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第510行： − − In order to deduce the scale invariance of these equations we specify an equation of state, relating the fluid pressure to the fluid density. The equation of state depends on the type of fluid and the conditions to which it is subjected. For example, we consider the isothermal ideal gas, which satisfies − :P=c_s^2\rho, − where c_s is the speed of sound in the fluid. Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformations − :x\rightarrow\lambda x, − :t\rightarrow\lambda^2 t, − :\rho\rightarrow\lambda^{-1} \rho, − :\mathbf{u}\rightarrow\mathbf{u}. − Given the solutions \mathbf{u}(\mathbf{x},t) and \rho(\mathbf{x},t), we automatically have that − \lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t) and \lambda\rho(\lambda\mathbf{x},\lambda^2 t) are also solutions.
→Other examples of scale invariance 标度不变性的其他实例
Given the solutions <math>\mathbf{u}(\mathbf{x},t)</math> and <math>\rho(\mathbf{x},t)</math>, we automatically have that
Given the solutions <math>\mathbf{u}(\mathbf{x},t)</math> and <math>\rho(\mathbf{x},t)</math>, we automatically have that
<math>\lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t)</math> and <math>\lambda\rho(\lambda\mathbf{x},\lambda^2 t)</math> are also solutions.
<math>\lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t)</math> and <math>\lambda\rho(\lambda\mathbf{x},\lambda^2 t)</math> are also solutions.
为了推导这些方程的尺度不变性，我们指定一个状态方程，将流体压力与流体密度联系起来。状态方程取决于流体的类型及其所处的条件。例如，我们考虑等温理想气体，它满足：
为了推导这些方程的尺度不变性，我们指定一个状态方程，将流体压力与流体密度联系起来。状态方程取决于流体的类型及其所处的条件。例如，我们考虑等温理想气体，它满足：