For two-layer system between rigid walls we arrive at the dispersion equation tanh gh∘ 1 ffiffiffiffiffiffiffiffiffiffiffifi 1 4A1ω2 g2 q 2A1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4A1ω2 g2 q þ tanh gh∘ 2 ffiffiffiffiffiffiffiffiffiffiffifi 1 4A2ω2 g2 q 2A2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4A2ω2 g2 q ¼0: ð18:5Þ For two-layer system between rigid wall at the bottom and pressure-free boundary at the top we arrive at the dispersion equation tanh gh∘ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4A1ω2 g2 s 2A1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4A1ω2 g2 s þ1 0 BBB BBB BBB B@ 1 CCC CCC CCC CA tanh gh∘ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4A2ω2 g2 s 2A2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4A2ω2 g2 s þ1 0 BBB BBB BBB B@ 1 CCC CCC CCC CA 4ω2A2 g2 tanh gh∘ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4A1ω2 g2 s 2A1 tanh gh∘ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4A2ω2 g2 s 2A2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4A1ω2 g2 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4A2ω2 g2 s ¼0: ð18:6Þ Both dispersion equations are explored analytically and numerically in [6]. 18.5 The Ehrenfest Theory of the Second Order Phase Transformations The Ehrenfest model of the second order phase transformation can be formulated as a certain type of singularity of thermodynamic potential. Let us dwell on the thermodynamic parameter space “specific volume v-specific entropy η” and the canonically associated thermodynamic potential, called the specific internal energy e ¼e(ν, η). Usually, the textbooks prefer the thermodynamic parameter space “specific volume ν-the absolute temperature T” and the canonically associated thermodynamic potential, called the specific free energyψ ¼ψ(ν, T). The textbook choice is dictated by the fact that for the first order phase transformations and for the statistical mechanics the thermodynamic space (ν, T) is much more convenient for the analysis that the space (ν, η). However, for the analysis of the second order phase transformations the variables (ν, T) do not deliver any advantages as compared with the variables (ν, η). At the same time, the variables (ν, η) are more convenient, than (ν, T) for thermodynamic analysis of shock waves. That is why in this paper we prefer to describe the Ehrenfest model in the variables (ν, η). Per Ehrenfest model, the potential e ¼e(ν, η) possesses singularities located along the smooth curve S ν; η ð Þ η Σ νð Þ¼0: ð18:7Þ The potential e(ν, η) and its first derivatives e(ν, η) and eη(ν, η) are all continuous everywhere, including the surface Σ. At the same time, the second derivates evv(ν, η), evη(ν, η), andeηη(ν, η) are continuous only outside the surfaceΣ. The second derivatives have finite one-sided limit values onΣ. However, those one-sided limit values are not equal to each other. For the second singularities the jumps of the one-sided limits must satisfy the following compatibility conditions: Δevv ¼HΣ 2 ν, Δevη ¼ HΣν, Δeηη ¼H, ð18:8Þ where the jump-function H(ν) is defined along the curve Σ(ν). The identities (18.8) express just a fact of calculus (see, for instance, [7])—there is no special physics, whatsoever, behind it. Eliminating two functions Hand Σν between the three identities (18.8) we arrive at the following relationship [8]: ΔevvΔeηη ¼ΔevηΔevη, ð18:9Þ expressed solely in terms of the second derivatives of the internal energy potential. This is the celebrated identity of Ehrenfest. 116 P. Grinfeld and M. Grinfeld
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