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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) 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, and in section 357 et seq. of Rayleigh (1896/1926)); waves on the surface of the sea break discontinuously when they reach the shore (Thom 1975). 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), also called penetrative convection (Lindzen 1977)

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) 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, and in section 357 et seq. of Rayleigh (1896/1926)); waves on the surface of the sea break discontinuously when they reach the shore (Thom 1975). 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), also called penetrative convection (Lindzen 1977)

经典的非平衡热力学理论大多关注流体的空间连续运动，但流体也可以在空间不连续的情况下运动。赫尔姆霍兹Helmholtz(1868)写道，在流动的流体中，可以出现流体压力为零的情况，这就看到了流体的断裂。这产生于流体流动的动量，显示出与热或电的传导不同的动力结构。因此，例如：从喷头喷出的水可以形成水滴雨（雷利1878年，以及在雷利（1896/1926）第357节等）；海面上的波浪到达岸边时，会不连续地断裂（Thom 1975）。Helmholtz指出，风琴管的声音一定产生于这种流动的不连续，是由空气经过一个尖锐的障碍物时引起的；否则，声波的振荡特性就会被阻尼掉，化为乌有。这种流动的熵产生率的定义，不是经典非平衡热力学的通常理论所能涵盖的。还有许多经常观察到的流体流动的不连续现象，也不在经典的非平衡热力学理论的范围内，如：沸腾液体中的气泡和泡腾饮料中的气泡；还有受保护的热带深层对流塔（Riehl，Malkus，1958），也叫穿透性对流（Lindzen，1977）。

经典的非平衡热力学理论大多关注流体的空间连续运动，但流体也可以在空间不连续的情况下运动。赫尔姆霍兹Helmholtz(1868)写道，在流动的流体中，可以出现流体压力为零的情况，这就看到了流体的断裂。这产生于流体流动的动量，显示出与热或电的传导不同的动力结构。因此，例如：从喷头喷出的水可以形成水滴雨（Rayleigh 1878年，以及在Rayleigh（1896/1926）第357节等）；海面上的波浪到达岸边时，会不连续地断裂（Thom 1975）。Helmholtz指出，风琴管的声音一定产生于这种流动的不连续，是由空气经过一个尖锐的障碍物时引起的；否则，声波的振荡特性就会被阻尼掉，化为乌有。这种流动的熵产生率的定义，不是经典非平衡热力学的通常理论所能涵盖的。还有许多经常观察到的流体流动的不连续现象，也不在经典的非平衡热力学理论的范围内，如：沸腾液体中的气泡和泡腾饮料中的气泡；还有受保护的热带深层对流塔（Riehl，Malkus，1958），也叫穿透性对流（Lindzen，1977）。

==Historical development==

==Historical development==