Normalization of Neutral Gas Equations

Let $\bar{n}$ , $\bar{v}_t$ , and $\bar{l}$ be typical values of the particle density, the thermal velocity, and the mean-free-path, respectively. Suppose that the typical flow velocity is $\lambda\,\bar{v}_t$ , and the typical variation lengthscale of macroscopic quantities is $L$ . Let us define the following normalized quantities:

$$\hat{n} =\frac{n}{\bar{n}}, \hat{v}_t =\frac{v_t}{\bar{v}_t}, \hat{l} = \frac{l}{\bar{l}}, \hat{{\bf r}} = \frac{{\bf r}}{L},$$

$$\widehat{\nabla} = L\,\nabla, \hat{t} = \frac{\lambda\,\bar{v}_t\,t}{L}, \widehat{\bf V} = \frac{{\bf V}}{\lambda\,\bar{v}_t}, \widehat{T} = \frac{T}{m\,\bar{v}_t^{\,2}},$$

$$\hat{\bf g} = \frac{L\,{\bf g}}{(1+\lambda^2)\,\bar{v}_t^{\,2}}, \hat{p} = \frac{p}{m\,\bar{n}\,\bar{v}_t^{\,2}}, \widehat{\mbox{\boldmath$\pi$}} =\frac{\mbox{\boldmath$\pi$}}{ \lambda\,\epsilon\,m\,\bar{n}\,\bar{v}_t^{\,2}}, \widehat{\bf q} = \frac{{\bf q}}{\epsilon\,m\,\bar{n}\,\bar{v}_t^{\,3}}.$$

Here,

$$\epsilon = \frac{\bar{l}}{L}\ll 1.$$ (4.70)

Note that

$$\widehat{\pi}_{\alpha\beta} = - A_v\,\hat{n}\,\,\hat{v}_t\,\hat{l}\,\left( \frac{\partial\wid... ...frac{2}{3}\,\widehat{\nabla} \cdot\widehat{\bf V}\,\delta_{\alpha\beta}\right),$$ (4.71)

$$\widehat{\bf q} = - A_t\,\hat{n}\,\,\hat{v}_t\,\hat{l}\,\,\,\widehat{\nabla}\widehat{T}.$$ (4.72)

All hatted quantities are designed to be ${\cal O}(1)$ . The normalized fluid equations are written:

$$\frac{d\hat{n}}{d\hat{t}} + \hat{n}\,\,\widehat{\nabla}\cdot\widehat{\bf V} =0,$$ (4.73)

$$\lambda^2\,\,\hat{n}\,\frac{d \widehat{\bf V}}{d\hat{t}} + \wideh... ...}\cdot \widehat{\mbox{\boldmath$\pi$}} + (1+\lambda^2) \,\hat{n}\,\,\hat{\bf g} ={\bf0},$$ (4.74)

$$\lambda\,\frac{3}{2}\frac{d \hat{p}}{d\hat{t}} + \lambda\,\frac{5... ...widehat{\nabla}\widehat{\bf V} + \epsilon\,\widehat{\nabla}\cdot\widehat{\bf q} =0,$$ (4.75)

where

$$\frac{d}{d\hat{t}}\equiv \frac{\partial}{\partial \hat{t}} + \widehat{\bf V}\cdot\widehat{\nabla}.$$ (4.76)

The only large or small quantities remaining in the previous equations are the parameters $\lambda$ and $\epsilon$ .

Suppose that $\lambda\gg 1$ . In other words, suppose that the flow velocity is much greater than the thermal speed. Retaining only the largest terms in Equations (4.73)-(4.75), our system of fluid equations reduces to (in unnormalized form):

$$\frac{dn}{dt} + n\,\nabla\cdot{\bf V} =0,$$ (4.77)

$$\frac{d{\bf V}}{dt} + {\bf g} \simeq {\bf0}.$$ (4.78)

These are called the cold-gas equations, because they can also be obtained by formally taking the limit $T\rightarrow 0$ . The cold-gas equations describe externally driven, highly supersonic, gas dynamics. The gas pressure (that is, the thermal energy density) can be neglected in the cold-gas limit, because the thermal velocity is much smaller than the flow velocity. Consequently, there is no need for an energy evolution equation. Furthermore, the viscosity can also be neglected, because the viscous diffusion velocity is also far smaller than the flow velocity.

Suppose that $\lambda\sim {\cal O}(1)$ . In other words, suppose the flow velocity is of similar magnitude to the thermal speed. Again, retaining only the largest terms in Equations (4.73)-(4.75), our system of fluid equations reduces to (in unnormalized form):

$$\frac{dn}{dt} + n\,\nabla\cdot{\bf V} =0,$$ (4.79)

$$m \,n\,\frac{d {\bf V}}{dt} + \nabla p + m\,n\,{\bf g} \simeq {\bf0},$$ (4.80)

$$\frac{3}{2}\frac{d p}{dt} + \frac{5}{2}\,p\,\nabla\cdot{\bf V} \simeq 0.$$ (4.81)

The previous equations can be rearranged to give:

$$\frac{dn}{dt} + n\,\nabla\cdot{\bf V} =0,$$ (4.82)

$$m \,n\,\frac{d {\bf V}}{dt} + \nabla p + m\,n\,{\bf g} \simeq {\bf0},$$ (4.83)

$$\frac{d}{dt}\!\left(\frac{p}{n^{5/3}}\right) \simeq 0.$$ (4.84)

These are called the hydrodynamic equations, because they are similar to the equations governing the dynamics of water. The hydrodynamic equations govern relatively fast, internally driven, gas dynamics--in particular, the dynamics of sound waves. The gas pressure is non-negligible in the hydrodynamic limit, because the thermal velocity is similar in magnitude to the flow speed. Consequently, an energy evolution equation is needed. However, the energy equation takes a particularly simple form, as Equation (4.84) is immediately recognizable as the adiabatic equation of state for a monatomic gas. This is not surprising, because the flow velocity is still much faster than the viscous and thermal diffusion velocities (which accounts for the absence of viscosity and thermal conductivity in the hydrodynamic equations), in which case the gas acts effectively like a perfect thermal insulator.

Suppose, finally, that $\lambda\sim \epsilon$ . In other words, suppose the flow velocity is of similar magnitude to the viscous and thermal diffusion velocities. Our system of fluid equations now reduces to a force balance criterion,

$$\nabla p + m\,n\,{\bf g} \simeq {\bf0},$$ (4.85)

to lowest order. To next order, we obtain a set of equations describing the relatively slow viscous and thermal evolution of the gas:

$$\frac{dn}{dt} + n\,\nabla\cdot{\bf V} =0,$$ (4.86)

$$m \,n\,\frac{d {\bf V}}{dt} + \nabla \cdot \mbox{\boldmath$\pi$} \simeq {\bf0},$$ (4.87)

$$\frac{3}{2}\frac{d p}{dt} + \frac{5}{2}\,p\,\nabla\cdot{\bf V} + \nabla\cdot{\bf q} \simeq 0.$$ (4.88)

Clearly, this set of equations is only appropriate to relatively quiescent, quasi-equilibrium, gas dynamics. Virtually all of the terms in our original fluid equations, (4.55)-(4.57), must be retained in this limit.

The previous investigation reveals an important truth in gas dynamics, which also applies to plasma dynamics. Namely, the form of the fluid equations crucially depends on the typical fluid velocity associated with the type of dynamics under investigation. As a general rule, the equations get simpler as the typical velocity gets faster, and vice versa.