Hydrodynamics of the non-relativistic compressible fluid in the curved spacetime is derived using the generalized framework of the stochastic variational method (SVM) for continuum medium. The fluid-stress tensor of the resultant equation becomes asymmetric for the exchange of the indices, different from the standard Euclidean one. Its incompressible limit suggests that the viscous term should be represented with the Bochner Laplacian. Moreover the modified Navier–Stokes–Fourier equation proposed by Brenner can be considered even in the curved spacetime. To confirm the compatibility with the symmetry principle, SVM is applied to the gauge-invariant Lagrangian of a charged compressible fluid and then the Lorentz force is reproduced as the interaction between the Abelian gauge fields and the viscous charged fluid.

中文翻译：

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弯曲时空中可压缩流体力学的变分公式和应力张量的对称性

使用连续变分介质的随机变分方法（SVM）的广义框架，得出了弯曲时空中非相对论性可压缩流体的流体动力学。与标准的欧几里得指数不同，合成方程的流体应力张量对于指数的交换变得不对称。它的不可压缩极限表明，粘性术语应用Bochner Laplacian表示。而且，即使在弯曲的时空中，也可以考虑Brenner提出的改进的Navier–Stokes–Fourier方程。为了确认与对称原理的兼容性，将SVM应用于带电可压缩流体的轨距不变拉格朗日，然后将洛伦兹力再现为阿贝尔测距场与粘性带电流体之间的相互作用。