Irreversibility of mechanical and hydrodynamic instabilities
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The document discusses the concept of self-oscillation and its relation to mechanical and hydrodynamic instabilities, emphasizing the role of feedback and energy dynamics in these systems. It critiques the lack of traditional mechanics in considering non-conservative forces and suggests that thermodynamics could inform the understanding of self-oscillators. The work presented seeks to establish a coherent theoretical framework for analyzing energy methods in relation to oscillatory systems and instabilities.
![Self-oscillation, cont.
• Le Corbeiller, “The non-linear theory of the
maintenance of oscillations”, Proc. Inst.
Electr. Engrs. 11, 292 (1936); “Theory of
prime movers”, in Non-Linear Mechanics
(Brown U., 1943), lecture 2
• Considerations of work extraction in
non-conservative (open) systems, and
irreversibility
• AJ, “Self-Oscillation”, Phys. Rep. 525,
167 (2013); The Physical Theory of
Engines & Self-Oscillators, (in
preparation)
• This work: C. D. Díaz-Marín & AJ, arXiv:
1709.07120 [physics.class-ph]
3
Philippe Emmanuel
Le Corbeiller (1891-1980)
© Collections École polytechnique (Palaiseau)](https://image.slidesharecdn.com/ajenkins-np-171214101603/85/Irreversibility-of-mechanical-and-hydrodynamic-instabilities-3-320.jpg)
![Energy method
5
arctan(w /r )0θ
θ
R
-Fdis
Ft
disk surfacedashpot =
sgn( ˙w)fposw✓
r0 ✓=⌦t, r=r0
Wt =
Z ⌧
0
Ftr0⌦dt = ⌦
Z ⌧
0
[sgn( ˙w)fposw✓]✓=⌦t, r=r0
dt
Work by external force:
Ft =
Fdis · w✓
r0 ✓=⌦t, r=r0
For ˙✓ = const. :
Wd =
Z ⌧
0
[Fdis ˙w]✓=⌦t, r=r0
dt =
Z ⌧
0
[sgn( ˙w)fpos ˙w]✓=⌦t, r=r0
dtWork by dashpot:
=
Z ⌧
0
[wt sgn( ˙w)fpos]✓=⌦t, r=r0
dt =
Z ⌧
0
[wtFdis]✓=⌦t, r=r0
dt
E = Wt Wd =
Z ⌧
0
[(⌦w✓ ˙w) sgn( ˙w)fpos]✓=⌦t, r=r0
dt
Energy absorbed by oscillation:](https://image.slidesharecdn.com/ajenkins-np-171214101603/85/Irreversibility-of-mechanical-and-hydrodynamic-instabilities-5-320.jpg)
![Energy method, cont.
6
E =
Z ⌧
0
[!A(r) cos (m✓ !t) sgn( ˙w)fpos]✓=⌦t, r=r0
dt
= !A(r0)
Z ⌧
0
cos( t) sgn [A(r0) cos( t)] fpos|✓=⌦t, r=r0
dt
= sgn( ) · !|A(r0)|
Z ⌧
0
| cos( t)| fpos|✓=⌦t, r=r0
dt
w = A(r) sin (m✓ !t)Traveling wave:
!/mphase velocity:
⌘ m⌦ !“detuning parameter”:](https://image.slidesharecdn.com/ajenkins-np-171214101603/85/Irreversibility-of-mechanical-and-hydrodynamic-instabilities-6-320.jpg)
![Kelvin-Helmholtz
Galileo:
@⇠
@t
!
@⇠
@t
+ V
@⇠
@x
= i!⇠ + V · ik⇠
= i!⇠
✓
1
V
v
◆
See: Y. B. Zel’dovich, JETP Lett. 14, 180 (1971);
Sov. Phys. JETP 35, 1085 (1971)
8
⇠ ⇠ exp [ik(x vt)] ! = kv
Air
Water
x](https://image.slidesharecdn.com/ajenkins-np-171214101603/85/Irreversibility-of-mechanical-and-hydrodynamic-instabilities-8-320.jpg)
Irreversibility of mechanical and hydrodynamic instabilities
- 1. Irreversibility of mechanical and hydrodynamic instabilities Alejandro Jenkins U. de Costa Rica & Academia Nacional de Ciencias 14th International Conference Dynamical Systems Theory & Applications Łódź, Poland 11 December 2017
- 2. Self-oscillation • Self-oscillators maintain regular, periodic motion at expense of power source with no corresponding periodicity • They make waves, by positive feedback between oscillation & power source • Many other names: maintained, sustained, self-induced, self- excited, spontaneous, autonomous, etc. • Usually approached from theory of differential eqs. (limit cycle, Hopf bifurcation), or as instabilities in feedback control • P. E. Le Corbeiller: Autonomous motors are self-oscillators (“prime movers”) • Theory of self-oscillators may benefit from thermodynamics and vice-versa 2
- 3. Self-oscillation, cont. • Le Corbeiller, “The non-linear theory of the maintenance of oscillations”, Proc. Inst. Electr. Engrs. 11, 292 (1936); “Theory of prime movers”, in Non-Linear Mechanics (Brown U., 1943), lecture 2 • Considerations of work extraction in non-conservative (open) systems, and irreversibility • AJ, “Self-Oscillation”, Phys. Rep. 525, 167 (2013); The Physical Theory of Engines & Self-Oscillators, (in preparation) • This work: C. D. Díaz-Marín & AJ, arXiv: 1709.07120 [physics.class-ph] 3 Philippe Emmanuel Le Corbeiller (1891-1980) © Collections École polytechnique (Palaiseau)
- 4. Spinning disk 4 wt ⌘ @w/@t ; w✓ ⌘ @w/@✓ w(t, r, ✓)transverse displacement: ˙✓ = ⌦ = const.dashpot rotation: ˙w = wt + ⌦w✓ for disk element in contact with dashpot: Fdis = sgn( ˙w)fpos force exerted by dashpot: fpos = c| ˙w|(for linear damping ) See: I. Y. Shen and C. D. Mote, J. Sound Vib. 148, 307 (1991) θ = Ωt Fr Ft r0 top: arctan(w /r )0θ θ R -Fdis Ft disk surfacedashpot side: fpos > 0
- 5. Energy method 5 arctan(w /r )0θ θ R -Fdis Ft disk surfacedashpot = sgn( ˙w)fposw✓ r0 ✓=⌦t, r=r0 Wt = Z ⌧ 0 Ftr0⌦dt = ⌦ Z ⌧ 0 [sgn( ˙w)fposw✓]✓=⌦t, r=r0 dt Work by external force: Ft = Fdis · w✓ r0 ✓=⌦t, r=r0 For ˙✓ = const. : Wd = Z ⌧ 0 [Fdis ˙w]✓=⌦t, r=r0 dt = Z ⌧ 0 [sgn( ˙w)fpos ˙w]✓=⌦t, r=r0 dtWork by dashpot: = Z ⌧ 0 [wt sgn( ˙w)fpos]✓=⌦t, r=r0 dt = Z ⌧ 0 [wtFdis]✓=⌦t, r=r0 dt E = Wt Wd = Z ⌧ 0 [(⌦w✓ ˙w) sgn( ˙w)fpos]✓=⌦t, r=r0 dt Energy absorbed by oscillation:
- 6. Energy method, cont. 6 E = Z ⌧ 0 [!A(r) cos (m✓ !t) sgn( ˙w)fpos]✓=⌦t, r=r0 dt = !A(r0) Z ⌧ 0 cos( t) sgn [A(r0) cos( t)] fpos|✓=⌦t, r=r0 dt = sgn( ) · !|A(r0)| Z ⌧ 0 | cos( t)| fpos|✓=⌦t, r=r0 dt w = A(r) sin (m✓ !t)Traveling wave: !/mphase velocity: ⌘ m⌦ !“detuning parameter”:
- 7. Interpretation • Power delivered to wave: • Supercritically ( ), wave moves backwards with respect to dashpot • Therefore Fdis leads wave, so that • Requires phase difference, and therefore dissipation in moving dashpot • Disk’s internal friction stabilizes high-m modes 7 P = Fdiswt ⌦ > !/m E > 0
- 8. Kelvin-Helmholtz Galileo: @⇠ @t ! @⇠ @t + V @⇠ @x = i!⇠ + V · ik⇠ = i!⇠ ✓ 1 V v ◆ See: Y. B. Zel’dovich, JETP Lett. 14, 180 (1971); Sov. Phys. JETP 35, 1085 (1971) 8 ⇠ ⇠ exp [ik(x vt)] ! = kv Air Water x
- 9. Surface waves 9 • Feedback between air pressure & wave • For V > v, air pressure higher in throughs, lower in peaks (Bernoulli’s theorem) • Air viscosity introduces phase delay • Since wave moves backwards in air’s frame, air pressure variation leads water wave: sin > 0 ) net W > 0 Images: E. Mollo-Christensen, “Flow Instabilities”, National Committee for Fluid Mechanics Films (1972)
- 10. Ω ω0 θ Dissipation-induced instability • treated as paradox in dynamical systems literature (e.g., Ziegler, 1952; Krechetnikov & Marsden, 2007) • damping potentially destabilizing whenever moving part “carries its dissipative mechanisms around with it” - Pippard, Physics of vibration (1979), ch. 8 • Irreversibility & connection to damping often overlooked in discussion of flow instabilities • cf. Thorne & Blandford, Modern Classical Physics (2017), sec. 14.6 10
- 11. Tidal acceleration 11 Circulatory force: dissipation-induced time lag: ˙Eorbital = 3Gm2 R5 r6 (! ˙↵) ˙↵⌧ x y tidal bulge Moon (mass m) G Earth O S F See: R. Brito, V. Cardoso y P. Pani, Superradiance (2015), sec. 2.6
- 12. Outlook, 1 • Important “blind spot” in theoretical physics • Non-conservative forces not considered mechanical (see Landau & Lifshitz) • Classical thermodynamics doesn’t use time as a variable (see literature on “finite-time thermo.”) • Statistical mechanics treats large systems in thermal & chemical equilibrium, fluctuating stochastically about equilibrium, or steadily relaxing to equilibrium • We lack adequate physical theory of self-oscillators & engines! 12
- 13. Outlook, 2 • This approach offers simpler, more general description of mechanical & hydrodynamic instabilities • Dispels paradox of dissipation-induced instability • Reveals underlying unity across diverse phenomena • “Energy method” has already proved useful in engineering • see, e.g., Ran, Besselink & Nijmeijer, “Energy analysis of the Von Schlippe tyre model with application to shimmy”, Vehicle Syst. Dyn. 53, 1795 (2015) 13