# Twice the speed of water carries objects 64 times larger. Why?

From a source1 I regard as reputable, I heard the following assertion (without explanation):

If the water in a stream flows twice as fast, it can carry objects (pebbles/etc) sixty four times larger.

Bearing in mind a related question: Power vs. Speed - Indoor rowing which shows that to double your speed, you require eight times the power.
And further, the energy generated from a wind-turbine grows with the cube of the air velocity.

But back to the river.
It seems analagous to the rowing question, which is drag (the rock or pebble) is subjected to eight times the energy with twice the flow rate.

I can't see what factors I am missing to account for the required energy that statement asserts.
What am I missing?

1 How To Read Water: Clues & Patterns from Puddles to the Sea - Tristan Gooley

• Does the author mention a plausible source for his assertion?
– Deep
Oct 8, 2020 at 6:01

By 64 times larger, I would expect you mean 64 time more massive, or 4 times longer, wider, and taller.

An object may be moved by rolling or sliding.

If it isn't perfectly round, it will tend to lie in an orientation where its center of gravity is as low as possible. Moving water will tip it over onto another side. If it is 4 times larger, the torque needed will be 64 times larger because of the bigger mass, with another factor of 4 because the center of mass is 4 times farther from the edge. So the torque must be 256 times bigger.

The source of the force is diverting water from a straight path for larger Reynolds numbers (larger objects, faster flow, lower viscosity) or surface friction for low Reynolds numbers (smaller objects, slower flow, more viscosity). A stream rolling a rock would likely flow fast enough to be turbulent, which is a sign of high Reynolds number.

A rock 4 times larger has a cross section 16 times larger, and diverts that much more water. For high Reynolds number, flowing water exerts a force proportional to $$v^3$$, so doubling the water speed increases the force by a factor of 8. So the force is 128 times bigger. The center of the cross sectional area is 4 times taller, so the torque is 512 times bigger. You could roll a slightly bigger rock than your source says.

• But friction does not depend on contact area in general, just on normal force. Oct 7, 2020 at 9:19
• @AgniusVasiliauskas - You are right. Since you have answered correctly for friction, I removed it from my answer. Oct 7, 2020 at 10:06

For water would be able to move a pebble, drag force of water should be equals to static friction of pebble on ground :

$$F_s = F_d$$ or $$\mu_smg = 1/2 \rho_sv^2C_DA$$

Expressing pebble mass in terms of it's density and volume, substituting cross-section area and solving for $$v^2$$ gives :

$$v^2 = \frac {8\mu_s\rho_kRg}{3\rho_sC_D}$$

or, marking non-important terms as coefficient $$\alpha$$ :

$$\boxed{ \alpha R = v^2}$$

Expressing this equation as ratios, gives :

$$\frac {R_2}{R_1} = \frac {v_2^{~2}}{v_1^{~2}}$$

or, noticing that $$v_2 = v_1 N$$, gives :

$$\frac {R_2}{R_1} = N^2$$

Thus, increasing water flow velocity two times ($$N=2$$), water roughly is able to carry pebbles $$4\times$$ bigger radius. Thus, I don't believe that it can be $$64\times$$ (or $$N^{6}$$) anyhow as your reference author claims. Of course there are subtle things as flow laminarity, turbulence, lubrication, etc, which may increase this number a bit, but it's very hard to believe that it may increase to author claimed level.

Objects “carried” by a stream are likely to be rocks which are rolling along the bottom. Larger rolling rocks would not require a much larger force and would be impeded less by irregularities in the bottom of the stream.

• This doesn't really explain anything Oct 6, 2020 at 16:06