特に意味はないですが、Rubeyの式とVan Rijnの式を比較してみました。

import numpy as np
import holoviews as hv

hv.extension('bokeh', logo=False)

Settling velocity Equation

Rubey

https://www.ajsonline.org/content/s5-25/148/325

$\begin{align} w_0 &= \sqrt{sgd}F \\ F &= \sqrt{ \dfrac{2}{3} + \dfrac{36 \nu^2}{sgd^3}} -\sqrt{\dfrac{36 \nu^2}{sgd^3}} \end{align}$

$\begin{align} \nu: \textrm{Kinematic viscosity of water} \end{align}$

def rubeyw0(dd, nu=10**(-6)):
    rhosw = 1.65 # Specific gravity in water of sand 
    g = 9.8 # acceleration of gravity
    tmp = 36.0*nu**2/rhosw/g/dd**3
    F = np.sqrt(2.0/3.0 + tmp) - np.sqrt(tmp)
    
    return np.sqrt(rhosw*g*dd)*F

Van Rijn

https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9429%281984%29110%3A11%281613%29

$\begin{align} w_0 &= \left\{ \begin{array}{ll} \dfrac{1}{18}\dfrac{sgd^2}{\nu} &( d < 0.1 \textrm{mm}) \\ 10 \dfrac{\nu}{d} \left[ \left(1+0.01\dfrac{sgd^3}{\nu^2} \right)^{1/2} - 1 \right] &( 0.1 \leq d \leq 1 \textrm{mm}) \\ 1.1\sqrt{sgd} &( d > 1\textrm{mm}) \end{array} \right. \end{align}$

$\begin{align} \nu: \textrm{Kinematic viscosity of water} \end{align}$

def vanRijinw0(dd, nu=10**(-6)):
    rhosw = 1.65 # Specific gravity in water of sand 
    g = 9.8 # acceleration of gravity
    if dd < 0.0001 :
        w0 = float(1/18)*rhosw*g*dd**2/nu
    elif dd <= 0.001 :
        w0 = float(10)*nu/dd*( (1+0.01*rhosw*g*dd**3/nu**2)**0.5 -1 )
    else:
        w0 = np.sqrt(rhosw*g*dd)*1.1
    
    return w0

figure

dm = np.logspace(-2,1,10000)
dm /= 1000

w01 = np.array( [rubeyw0(dmp) for dmp in dm] )
w02 = np.array( [vanRijinw0(dmp) for dmp in dm] )

g = hv.Curve((dm*1000,w01),label='Rubey') \
   *hv.Curve((dm*1000,w02),label='Van Rijin').options(line_dash='dashed', line_width=3) 
g.options(logx=True,  logy=True
          , xlabel='diameter[mm]', ylabel='setling velocity[m/s]'
          , legend_position='bottom_right', show_grid=True, width=500,height=450)