In this paper, we study two different cases of inhomogeneous EOS of the form pd = w(t)ρd + w1f(H,t). We derive the energy density of dark fluid and dark matter component for both the cases. Further, we calculate the evolution of energy density, gravitational constant and cosmological constant. We also explore the finite time singularity and thermodynamic stability conditions for the two cases of EOS. Finally, we discuss the thermodynamics of inhomogeneous EOS with the derivation of internal energy, Temperature and entropy and also show that all the stability conditions are satisfied for the two cases of EOS.

中文翻译：

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非均匀 EOS 的能量条件及其热力学分析及有限时间未来奇点问题的求解

在本文中，我们研究了两种不同形式的非均匀 EOS 案例pd = w(吨)ρd + w1F(H,吨). 我们推导出这两种情况下暗流体和暗物质成分的能量密度。此外，我们计算了能量密度、引力常数和宇宙学常数的演变。我们还探讨了 EOS 两种情况的有限时间奇点和热力学稳定性条件。最后，我们讨论了非均匀 EOS 的热力学与内能、温度和熵的推导，并证明了 EOS 的两种情况都满足了所有的稳定性条件。