The document discusses thermodynamic approaches to mechanical and flow instabilities. It provides examples of self-oscillators that maintain periodic motion by extracting energy from an external source, such as surface waves driven by the motion of air above water. The document argues that irreversibility and dissipation, which are often overlooked, can potentially destabilize systems and induce oscillations whenever a moving part transports its dissipative mechanisms.

1. Thermodynamic approach
to mechanical and flow
instabilities
Alejandro Jenkins
U. de Costa Rica &
Academia Nacional de Ciencias
in collaboration with Carlos D. Díaz-Marín (U. de Costa Rica)
10th European Solid Mechanics Conference
Mini-symposium 7-4
Bologna, Italy
Monday 2 June 2018

2. Self-oscillation
• Self-oscillators maintain regular, periodic motion at
expense of power source with no corresponding
periodicity
• Make waves, by positive feedback between
oscillation & energy 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
2

3. Self-oscillation, cont.
• Le Corbeiller: autonomous engines
are self-oscillators (“prime movers”)
• Mathematical theory of self-
oscillators may benefit from
thermodynamics and vice-versa
• AJ, Phys. Rep. 525, 167 (2013)
• AJ, The Physical Theory of Engines &
Self-Oscillators, (in preparation)
• This work: C. D. Díaz-Marín & AJ,
arXiv:1806.01527
3
Philippe Emmanuel
Le Corbeiller (1891-1980)
© Collections École polytechnique (Palaiseau)

4. Thermodynamics
• Time not a variable in classical thermodynamics
• Engine’s output W > 0 implies active, non-conservative force
• Dissipation makes both passive and active non-conservation
possible
• Carnot’s bound on efficiency of heat engines corresponds to
zero power (see “finite-time thermo.”)
• Extensive theoretical literature treats current pumps as anti-
resistors or “Maxwell demons”;
• in reality, engines always run irreversibly
4

5. Shear flow instability
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)
5
⇠ ⇠ exp [ik(x vt)] ! = kv
Air
Water
x

6. Surface waves
6
• 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)

7. Non-potential flow
when the air is moving in the direction
in which the wave-form is travelling, but
with a greater velocity, there will
evidently be an excess of pressure in
the rear slopes, as well as a
tangential drag on the exposed
crests… Hence the tendency will be to
increase the amplitude of the waves to
such a point that the dissipation
balances the work done by the surface
forces.
— Lamb, Hydrodynamics, 6th ed.
(1932), sec. 350
7

8. Irreversibility, cont.
• We knew air at rest damps water wave
• For V > v, air can transfer energy to wave and
produce entropy
• Process will occur, by basic principle of non-
equilibrium thermodynamics
see: Maghrebi, Jaffe & Kardar, Phys. Rev. A 90, 012515 (2014)
• Gil Refael: “Thermodynamics is saying something
[useful] without knowing anything”
8

9. Ω
ω0
θ
Dissipation-induced
instabilities
• 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 theory of flow instabilities
• see, e.g., Thorne and Blandford, Modern
Classical Physics (2017), sec. 14.6
9

10. D’Alembert’s paradox

11. Garden hose instability
11
fixed end
free end
y
x
R
joint
joint
joint
O
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discharge velocity:
F = MUu = MU ˙R MU2
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W =
Z T
0
F · ˙R dt =
Z T
0
MU
⇣
˙R2
+ U ˆ⌧ · ˙R
⌘
dt
<latexit sha1_base64="W6U9pwdl3S4kIbwpjQRO44vXnhk=">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</latexit><latexit sha1_base64="W6U9pwdl3S4kIbwpjQRO44vXnhk=">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</latexit><latexit sha1_base64="W6U9pwdl3S4kIbwpjQRO44vXnhk=">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</latexit><latexit sha1_base64="W6U9pwdl3S4kIbwpjQRO44vXnhk=">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</latexit>
— Benjamin, Proc. R. Soc. Lond. A 261, 457 (1961)

12. Hopf bifurcation
Garden hose instability, cont.
12
deflection of elastic, cantilevered pipe: w(t, x) ; 0  x  L<latexit sha1_base64="GHzl4okln3CAzAMHef1KBsuiG4U=">AAACBnicbVC7SgNBFJ31GeNr1VKEwSBEkLArgoJN0MbCIoJ5QLKE2clNMmT24cysJizZxsZfsbFQxNZvsPNvnCRbaOKByz2ccy8z97ghZ1JZ1rcxN7+wuLScWcmurq1vbJpb2xUZRIJCmQY8EDWXSODMh7JiikMtFEA8l0PV7V2O/Oo9CMkC/1YNQnA80vFZm1GitNQ09x7y6qh/iJNznCQJtnCDwx3uT9p108xZBWsMPEvslORQilLT/Gq0Ahp54CvKiZR12wqVExOhGOUwzDYiCSGhPdKBuqY+8UA68fiMIT7QSgu3A6HLV3is/t6IiSflwHP1pEdUV057I/E/rx6p9pkTMz+MFPh08lA74lgFeJQJbjEBVPGBJoQKpv+KaZcIQpVOLqtDsKdPniWV44JtFeybk1zxIo0jg3bRPsojG52iIrpCJVRGFD2iZ/SK3own48V4Nz4mo3NGurOD/sD4/AEO7Jbw</latexit><latexit sha1_base64="GHzl4okln3CAzAMHef1KBsuiG4U=">AAACBnicbVC7SgNBFJ31GeNr1VKEwSBEkLArgoJN0MbCIoJ5QLKE2clNMmT24cysJizZxsZfsbFQxNZvsPNvnCRbaOKByz2ccy8z97ghZ1JZ1rcxN7+wuLScWcmurq1vbJpb2xUZRIJCmQY8EDWXSODMh7JiikMtFEA8l0PV7V2O/Oo9CMkC/1YNQnA80vFZm1GitNQ09x7y6qh/iJNznCQJtnCDwx3uT9p108xZBWsMPEvslORQilLT/Gq0Ahp54CvKiZR12wqVExOhGOUwzDYiCSGhPdKBuqY+8UA68fiMIT7QSgu3A6HLV3is/t6IiSflwHP1pEdUV057I/E/rx6p9pkTMz+MFPh08lA74lgFeJQJbjEBVPGBJoQKpv+KaZcIQpVOLqtDsKdPniWV44JtFeybk1zxIo0jg3bRPsojG52iIrpCJVRGFD2iZ/SK3own48V4Nz4mo3NGurOD/sD4/AEO7Jbw</latexit><latexit sha1_base64="GHzl4okln3CAzAMHef1KBsuiG4U=">AAACBnicbVC7SgNBFJ31GeNr1VKEwSBEkLArgoJN0MbCIoJ5QLKE2clNMmT24cysJizZxsZfsbFQxNZvsPNvnCRbaOKByz2ccy8z97ghZ1JZ1rcxN7+wuLScWcmurq1vbJpb2xUZRIJCmQY8EDWXSODMh7JiikMtFEA8l0PV7V2O/Oo9CMkC/1YNQnA80vFZm1GitNQ09x7y6qh/iJNznCQJtnCDwx3uT9p108xZBWsMPEvslORQilLT/Gq0Ahp54CvKiZR12wqVExOhGOUwzDYiCSGhPdKBuqY+8UA68fiMIT7QSgu3A6HLV3is/t6IiSflwHP1pEdUV057I/E/rx6p9pkTMz+MFPh08lA74lgFeJQJbjEBVPGBJoQKpv+KaZcIQpVOLqtDsKdPniWV44JtFeybk1zxIo0jg3bRPsojG52iIrpCJVRGFD2iZ/SK3own48V4Nz4mo3NGurOD/sD4/AEO7Jbw</latexit><latexit sha1_base64="GHzl4okln3CAzAMHef1KBsuiG4U=">AAACBnicbVC7SgNBFJ31GeNr1VKEwSBEkLArgoJN0MbCIoJ5QLKE2clNMmT24cysJizZxsZfsbFQxNZvsPNvnCRbaOKByz2ccy8z97ghZ1JZ1rcxN7+wuLScWcmurq1vbJpb2xUZRIJCmQY8EDWXSODMh7JiikMtFEA8l0PV7V2O/Oo9CMkC/1YNQnA80vFZm1GitNQ09x7y6qh/iJNznCQJtnCDwx3uT9p108xZBWsMPEvslORQilLT/Gq0Ahp54CvKiZR12wqVExOhGOUwzDYiCSGhPdKBuqY+8UA68fiMIT7QSgu3A6HLV3is/t6IiSflwHP1pEdUV057I/E/rx6p9pkTMz+MFPh08lA74lgFeJQJbjEBVPGBJoQKpv+KaZcIQpVOLqtDsKdPniWV44JtFeybk1zxIo0jg3bRPsojG52iIrpCJVRGFD2iZ/SK3own48V4Nz4mo3NGurOD/sD4/AEO7Jbw</latexit>
˙w = wt + Uwx<latexit sha1_base64="n0Nh/RwLs2zEO+p9ZB/AdLdujF8=">AAAB/nicbVDLSgMxFM3UV62vUXHlJlgEQSgzIuhGKLpxWcFpC+0wZNK0Dc1khuSOtQwFf8WNC0Xc+h3u/BvTdhbaeiDkcM69NzcnTATX4DjfVmFpeWV1rbhe2tjc2t6xd/fqOk4VZR6NRayaIdFMcMk84CBYM1GMRKFgjXBwM/EbD0xpHst7GCXMj0hP8i6nBIwU2AftTgx4iK/wMAB8ij1zP+LALjsVZwq8SNyclFGOWmB/mTk0jZgEKojWLddJwM+IAk4FG5faqWYJoQPSYy1DJYmY9rPp+mN8bJQO7sbKHAl4qv7uyEik9SgKTWVEoK/nvYn4n9dKoXvpZ1wmKTBJZw91U4EhxpMscIcrRkGMDCFUcbMrpn2iCAWTWMmE4M5/eZHUzyquU3HvzsvV6zyOIjpER+gEuegCVdEtqiEPUZShZ/SK3qwn68V6tz5mpQUr79lHf2B9/gBem5PM</latexit><latexit sha1_base64="n0Nh/RwLs2zEO+p9ZB/AdLdujF8=">AAAB/nicbVDLSgMxFM3UV62vUXHlJlgEQSgzIuhGKLpxWcFpC+0wZNK0Dc1khuSOtQwFf8WNC0Xc+h3u/BvTdhbaeiDkcM69NzcnTATX4DjfVmFpeWV1rbhe2tjc2t6xd/fqOk4VZR6NRayaIdFMcMk84CBYM1GMRKFgjXBwM/EbD0xpHst7GCXMj0hP8i6nBIwU2AftTgx4iK/wMAB8ij1zP+LALjsVZwq8SNyclFGOWmB/mTk0jZgEKojWLddJwM+IAk4FG5faqWYJoQPSYy1DJYmY9rPp+mN8bJQO7sbKHAl4qv7uyEik9SgKTWVEoK/nvYn4n9dKoXvpZ1wmKTBJZw91U4EhxpMscIcrRkGMDCFUcbMrpn2iCAWTWMmE4M5/eZHUzyquU3HvzsvV6zyOIjpER+gEuegCVdEtqiEPUZShZ/SK3qwn68V6tz5mpQUr79lHf2B9/gBem5PM</latexit><latexit sha1_base64="n0Nh/RwLs2zEO+p9ZB/AdLdujF8=">AAAB/nicbVDLSgMxFM3UV62vUXHlJlgEQSgzIuhGKLpxWcFpC+0wZNK0Dc1khuSOtQwFf8WNC0Xc+h3u/BvTdhbaeiDkcM69NzcnTATX4DjfVmFpeWV1rbhe2tjc2t6xd/fqOk4VZR6NRayaIdFMcMk84CBYM1GMRKFgjXBwM/EbD0xpHst7GCXMj0hP8i6nBIwU2AftTgx4iK/wMAB8ij1zP+LALjsVZwq8SNyclFGOWmB/mTk0jZgEKojWLddJwM+IAk4FG5faqWYJoQPSYy1DJYmY9rPp+mN8bJQO7sbKHAl4qv7uyEik9SgKTWVEoK/nvYn4n9dKoXvpZ1wmKTBJZw91U4EhxpMscIcrRkGMDCFUcbMrpn2iCAWTWMmE4M5/eZHUzyquU3HvzsvV6zyOIjpER+gEuegCVdEtqiEPUZShZ/SK3qwn68V6tz5mpQUr79lHf2B9/gBem5PM</latexit><latexit sha1_base64="n0Nh/RwLs2zEO+p9ZB/AdLdujF8=">AAAB/nicbVDLSgMxFM3UV62vUXHlJlgEQSgzIuhGKLpxWcFpC+0wZNK0Dc1khuSOtQwFf8WNC0Xc+h3u/BvTdhbaeiDkcM69NzcnTATX4DjfVmFpeWV1rbhe2tjc2t6xd/fqOk4VZR6NRayaIdFMcMk84CBYM1GMRKFgjXBwM/EbD0xpHst7GCXMj0hP8i6nBIwU2AftTgx4iK/wMAB8ij1zP+LALjsVZwq8SNyclFGOWmB/mTk0jZgEKojWLddJwM+IAk4FG5faqWYJoQPSYy1DJYmY9rPp+mN8bJQO7sbKHAl4qv7uyEik9SgKTWVEoK/nvYn4n9dKoXvpZ1wmKTBJZw91U4EhxpMscIcrRkGMDCFUcbMrpn2iCAWTWMmE4M5/eZHUzyquU3HvzsvV6zyOIjpER+gEuegCVdEtqiEPUZShZ/SK3qwn68V6tz5mpQUr79lHf2B9/gBem5PM</latexit>
vel. of fluid element:
⇠0 = A · ei(kx !t)
, for ! = kv<latexit sha1_base64="Lc1C9rBSstVRGLp5da0J17PDuPc=">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</latexit><latexit sha1_base64="Lc1C9rBSstVRGLp5da0J17PDuPc=">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</latexit><latexit sha1_base64="Lc1C9rBSstVRGLp5da0J17PDuPc=">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</latexit><latexit sha1_base64="Lc1C9rBSstVRGLp5da0J17PDuPc=">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</latexit>
traveling wave:
W = MU
Z T
0
[wt(wt + Uwx)]x=L dt = MU
Z T
0
wt ˙w|x=L dt
<latexit sha1_base64="zUIEgCSfCpyq7pLx+wiplN5wzDA=">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</latexit><latexit sha1_base64="zUIEgCSfCpyq7pLx+wiplN5wzDA=">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</latexit><latexit sha1_base64="zUIEgCSfCpyq7pLx+wiplN5wzDA=">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</latexit><latexit sha1_base64="zUIEgCSfCpyq7pLx+wiplN5wzDA=">AAACWXicbZFdT9swFIadjI+usNGNy90cUSGBEFWCkOAGqRo3uxgSkwhFakLkuE5r1XEi+2S0yvonuZiE+Cu7wA25GB9Hsv3qnPP443VSSGHQ8x4c98PK6tp662N7Y/PT563Ol6/XJi814wHLZa5vEmq4FIoHKFDym0JzmiWSD5Lp+bI++M21Ebm6wnnBo4yOlUgFo2hTcacYwBkcwkUAoVAYe7dXEEqe4vAuRtiD5XwAgV1n+xBqMZ5gFFczy/xcwAjfZ3s1Fo5yhLsG+mOhsxqJO12v59UBb4XfiC5p4jLu3NudWJlxhUxSY4a+V2BUUY2CSb5oh6XhBWVTOuZDKxXNuImq2pkF7NrMCNJc26EQ6uz/REUzY+ZZYjszihPzurZMvlcblpieRpVQRYlcseeD0lIC5rC0GUZCc4ZybgVlWti7AptQTRnaz2hbE/zXT34rro96vtfzfx13+98bO1rkG9khe8QnJ6RPfpBLEhBG/pJ/zqqz5jy6jtty28+trtMw2+RFuNtPXiqvKw==</latexit>
= MU
✓
1
U
v
◆
A2
!⇡ = MU(U v)A2
k⇡
<latexit sha1_base64="0m3Im+9uZXoTMAAx4a3koKBH/SE=">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</latexit><latexit sha1_base64="0m3Im+9uZXoTMAAx4a3koKBH/SE=">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</latexit><latexit sha1_base64="0m3Im+9uZXoTMAAx4a3koKBH/SE=">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</latexit><latexit sha1_base64="0m3Im+9uZXoTMAAx4a3koKBH/SE=">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</latexit>
wt = Re ( i!⇠0) ; ˙w = Re [( i! + U · ik)⇠0] = Re

i!
✓
1
U
v
◆
⇠0
<latexit sha1_base64="+V4paQOtIK7ioyhImXp/Hn3YGFk=">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</latexit><latexit sha1_base64="+V4paQOtIK7ioyhImXp/Hn3YGFk=">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</latexit><latexit sha1_base64="+V4paQOtIK7ioyhImXp/Hn3YGFk=">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</latexit><latexit sha1_base64="+V4paQOtIK7ioyhImXp/Hn3YGFk=">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</latexit>
W = MU
✓
1
U
v
◆ Z 2⇡/!
0
[Re ( i!⇠0)]
2
dt
<latexit sha1_base64="WHh9Y0bIbzrV6PjH+4q/gNbhu/A=">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</latexit><latexit sha1_base64="WHh9Y0bIbzrV6PjH+4q/gNbhu/A=">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</latexit><latexit sha1_base64="WHh9Y0bIbzrV6PjH+4q/gNbhu/A=">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</latexit><latexit sha1_base64="WHh9Y0bIbzrV6PjH+4q/gNbhu/A=">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</latexit>

13. 13
Source: Doaré & de Langre, Eur. J. Mech. A Solids 21, 857 (2002)

14. Interpretation
• Garden hose instability doesn’t require thermal dissipation
• does require energy loss by outflow
• → damping term
• as in Zel’dovich’s analysis, Hopf bifurcation may be obtained
from sign of work on a period of free, linear wave
• cf. Ziegler pendulum, with non-conservative follower force
plus damping at joints
• implementation of pure follower force does require thermal
dissipation; see Bigoni et al., J. Mech. Phys. Solids 116, 99
(2018); Phys. Rev. E 97, 023003 (2018)
14
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15. Outlook, 1
• Blind spot in theoretical physics:
• non-conservative forces not regarded as mechanical
(see, e.g., Landau & Lifshitz);
• classical thermodynamics doesn’t use time as variable
(cf. “finite-time thermo.”)
• Statistical mechanics treats large systems in thermal &
chemical equilibrium, fluctuating stochastically about
equilibrium, or steadily relaxing to equilibrium
• We still need adequate physical theory of self-oscillators,
engines, and pumps
15

16. Outlook, 2
• This approach offers simpler, more general
description of non-conservative mechanical &
hydrodynamic instabilities
• Dispels much of the paradox of dissipation-induced
instability
• Reveals underlying unity across diverse phenomena
• “energy flow method” has already proved useful in
engineering; see, e.g., Ran, Besselink & Nijmeijer,
Vehicle Syst. Dyn. 53, 1795 (2015)
16