Kelvin’s theorem

From Kelvin s theorem, inviscid motions in a gravity (conservative) field which are initially irrotational remain so. We may, therefore, write... 

This definition is consistent with the definition of the chirality of rigid molecules and forms a sufficient and necessary condition for the optical activity of NRMs. The generalization of Kelvin s theorem for NRMs may be stated as a NRM is chiral, if the group H3) 3 is properly orthogonal. 

The practical interest of discriminating between flows based on their rotational or irrotational character results from Kelvin s theorem, one main consequence of which is that, in the absence of viscosity, it is impossible for an irrotational flow to spontaneously become rotational. Vorticity can only be produced in a flow at the boundaries of the domain, through the action of viscosity. It only appears within the fluid by transport or diffusion from the boundaries. Vorticity is produced in boundary layers near the walls. 

The second restriction within which we shall place ourselves is that of high-Reynolds-number flows, that is, Re = FR / v 1. V is the velocity scale of the azimuthal component and R is a characteristic dimension of the flow in the Oxy plane. Viscosity will therefore be negligible outside the boimdary layers, and the properties resulting from Kelvin s theorem will be applicable. 

At high Reynolds numbers, if transit through the apparatus is fast, vorticity does not have enough time to diffuse. Kelvin s theorem resirlts in the flow being irrotational in the part of the apparatus through which the flow passes. It is therefore also irrotational in the outlet sectiorts (spigot and overflow). This resirlt has two consequences ... 

A proof of Kelvin s theorem is as follows. Under the conditions specified, the equation of motion of an inviscid liquid is obtained by combining equations [5.16] and F = —V12 to give ... 

The concept of the vortex line can be usefully extended to that of the vortex tube as follows. Take an element of surface dS within a region of liquid undergoing rotational motion and construct the vortex line passing through each point on the boundary. These lines form the surface of a vortex tube. From Kelvin s theorem ... 

Gibbsitic [14762-49-3] Gibbs-Kelvin equation Gibbs phase rule Gibbs s phase rule Gibbs s theorem Gibbs-Thomson equation... 

In 1879 Lord Kelvin introduced the term nwtivity for the possession, the waste of which is called dissipation at constant temperature this is identical with Maxwell s available energy. He showed in a paper On Thermodynamics founded on Motivity and Energy Phil. Mag., 1898), that all the thermodynamic equations could be derived from the properties of motivity which follow directly from Carnot s theorem, without any explicit introduction of the entropy. 

Wulff s theorem [25] states that cr/r is invariant for all faces. Therefore, the result obtained from the Kelvin s equation must be independent of the choice of a face. 

The theory behind the third law of thermodynamics was initially formulated by Walther Nemst in 1906, which was known as Nemst theorem (https //www.sussex. ac.uk/webteam/gateway/file.php name=a-thermodynamicshistory. pdf site=35). The third law of thermodynamics was conceived from the fact that attaining absolute zero temperature is practically impossible. Lord Kelvin deduced this fact from the second law of thermodynamics with his study of heat transfer, work done, and efficiency of a number of heat engines in series. Kelvin s work was the foundation for the formulation of the third law. It can be stated as follows Absolute zero temperature is not attainable in thermodynamic processes. Another noted scientist, Max Planck, put forward the third law of thermodynamics from his observations in 1913. It states that The entropy of a pure substance is zero at absolute zero temperature. Plank observed that only pure, perfectly crystalline stmctures would have zero entropy at absolute zero temperamre. All other substances attain a state of minimum energy at absolute zero temperature as the molecules of the substance are arranged in their lowest possible energy state. 

This leads to Kelvin s circulation theorem, which states that for a barotropic fluid with no frictional forces acting, the absolute circulation is conserved following the motion. 

The vorticity equation describes how vorticity is changed by various properties of the flow. Only in very special circumstances would the vorticity be conserved following the flow. Kelvin s circulation theorem describes how an integral measure of vorticity is conserved but is valid only for barotropic flow and furthermore requires a knowledge of the time evolution of material surfaces. There does exist a quantity, referred to as the Ertel potential vorticity, that is conserved under more general conditions than either the vorticity or the circulation. It may be shown by combining the curl of the momentum equation [Eq. (26a)] with the continuity equation [Eq. (26c)] and the thermodynamic equation [Eq. (26b)] expressed in terms of potential temperature 0 that... 

Assuming that H(Y X) < 0 [information variant of Kelvin s formulation of the 11. Principle of Thermodynamics for irreversible cyclical transfer O when the respective relation (34) is valid], we also have the information variant of the second part of Carnot s theorem... 

An important property of the circulation is contained in the result known as Kelvin s circulation theorem. This states that, when the forces acting on an inviscid liquid are conservative and derived from a single-valued potential function, and the liquid density is a function of pressure only, the time rate of change of the circulation around any closed curve moving with the liquid is zero. Here, moving with the liquid means that the contour must always be drawn through the same liquid particles. 

The ratio 7a/ha can advantageously be expressed by averaged quantities , as 7/f, where 7 = Sjaj7j/Sjaj and f = Sja hj/Eja with j being a running index of the equilibrium planes. Note that Sjaj7j/a = (7j/hj)Sjhjaj/a on accoimt of Wulff s theorem. For spherical bodies (fluid phases) the well-known Kelvin-equation (7/f = 7/r) follows from Eq. (5.73). 

Jochmann s equation, 164 Joule, 31 experiments with gases, 137 Kelvin effect, 164, 225 researches, 28, 51 theorem, 136... 

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