弹性力学与流体力学基本方程的统一性

UNITY OF BASIC EQUATIONS OF ELASTICITY AND FLUID MECHANICS

本文将流体微元看成固体微元的特殊情况,并结合流体静止时无切应力的性质,首先利用弹性力学的柯西公式得到流体静压强的概念,其次利用弹性力学的平衡微分方程得到流体静力学的欧拉方程。然后通过考虑弹性体与不可压缩流体的共性及差异,利用弹性体动力学的基本方程直接推导出流体动力学中不可压缩黏性流体的N–S方程。本文方法避开了流体力学基本方程的传统推导方法,证明了流体力学与弹性力学的基本方程具有高度的统一性。利用该研究结果可以进一步改善力学课程的教学内容,加深理解力学基本原理,有效提高力学课程教学水平。

This paper regards the fluid micro-element as a special case of solid micro-element and considers the nature of no shear stress when the fluid is at rest, the concept of hydrostatic pressure is obtained by using the Cauchy formula of elasticity, then, the equilibrium differential equation of elasticity is used to derive the Euler equation of hydrostatics. By considering the similarities and differences between elastic solid and incompressible viscous fluid, the basic equation of elasticity is used to directly derive the N– S equations for incompressible viscous fluid in fluid dynamics. This method avoids the traditional derivation method of the basic equations of fluid mechanics and proves that the basic equations of elasticity and fluid mechanics are highly unified. The research results can further improve the teaching content of mechanics course, promote the deep understanding of the basic principles of mechanics, and effectively improve the teaching quality of mechanics course.