Editing Scale invariance (section)

===Newtonian fluid mechanics with no applied forces===

Under certain circumstances, [[fluid mechanics]] is a scale-invariant classical field theory. The fields are the velocity of the fluid flow, <math>\mathbf{u}(\mathbf{x},t)</math>, the fluid density, <math>\rho(\mathbf{x},t)</math>, and the fluid pressure, <math>P(\mathbf{x},t)</math>. These fields must satisfy both the [[Navier–Stokes equation]] and the [[continuity equation#Fluid dynamics|continuity equation]]. For a [[Newtonian fluid]] these take the respective forms

<math display="block">\rho\frac{\partial \mathbf{u}}{\partial t}+\rho\mathbf{u}\cdot\nabla \mathbf{u} = -\nabla P+\mu \left(\nabla^2 \mathbf{u}+\frac{1}{3}\nabla\left(\nabla\cdot\mathbf{u}\right)\right)</math>
:<math>\frac{\partial \rho}{\partial t}+\nabla\cdot \left(\rho\mathbf{u}\right)=0</math>
where <math>\mu</math> is the [[dynamic viscosity#Viscosity .28dynamic viscosity.29: .CE.BC|dynamic viscosity]].

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
:<math>P=c_s^2\rho,</math>
where <math>c_s</math> is the speed of sound in the fluid. Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformations
:<math>x\rightarrow\lambda x,</math>
:<math>t\rightarrow\lambda^2 t,</math>
:<math>\rho\rightarrow\lambda^{-1} \rho,</math>
:<math>\mathbf{u}\rightarrow\lambda^{-1}\mathbf{u}.</math>
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.