Fig 9 Above is the plot of Force vs. Flow rate at a constant ball speed of 0.38 m/sec. The line tends to diverge to a critical gas flow rate of 78. This plot resembles one of a second order phase transition.

Distnace From Top Surface (in)

16 14 12 10 8 6 4 2 0

solid

liquid

0

50

100

150

Flow Rate

# Fig 10

This graph is distance vs. flow rate. This shows how far the ball has sunk at different flow rates. This graph has the same critical flow rate at about 90 as in figure 7. The graph shows the phase transition line where the sand is a solid or a liquid.

# The data lead us to a probable theoretical conclusion. Knowing that the function

for the viscous force is 2.3 ((W-Wcrit)/Wcrit) ^{–1}, the drag force factors into the product of

two equations. The drag force is equal to a function of velocity times the viscous force

function of the flow rate. The calculated function turns out to be

# F ( v,w ) = ( 1.9 + 0.064 log (v))

2.3 ((W-78)/78)

–

1

where F is a function of velocity and flow rate. We are unsure about the log part of this

equation, but the viscous function of the flow rate seems to agree with our data quite

well. This allows us to cover all the data points with this single function of velocity and

gas flow rate. (shown in figure 11).

12