Fluid Closure

No amount of manipulation, or rearrangement, can cure our fluid equations of their most serious defect: the fact that they are incomplete. In their present form, (220)-(222), our equations relate interesting fluid quantities, such as the density, $n_s$, the flow velocity, ${\bf V}_s$, and the scalar pressure, $p_s$, to unknown quantities, such as the viscosity tensor, $\mbox{\boldmath$\pi$}_s$, the heat flux density, ${\bf q}_s$, and the moments of the collision operator, ${\bf F}_s$ and $W_s$. In order to complete our set of equations, we need to use some additional information to express the latter quantities in terms of the former. This process is known as closure.

Lack of closure is an endemic problem in fluid theory. Since each moment is coupled to the next higher moment (e.g., the density evolution depends on the flow velocity, the flow velocity evolution depends on the viscosity tensor, etc.), any finite set of exact moment equations is bound to contain more unknowns than equations.

There are two basic types of fluid closure schemes. In truncation schemes, higher order moments are arbitrarily assumed to vanish, or simply prescribed in terms of lower moments. Truncation schemes can often provide quick insight into fluid systems, but always involve uncontrolled approximation. Asymptotic schemes depend on the rigorous exploitation of some small parameter. They have the advantage of being systematic, and providing some estimate of the error involved in the closure. On the other hand, the asymptotic approach to closure is mathematically very demanding, since it inevitably involves working with the kinetic equation.

The classic example of an asymptotic closure scheme is the Chapman-Enskog theory of a neutral gas dominated by collisions. In this case, the small parameter is the ratio of the mean-free-path between collisions to the macroscopic variation length-scale. It is instructive to briefly examine this theory, which is very well described in a classic monograph by Chapman and Cowling.

Consider a neutral gas consisting of identical hard-sphere molecules of mass $m$ and diameter $\sigma$. Admittedly, this is not a particularly physical model of a neutral gas, but we are only considering it for illustrative purposes. The fluid equations for such a gas are similar to Eqs. (220)-(222):

$$\frac{dn}{dt} + n\,\nabla\!\cdot\!{\bf V} = 0,$$ (228)

$$m n\,\frac{d {\bf V}}{dt} + \nabla p + \nabla\!\cdot \!\mbox{\boldmath$\pi$}+ mn\,{\bf g} = {\bf0},$$ (229)

$$\frac{3}{2}\frac{d p}{dt} + \frac{5}{2}\,p\,\nabla\!\cdot\!{\bf V} + \mbox{\boldmath$\pi$}:\nabla{\bf V} + \nabla\!\cdot\!{\bf q} = 0.$$ (230)

Here, $n$ is the (particle) density, ${\bf V}$ the flow velocity, $p$ the scalar pressure, and ${\bf g}$ the acceleration due to gravity. We have dropped the subscript $s$ because, in this case, there is only a single species. Note that there is no collisional friction or heating in a single species system. Of course, there are no electrical or magnetic forces in a neutral gas, so we have included gravitational forces instead. The purpose of the closure scheme is to express the viscosity tensor, $\mbox{\boldmath$\pi$}$, and the heat flux density, ${\bf q}$, in terms of $n$, ${\bf V}$, or $p$, and, thereby, complete the set of equations.

The mean-free-path $l$ for hard-sphere molecules is given by

$$l = \frac{1}{\sqrt{2}\,\pi\,n\,\sigma^2}.$$ (231)

This formula is fairly easy to understand: the volume swept out by a given molecule in moving a mean-free-path must contain, on average, approximately one other molecule. Note that $l$ is completely independent of the speed or mass of the molecules. The mean-free-path is assumed to be much smaller than the variation length-scale $L$ of macroscopic quantities, so that

$$\epsilon = \frac{l}{L} \ll 1.$$ (232)

In the Chapman-Enskog scheme, the distribution function is expanded, order by order, in the small parameter $\epsilon$:

$$f({\bf r}, {\bf v}, t) = f_0({\bf r}, {\bf v}, t) + \epsilon... ... {\bf v}, t) + \epsilon^2\, f_2({\bf r}, {\bf v}, t) + \cdots.$$ (233)

Here, $f_0$, $f_1$, $f_2$, etc., are all assumed to be of the same order of magnitude. In fact, only the first two terms in this expansion are ever calculated. To zeroth order in $\epsilon$, the kinetic equation requires that $f_0$ be a Maxwellian:

$$f_0({\bf r}, {\bf v}, t) = n({\bf r}) \left(\frac{m}{2\pi\,T... ...!\left[-\frac{m\,({\bf v}- {\bf V})^2} {2\,T({\bf r})}\right].$$ (234)

Recall that $p=n\,T$. Note that there is zero heat flow or viscous stress associated with a Maxwellian distribution function. Thus, both the heat flux density, ${\bf q}$, and the viscosity tensor, $\mbox{\boldmath$\pi$}$, depend on the first-order non-Maxwellian correction to the distribution function, $f_1$.

It is possible to linearize the kinetic equation, and then rearrange it so as to obtain an integral equation for $f_1$ in terms of $f_0$. This rearrangement depends crucially on the bilinearity of the collision operator. Incidentally, the equation is integral because the collision operator is an integral operator. The integral equation is solved by expanding $f_1$ in velocity space using Laguerre polynomials (sometime called Sonine polynomials). It is possible to reduce the integral equation to an infinite set of simultaneous algebraic equations for the coefficients in this expansion. If the expansion is truncated, after $N$ terms, say, then these algebraic equations can be solved for the coefficients. It turns out that the Laguerre polynomial expansion converges very rapidly. Thus, it is conventional to only keep the first two terms in this expansion, which is usually sufficient to ensure an accuracy of about $1\%$ in the final result. Finally, the appropriate moments of $f_1$ are taken, so as to obtain expression for the heat flux density and the viscosity tensor. Strictly speaking, after evaluating $f_1$, we should then go on to evaluate $f_2$, so as to ensure that $f_2$ really is negligible compared to $f_1$. In reality, this is never done because the mathematical difficulties involved in such a calculation are prohibitive.

The Chapman-Enskog method outlined above can be applied to any assumed force law between molecules, provided that the force is sufficiently short-range (i.e., provided that it falls off faster with increasing separation than the Coulomb force). For all sensible force laws, the viscosity tensor is given by

$$\pi_{\alpha\beta} =- \eta \left( \frac{\partial V_\alpha}{\p... ...c{2}{3}\,\nabla\!\cdot\!{\bf V}\,\delta_{\alpha\beta} \right),$$ (235)

whereas the heat flux density takes the form

$${\bf q} = - \kappa\,\nabla T.$$ (236)

Here, $\eta$ is the coefficient of viscosity, and $\kappa$ is the coefficient of thermal conduction. It is convenient to write

$$\eta = m n \,{\mit\chi}_v,$$ (237)

$$\kappa = n\,{\mit\chi}_t,$$ (238)

where ${\mit\chi}_v$ is the viscous diffusivity and ${\mit\chi}_t$ is the thermal diffusivity. Note that both ${\mit\chi}_v$ and ${\mit\chi}_t$ have the dimensions ${\rm m}^2\,{\rm s}^{-1}$ and are, effectively, diffusion coefficients. For the special case of hard-sphere molecules, Chapman-Enskog theory yields:

$${\mit\chi}_v = \frac{75\,\pi^{1/2}}{64}\,\left[1+ \frac{3}{202} +\cdots\right]\,\nu\,l^2 = A_v\,\nu\,l^2,$$ (239)

$${\mit\chi}_t = \frac{5\,\pi^{1/2}}{16}\,\left[ 1+ \frac{1}{44}+\cdots\right]\,\nu\,l^2 = A_t\,\nu\,l^2.$$ (240)

Here,

$$\nu \equiv \frac{v_t}{l} \equiv \frac{\sqrt{2\,T/m}}{l}$$ (241)

is the collision frequency. Note that the first two terms in the Laguerre polynomial expansion are shown explicitly (in the square brackets) in Eqs. (239)-(240).

Equations (239)-(240) have a simple physical interpretation: the viscous and thermal diffusivities of a neutral gas can be accounted for in terms of the random-walk diffusion of molecules with excess momentum and energy, respectively. Recall the standard result in stochastic theory that if particles jump an average distance $l$, in a random direction, $\nu$ times a second, then the diffusivity associated with such motion is $\chi\sim\nu\,l^2$. Chapman-Enskog theory basically allows us to calculate the numerical constants $A_v$ and $A_t$, multiplying $\nu\,l^2$ in the expressions for $\chi_v$ and $\chi_t$, for a given force law between molecules. Obviously, these coefficients are different for different force laws. The expression for the mean-free-path, $l$, is also different for different force laws.

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 length-scale is $L$. Let us define the following normalized quantities: $\hat{n}=n/\bar{n}$, $\hat{v}_t= v_t/\bar{v}_t$, $\hat{l} = l/\bar{l}$, $\hat{{\bf r}}= {\bf r}/ L$, $\hat{\nabla} = L\,\nabla$, $\hat{t} = \lambda\,\bar{v}_t\,t/L$, $\hat{\bf V} = {\bf V}/\lambda\,\bar{v}_t$, $\hat{T} = T/m\,\bar{v}_t^{~2}$, $\hat{\bf g} = L\,{\bf g}/ (1+\lambda^2)\,\bar{v}_t^{~2}$, $\hat{p}= p/m\,\bar{n}\,\bar{v}_t^{~2}$, $\hat{\mbox{\boldmath$\pi$}} = \mbox{\boldmath$\pi$}/ \lambda\,\epsilon\,m\,\bar{n}\,\bar{v}_t^{~2}$, $\hat{\bf q}= {\bf q} / \epsilon\,m\,\bar{n}\,\bar{v}_t^{~3}$. Here, $\epsilon = \bar{l}/L\ll 1$. Note that

$$\hat{\mbox{\boldmath$\pi$}} = - A_v\,\hat{n}\,\hat{v}_t\,\hat{l}\left( \frac{\partial\hat{V}_\a... ...a}-\frac{2}{3}\,\hat{\nabla}\! \cdot\!\hat{\bf V}\,\delta_{\alpha\beta}\right),$$ (242)

$$\hat{\bf q} = - A_t\,\hat{n}\,\hat{v}_t\,\hat{l}\,\,\hat{\nabla}\hat{T}.$$ (243)

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

$$\frac{d\hat{n}}{d\hat{t}} + \hat{n}\,\hat{\nabla}\!\cdot\!\hat{\bf V} = 0,$$ (244)

$$\lambda^2\,\hat{n}\,\frac{d \hat{\bf V}}{d\hat{t}} + \hat{\nabla}... ...la}\!\cdot \!\hat{\mbox{\boldmath$\pi$}} + (1+\lambda^2) \,\hat{n}\,\hat{\bf g} = {\bf0},$$ (245)

$$\lambda\,\frac{3}{2}\frac{d \hat{p}}{d\hat{t}} + \lambda\,\frac{... ...th$\pi$}}:\hat{\nabla}\hat{\bf V} + \epsilon\,\hat{\nabla}\!\cdot\!\hat{\bf q} = 0,$$ (246)

where

$$\frac{d}{d\hat{t}}\equiv \frac{\partial}{\partial \hat{t}} + \hat{\bf V}\!\cdot\!\hat{\nabla}.$$ (247)

Note that the only large or small quantities in the above equations are the parameters $\lambda$ and $\epsilon$.

Suppose that $\lambda\gg 1$. In other words, the flow velocity is much greater than the thermal speed. Retaining only the largest terms in Eqs. (244)-(246), our system of fluid equations reduces to (in unnormalized form):

$$\frac{dn}{dt} + n\,\nabla\!\cdot\!{\bf V} = 0,$$ (248)

$$\frac{d{\bf V}}{dt} + {\bf g} \simeq {\bf0}.$$ (249)

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. Note that the gas pressure (i.e., energy density) can be neglected in the cold-gas limit, since the thermal velocity is much smaller than the flow velocity, and so there is no need for an energy evolution equation. Furthermore, the viscosity can also be neglected, since the viscous diffusion velocity is also far smaller than the flow velocity.

Suppose that $\lambda\sim O(1)$. In other words, the flow velocity is of order the thermal speed. Again, retaining only the largest terms in Eqs. (244)-(246), our system of fluid equations reduces to (in unnormalized form):

$$\frac{dn}{dt} + n\,\nabla\!\cdot\!{\bf V} = 0,$$ (250)

$$m n\,\frac{d {\bf V}}{dt} + \nabla p + mn\,{\bf g} \simeq {\bf0},$$ (251)

$$\frac{3}{2}\frac{d p}{dt} + \frac{5}{2}\,p\,\nabla\!\cdot\!{\bf V} \simeq 0.$$ (252)

The above equations can be rearranged to give:

$$\frac{dn}{dt} + n\,\nabla\!\cdot\!{\bf V} = 0,$$ (253)

$$m n\,\frac{d {\bf V}}{dt} + \nabla p + mn\,{\bf g} \simeq {\bf0},$$ (254)

$$\frac{d}{dt}\!\left(\frac{p}{n^{5/3}}\right) \simeq 0.$$ (255)

These are called the hydrodynamic equations, since 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. Note that the gas pressure is non-negligible in the hydrodynamic limit, since the thermal velocity is of order the flow speed, and so an energy evolution equation is needed. However, the energy equation takes a particularly simple form, because Eq. (255) is immediately recognizable as the adiabatic equation of state for a monatomic gas. This is not surprising, since the flow velocity is still much faster than the viscous and thermal diffusion velocities (hence, 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, the flow velocity is of order the viscous and thermal diffusion velocities. Our system of fluid equations now reduces to a force balance criterion,

$$\nabla p + mn\,{\bf g} \simeq {\bf0},$$ (256)

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

$$\frac{dn}{dt} + n\,\nabla\!\cdot\!{\bf V} = 0,$$ (257)

$$m n\,\frac{d {\bf V}}{dt} + \nabla\!\cdot \!\mbox{\boldmath$\pi$} \simeq {\bf0},$$ (258)

$$\frac{3}{2}\frac{d p}{dt} + \frac{5}{2}\,p\,\nabla\!\cdot\!{\bf V} + \nabla\!\cdot\!{\bf q} \simeq 0.$$ (259)

Clearly, this set of equations is only appropriate to relatively quiescent, quasi-equilibrium, gas dynamics. Note that virtually all of the terms in our original fluid equations, (228)-(230), must be retained in this limit.

The above investigation reveals an important truth in gas dynamics, which also applies to plasma dynamics. Namely, the form of the fluid equations depends crucially 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 get faster, and vice versa.