{wasysym} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$${\mathrm{w_{s} =0.0790 \frac{1}{\mu}(\rho_{s}-\rho)gL^{2}\left({\frac{L}{D}}\right)^{-1.664}}}$$\end{document} where μ is the fluid's absolute viscosity, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{wasysym} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$$\rho_{{\mathrm{s}}}$$\end{document} and p are respectively the grain and fluid densities, and L and D are the length and diameter of the circular cylinder. This equation is applicable only at low Reynolds numbers, approximately \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{wasysym} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$${\mathrm{Re = \rho w_{s}L/\mu < 2}}$$\end{document}, analogous to the Stokes region for the settling of spheres. If ellipsoidal-shaped grains are considered as well as cylinders, then the settling velocity can be calculated with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{wasysym} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$${\mathrm{w_{s} = \frac{1}{18}\frac{1}{\mu}(\rho_{s}-\rho)gD_{n}^{2}E^{0.380}}}$$\end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{wasysym} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$${\mathrm{D_{n}}}$$\end{document} is the grain nominal diameter (diameter of sphere having the same volume) and E is a measure of grain shape as defined by Janke (1966).