1. A Study In Forced Submission
Crowd Control
A Study In Forced Submission
Brant
Brant Bennett, Mariana Dionisio, Sami Karam, Amy Robertson, Michael Steele

2. Rioting and Anarchy Order = Safety
Coercion May Be Necessary to Maintain Order
Brant
Police and Protesters Clash
April 20, 2001 Quebec

3. Why Water Cannons? Police’s job - maximize safety Guns Are Lethal
Less Lethal = Better
Brant

4. Why Water Cannons?
Police must protect themselves as well

5. Safe For Public & Police
Why Water Cannons?
Water cannons
effective
non-lethal
Safe For Public & Police

6. Problem: Given: v = 42 ft/s A = 0.1 ft2/s vertical shield
horizontal, steady flow
Find: Force necessary to hold a shield in place against a horizontal jet of water

7. Problem: Assume: Water Density (ρ) = 1.94 slug/ft3
Steady, Incompressible, Plug Flow
Plan:
Momentum Balance Equation
Michael

8. Solution: = Rate of accumulation of momentum
+ Rate of momentum IN
- Rate of momentum OUT
Sum of Forces Acting On Control Volume =
Sami

9. Solution: ΣFx = б/бt òòòCV ρdV + òòCS ρvx (v.n)dA
Momentum Balance Equation
ΣFx = б/бt òòòCV ρdV + òòCS ρvx (v.n)dA
ρ - density vx - speed, x axis
v - velocity vector n - area vector
CV - control volume CS - control area(s)
Area vector points out by convention

10. Solution: 0 (Steady State) Integrate only where there is mass flux
ΣFx = б/бt òòòCV ρdV + òòCS ρvx (v.n)dA
0 (Steady State)
Integrate only where there is mass flux
(water crossing boundary)

11. Solution: ΣFx = òòCS ρvx (v.n)dA = òòCS1 ρvx (v.n)dA
Mass flux only at: CS1, CS2, and CS3
*Red indicates boundary of the flow

12. Solution: ΣFx = òòCS1 ρvx (v.n)dA + òòCS2 ρvx (v.n)dA
v.n = |v|*|n| cosΘ (where Θ is the angle between v and n) so…

13. Solution: Is identically equal to
òòCS1 ρvx (v.n)dA + òòCS2 ρvx (v.n)dA
+ òòCS3 ρvx (v.n)dA
Is identically equal to
òòCS1 ρvxvncosΘ1 dA + òòCS2 ρvxvncosΘ2 dA
+ òòCS3 ρvxvncosΘ3 dA

14. òòCS1 ρvxvncosΘ1 dA + òòCS2 ρvxvncosΘ2 dA
Solution:
vx (CS2) = 0 vx (CS3) = 0
so ΣFx =
òòCS1 ρvxvncosΘ1 dA + òòCS2 ρvxvncosΘ2 dA
+ òòCS3 ρvxvncosΘ3 dA

15. ρcosΘ1 constant with respect to x and y
Solution:
ΣFx = òòCS1 ρvxvncosΘ1 dA
ρcosΘ1 constant with respect to x and y
ΣFx = ρcosΘ1òòCS1 vxvn dA
Θ1 = Π => cos Θ1 = -1 1

16. Solution: ΣFx = ρcosΘ1òòCS1 vxv dA = -ρòòCS1 vxv dA
v constant with x & y (plug flow, steady state) -1

17. Solution: ΣFx = -ρvxvòòCS1 dA = -ρvxvA vx = v (flow on x axis only)
Resultant Force (R) = ΣFx = -ρv2A

18. Effective means of submission less lethal than gun
Solution: R = -ρv2A = -(1.94 slug/ft 3)(42ft/s)2(0.1ft2)(lbfs2/slug/ft)
Minus sign indicates momentum flux in negative x-direction, would have to push on the shield with ~340 lb of force.
Not many people can do that
-342 lbf
Effective means of submission less lethal than gun