Dissipative Descent: Rocking and Rolling down an incline. Bill Young, Neil Balmforth, John Bush & David Vener

Transcription

1 Dissipative Descent: Rocking and Rolling down an incline Bill Young, Neil Balmforth, John Bush & David Vener

2 The snail ball from Grand Illusions - $50

3 A small metallic gold ball just over cm in diameter... the ball does roll, but does so incredibly slowly. To an audience it seems baffling... inside the ball, which is actually hollow, there is a viscous liquid and a smaller ball which is heavy... it is the smaller heavier ball which determines the pace and this is slow because of the viscous liquid.

4 The Snail Cylinder from UBC - $0,000

5 The state of the snail cylinder is specified by ɛt, χt, Ω a t, and Ω b t Ω b Ω a α g The flow in the gap is analyzed with the lubrication approximation. φ = α + χ C B χ A φ O k^ b a = δ a M m a + m b + m f. m a m a m a, πa Lρ m a X a = X b + ɛ, α ^ ɛ ɛ sin χ î + cos χ ˆk

6 The equations of motion Ω b Ω a B α χ A φ g m a + m a m f m aa Ωa = T a ɛ χ + ɛ χ = f χ + m a g sin φ m a b Ω b cos χ, C O k^ m a + m a m f ɛ ɛ χ = f ɛ m ag cos φ m ab Ω b sin χ, α ^ d dt [ m aa Ω a + M + m b b Ω b + m a + m a m f ɛ χ + m a b ddt ɛ sin χ ] Recall: X a = X b + ɛ, ɛ ɛ sin χ î + cos χ ˆk +m ab Ω b ɛ cos χ Mgb sin α + m agɛ sin φ, The only approximation so far The lubrication approximation in the gap expresses the hydrodynamic forces and torques in terms of the four independent variables.

7 Lubrication in the gap: use the c. of m. frame of the fluid. a # ~ a ~ b# b r=a+z hθ = δ ɛ cos θ, d! /dt ~ b# b ~ a # a B! A " P O d! /dt ρνu zz = a p θ, p z = 0, a u θ + w z = 0 d! /dt The rotational part: u R z, θ = z a Ω a + z h h a Ω b pr θ zh z. aρν The squeeze part: u S z, θ = z h ɛ sin θ ps θ zh z aρν The pressure follows from global mass conservation. Integration round the inner cylinder gives the forces and torques.

8 d dt [ m a + m a m f m a + m a m f m aa Ω a + M + m b b Ω b + The full Monty ɛ χ + ɛ χ = f χ + m a g sin φ m a b Ω b cos χ, ɛ ɛ χ = f ɛ m ag cos φ m ab Ω b sin χ, m aa Ωa = T a m a + m a m f ɛ χ + m a b ddt ɛ sin χ ] +m ab Ω b ɛ cos χ Mgb sin α + m agɛ sin φ, f ɛ = νam a δ κ κ 3/, f χ = νam a δ κω a + Ω b χ + κ κ T a = νam a δ κ Ω b χ + κ Ω a χ 3 + κ κt ɛt κ δ

9 A non-accelerating solution g Ω a = Ω a = ɛ = χ = 0 O A B Ω b & Ω a sin α this is crazy C α We are relieved to discover that this solution is linearly unstable The c. of m. lies directly above the point of contact and the line of centers is horizontal.

10 The main approximation: δ a and sin α δ a m a + m a m f m a + m a m f ɛ χ + ɛ χ = f χ + m a g sin φ m a b Ω b cos χ, ɛ ɛ χ = f ɛ m ag cos φ m ab Ω b sin χ, m aa Ωa = T a κt ɛt δ d dt [ m aa Ω a + M + m b b Ω b + m a + m a m f ɛ χ + m a b ddt ɛ sin χ ] +m ab Ω b ɛ cos χ Mgb sin α + m agɛ sin φ, f ɛ = νam a δ κ κ 3/, f χ = νam a δ κω a + Ω b χ + κ κ T a = νam a δ κ Ω b χ + κ Ω a χ 3 + κ κ

11 Solution of the reduced equations If the slope is not too large, we find rocking solutions in which the inner cylinder slowly sediments towards the outer cylinder. If the slope is large, the system locks onto a runaway rolling solution with concentric cylinders: X b = Mg sin α M + m b + m a t There is a range of slopes for which both rocking and rolling solutions co-exist depending on ICs. The decisive slope parameter is: s a δ M m a sin α

12 Some details of the rocking solutions: outer cylinder center of the inner cylinder Limiting sedimentation point!t κt ɛt δ Time, t 6 4 0! "t and X b t Time, t # a t and # b t! Time, t The gap closes: κt t Power-law deceleration: Ω b t q, X b t t q, q 3 + 4µ + µ, µ M + m b m a

13 Experiments with the snail-cylinder 0 N. J. Balmforth, J. W. M. Bush, D. Vener and W. R. Young Distance, X a ν= 0 4, Slope Time, t b ν= There is no indication of power-law deceleration. The experimental results are far simpler than the theoretical model...

14 Hypothesis Asperities prevent the closure of the gap and maintain an effective minimum separation: κt ɛt δ κ = ɛ δ We include a contact force with a friction angle : C χ C ɛ tan ψ, The radial equation and the torque equation must be modified once contact occurs.

15 Microscope images of the surface Perspex Steel Aluminium The scale is 50 microns in total length Suggests the roughness scale is indeed of the order of tens of microns.

16 offers insight into the variations of κ. The fixed point to which the solution converges is given by Scaled rotation rates Predictions of the contact theory Ω b /3ζ * κ Ω b + κ Ω a 3 + κ κ Ω a /3ζ * 0.05 Locking Sliding 0 κ s κ sin φ = s, κ Ω a + Ω b sin φ = + κ, κ / cos φ = C ɛ, = C χ. The condition 5.5 for locking now reduces to s < s /κ tan φ Otherwise, Ω a = Ω b = s + κ κ. V bmδg Ω 4νm, a = s κ 3/ κ 3 κ cos φ tan ψ, Ω b = s + κ s V = V sin α ζ Max, a κ tan ψ. where s When the minimum gap is relatively narrow, and ζ κ, w solutions in the compact form,

17 Wrap the inner-cylinder αwith sandpaper grit CAMI vs σ mm a Scaled speed, V / V 80 grit b * κt t.5 0 grit grit 0 grit 0.5 Smooth Ω b t q, X b t t q, q 3 + 4µ Slope degrees For small slopes the contact force prevents slipping and the system rolls with constant speed X b = 0.05 Mg sin α M + m b + m a s a δ Grade Smooth Inferred mm Expected mm κt ɛt δ M sin α m a Expected = average particle diameter quoted by the CAMI standard V = bmgδ 4νm a }{{} V + µ, µ M + m κ = ɛ δ sin α κ }{{} ζ t sinα b V b / ζ * V * m a

18 Conclusions Sandpaper experiments show that surface roughness controls the speed of descent in rocking regime. The κ_ theory has some success in rationalizing experimental results. But κ_ is not completely convincing. V = bmgδ 4νm a }{{} V sin α κ }{{} ζ For example, the experimental dependence on α is not this simple. Nonetheless, I am now officially declaring victory over the snail cylinder and the snail ball. Other inclined-plane problems are diverting...

19 A Granular Snail Cylinder