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

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

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

Why Water Cannons? Police must protect themselves as well

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

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

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

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

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

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)

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

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…

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

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

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

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

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

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