Navier-Stokes方程变分形式的拉格朗日量
Lagrangians for variational formulations of the Navier-Stokes equation
AI总结：
本文研究了粘性流动的变分形式，通过引入热自由度或基于最小能量耗散原理构建拉格朗日量，成功导出Navier-Stokes方程，证明了拉格朗日、Hamilton与牛顿力学在推导该方程时的等价性。
摘要：
本文研究了导致Navier-Stokes方程的粘性流动变分形式。由于粘性会导致耗散，从而引起机械能向热能的不可逆转移，因此在构建粘性耗散拉格朗日量时，通过嵌入流动的热力学方面（如热质和流动火用）引入了热自由度。另一种方法基于以下假设：压力梯度力是一种约束力，其唯一作用是维持连续性约束，且其大小在每一时刻均最小。基于这些考虑，根据最小能量耗散原理构建了拉格朗日量，应用Euler-Lagrange方程可直接导出Navier-Stokes方程的标准形式，或至少能够为简单的一维定常和非定常粘性流动生成相同的运动方程。这些研究表明，就Navier-Stokes方程的推导而言，拉格朗日力学、Hamilton力学与牛顿力学之间存在等价性。然而，结论之一是，在更复杂的情况下变分方法的吸引力对于应用流体力学工作者而言仍是一个悬而未决的问题。
英文摘要：
Variational formulations for viscous flows which lead to the Navier-Stokes equation are examined. Since viscosity leads to dissipation and, therefore, to the irreversible transfer of mechanical energy to heat, thermal degrees of freedom have been included in the construction of viscous dissipative Lagrangians, by embedding of thermodynamics aspects of the flow, such as thermasy and flow exergy. Another approach is based on the presumption that the pressure gradient force is a constrained force, whose sole role is to maintain the continuity constraint, with a magnitude that is minimum at every instant. From these considerations, Lagrangians based on the minimal energy dissipation principal have been constructed from which the application of the Euler-Lagrange equation leads to the standard form of the Navier-Stokes equation directly, or at least they are capable of generating the same equations of motion for simple steady and unsteady 1 D viscous flows. These efforts show that there is equivalence between Lagrangian, Hamiltonian, and Newtonian mechanics as far as the derivation of the Navier-Stokes equation is concerned. However, one of the conclusions is that the attractiveness of the variational approach in more complex situations is still an open question for the applied fluid mechanician.
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