In this paper, the fundamental equations of fluid dynamics are investigated under the principle of invariance against averaging over arbitrary control volumes and invariance against multiple levels of averaging. In addition to the balance equations for mass and momentum new balance equations are introduced for the first spatial moments of these quantities. These equations are discussed in comparison to micro-polar fluid theory. For small but finite control volumes a closure of these balance equations is achieved using third order Taylor series expansion. The most important result is that the conventional momentum balance equation should be extended to read partial derivative rho ui/partial derivative t + partial derivative rho unui/mu xn + partial derivative/2 partial derivative xn (rho(s(nm) - u(n)o(m)) partial derivative ui/partial derivative xm +rho(s(im) -u(i)o(m)) partial derivative un/partial derivative sm) = rho gi + partial derivative tau ni/partial derivative xn, where the new quantities o(m) and s(nm), which are described by new balance equations. account for the fact that even the smallest quantities of fluid material are of finite size and may contain significant variations in density and velocity. Examples of constitutive relations and boundary conditions show the difference compared to what is commonly used. [References: 11]
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机译：在本文中，研究了流体动力学的基本方程式，其原理是在任意控制量上均值不变，而在多个平均水平上均不变。除了质量和动量的平衡方程之外，还为这些量的第一个空间矩引入了新的平衡方程。与微极性流体理论比较讨论了这些方程。对于较小但有限的控制量，使用三阶泰勒级数展开式可以实现这些平衡方程的关闭。最重要的结果是常规动量平衡方程应扩展为读取偏导数rho ui /偏导数t +偏导数rho uiui / mu xn +偏导数/ 2偏导数xn（rho（s（nm）-u（ n）o（m））偏导数ui /偏导数xm + rho（s（im）-u（i）o（m））偏导数un /偏导数sm）= rho gi +偏导数tau ni /偏导数xn，其中新的量o（m）和s（nm），由新的平衡方程式描述。考虑到以下事实：即使最小量的流体材料也具有有限的大小，并且可能在密度和速度上包含很大的变化。本构关系和边界条件的示例显示了与通常使用的区别。 [参考：11]
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