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==Continuous and discontinuous motions of fluids==

==Continuous and discontinuous motions of fluids==

Much of the theory of classical non-equilibrium thermodynamics is concerned with the spatially continuous motion of fluids, but fluids can also move with spatial discontinuities. Helmholtz (1868)<ref name="Helmholtz 1868">Helmholtz, H. (1868). On discontinuous movements of fluids, ''Philosophical Magazine'' series 4, vol. '''36''': 337-346, translated by F. Guthrie from ''Monatsbericht der koeniglich preussischen Akademie der Wissenschaften zu Berlin'' April 1868, page 215 et seq.</ref> wrote about how in a flowing fluid, there can arise a zero fluid pressure, which sees the fluid broken asunder. This arises from the momentum of the fluid flow, showing a different kind of dynamical structure from that of the conduction of heat or electricity. Thus for example: water from a nozzle can form a shower of droplets (Rayleigh 1878,<ref name="Rayleigh 1878">{{cite journal | last1 = Strutt | first1 = J.W. | year = 1878 | title = On the instability of jets | url = https://zenodo.org/record/2095384| journal = Proceedings of the London Mathematical Society | volume = 10 | issue = | pages = 4–13 | doi=10.1112/plms/s1-10.1.4}}</ref> and in section 357 et seq. of Rayleigh (1896/1926)<ref name="Rayleigh 1896/1926">Strutt, J.W. (Baron Rayleigh) (1896/1926). Section 357 et seq. ''The Theory of Sound'', Macmillan, London, reprinted by Dover, New York, 1945.</ref>); waves on the surface of the sea break discontinuously when they reach the shore (Thom 1975<ref name="Thom 1975">Thom, R. (1975). ''Structural Stability and Morphogenesis: An outline of a general theory of models'', translated from the French by D.H. Fowler, W.A. Benjamin, Reading Ma, {{ISBN|0-8053-9279-3}}</ref>). Helmholtz pointed out that the sounds of organ pipes must arise from such discontinuity of flow, occasioned by the passage of air past a sharp-edged obstacle; otherwise the oscillatory character of the sound wave would be damped away to nothing. The definition of the rate of entropy production of such a flow is not covered by the usual theory of classical non-equilibrium thermodynamics. There are many other commonly observed discontinuities of fluid flow that also lie beyond the scope of the classical theory of non-equilibrium thermodynamics, such as: bubbles in boiling liquids and in effervescent drinks; also protected towers of deep tropical convection (Riehl, Malkus 1958<ref name="Riehl Malkus 1958">{{cite journal | last1 = Riehl | first1 = H. | last2 = Malkus | first2 = J.S. | year = 1958 | title = On the heat balance in the equatorial trough zone | url = | journal = Geophysica | volume = 6 | issue = | pages = 503–538 }}</ref>), also called penetrative convection (Lindzen 1977<ref name="Lindzen 1977">Lindzen, R.S. (1977). Some aspects of convection in meteorology, pp. 128-141 in ''Problems of Stellar Convection'', volume 71 of ''Lecture Notes in Physics'', Springer, Berlin, {{ISBN|978-3-540-08532-4}}.</ref>).

Much of the theory of classical non-equilibrium thermodynamics is concerned with the spatially continuous motion of fluids, but fluids can also move with spatial discontinuities. Helmholtz (1868)<ref name="Helmholtz 1868">Helmholtz, H. (1868). On discontinuous movements of fluids, ''Philosophical Magazine'' series 4, vol. '''36''': 337-346, translated by F. Guthrie from ''Monatsbericht der koeniglich preussischen Akademie der Wissenschaften zu Berlin'' April 1868, page 215 et seq.</ref> wrote about how in a flowing fluid, there can arise a zero fluid pressure, which sees the fluid broken asunder. This arises from the momentum of the fluid flow, showing a different kind of dynamical structure from that of the conduction of heat or electricity. Thus for example: water from a nozzle can form a shower of droplets (Rayleigh 1878,<ref name="Rayleigh 1878">{{cite journal | last1 = Strutt | first1 = J.W. | year = 1878 | title = On the instability of jets | url = https://zenodo.org/record/2095384| journal = Proceedings of the London Mathematical Society | volume = 10 | issue = | pages = 4–13 | doi=10.1112/plms/s1-10.1.4}}</ref> and in section 357 et seq. of Rayleigh (1896/1926)<ref name="Rayleigh 1896/1926">Strutt, J.W. (Baron Rayleigh) (1896/1926). Section 357 et seq. ''The Theory of Sound'', Macmillan, London, reprinted by Dover, New York, 1945.</ref>); waves on the surface of the sea break discontinuously when they reach the shore (Thom 1975<ref name="Thom 1975">Thom, R. (1975). ''Structural Stability and Morphogenesis: An outline of a general theory of models'', translated from the French by D.H. Fowler, W.A. Benjamin, Reading Ma, {{ISBN|0-8053-9279-3}}</ref>). Helmholtz pointed out that the sounds of organ pipes must arise from such discontinuity of flow, occasioned by the passage of air past a sharp-edged obstacle; otherwise the oscillatory character of the sound wave would be damped away to nothing. The definition of the rate of entropy production of such a flow is not covered by the usual theory of classical non-equilibrium thermodynamics. There are many other commonly observed discontinuities of fluid flow that also lie beyond the scope of the classical theory of non-equilibrium thermodynamics, such as: bubbles in boiling liquids and in effervescent drinks; also protected towers of deep tropical convection (Riehl, Malkus 1958<ref name="Riehl Malkus 1958">{{cite journal | last1 = Riehl | first1 = H. | last2 = Malkus | first2 = J.S. | year = 1958 | title = On the heat balance in the equatorial trough zone | url = | journal = Geophysica | volume = 6 | issue = | pages = 503–538 }}</ref>), also called penetrative convection (Lindzen 1977<ref name="Lindzen 1977">Lindzen, R.S. (1977). Some aspects of convection in meteorology, pp. 128-141 in ''Problems of Stellar Convection'', volume 71 of ''Lecture Notes in Physics'', Springer, Berlin, {{ISBN|978-3-540-08532-4}}.</ref>).