Viscosity

Consider a gas of uniform number density, $n$, that has a gradient in the $x$-component of its mean flow velocity, $V_x$, along the $z$-axis, such that

$$V_x(z)= V_{x\,0}+ \frac{\partial V_x}{\partial z}\,dz.$$ (5.261)

It is assumed that the mean flow velocity is much less than the mean molecular speed. Slightly modifying the analysis of the previous two sections, the $x$-momentum per unit area, per second, carried by molecules whose speeds lie between $v$ and $v+dv$, and whose directions of motion subtend an angle lying between $\theta$ and $\theta+d\theta$ with the $z$-axis, that cross the $x$-$y$ plane, is

$$dP_{xz}= \left[m\,V_x'\right]\left[n\,F(v)\,dv\right]\left[\frac{1}{2}\,\sin\theta\,d\theta\right][v\,\cos\theta],$$ (5.262)

where $F(v)$ is the distribution of molecular speeds, $m$ the molecular mass, and $V_x'$ the $x$-component of flow velocity where the molecules last made a collision. On average, the molecules move a distance $l$ (i.e., the mean free path) between collisions. Hence, $dz =- l\,\cos\theta$, and

$$V_x' = V_{x\,0} - \frac{\partial V_x}{\partial z}\,l\,\cos\theta.$$ (5.263)

Thus, the net flux of $x$-momentum across the $x$-$y$ plane is

$$P_{xz} =\frac{m\,n}{2}\int_0^\pi\left[V_{x\,0}-\frac{\partial V_x... ...z}\,l\,\cos\theta\right]\cos\theta\,\sin\theta\,d\theta \int_0^\infty F(v)\,dv,$$ (5.264)

which gives

$$P_{xz} =-\eta\,\frac{\partial V_x}{\partial z},$$ (5.265)

where

$$\eta = \frac{1}{3}\,m\,n\,l\,\langle v\rangle$$ (5.266)

is termed the viscosity. Here, $\langle v\rangle$ is the mean molecular speed. (See Section 5.5.9.) Thus, we conclude that the flux of $x$-momentum in the $z$-direction is proportional to minus the gradient of the $x$-component of the flow velocity with respect to $z$. This is yet another example of Fick's law.

Equations (5.219), (5.244), and (5.266) imply that yield

$$\eta = \frac{2}{3\,\pi^{3/2}}\,\frac{1}{R^{\,2}}\sqrt{k_B\,T\,m},$$ (5.267)

where $R$ is the molecular diameter, and $T$ the gas temperature. It can be seen that the viscosity of an ideal gas scales as $T^{1/2}$, and is independent of the pressure at constant temperature. The previous formula yields the following estimate for the viscosity of air (assuming that $T=15^\circ$ C, $R=2\times 10^{-10}\,{\rm m}$, and $m=29\,m_p$),

$$\eta = 4\times 10^{-5}\,{\rm N\,s\,m^{-2}},$$ (5.268)

which turns out to be too large by a factor 2 (because of the approximate nature of our calculation).

Consider a slab of gas lying between $z$ and $z+dz$. The rate of change of the $x$-momentum contained in the slab is the difference between the flux of momentum into the slab and the flux of momentum out of the slab. In other words,

$$\frac{\partial (n\,A\,dz\,m\,V_x)}{\partial t} = \left[P_{xz}(z,t)- P_{xz}(z+dz,t)\right]A,$$ (5.269)

where $A$ is the cross-sectional area of the slab. It follows that

$$\frac{\partial V_x}{\partial t} = -\frac{1}{n\,m}\,\frac{\partial P_{xz}}{\partial z}.$$ (5.270)

Making use of Equations (5.265) and (5.266), we get

$$\frac{\partial V_x}{\partial t} = D_\eta\,\frac{\partial^2 V_x}{\partial z^2},$$ (5.271)

where

$$D_\eta = \frac{\eta}{n\,m} = \frac{1}{3}\,l\,\langle v\rangle.$$ (5.272)

Of course, Equation (5.271) is the diffusion equation, and $D_\eta$ is the associated diffusivity. Thus, we conclude, by comparison with the analysis in the previous two sections, that momentum diffuses through an ideal gas at the same very slow rate at which molecules and heat diffuse. Moreover, it is clear, from Sections 5.1.5 and 5.1.7, that momentum diffusion is due to a random walk of molecules with excess momentum under the action of molecular collisions.

Equations (5.235), (5.260), and (5.272) lead to the prediction that

$$D = \frac{\kappa}{n\,{\cal C}} = \frac{\eta}{n\,m}$$ (5.273)

for an ideal gas, where $D$ is the molecular diffusivity, $\kappa$ the thermal conductivity, $\eta$ the viscosity, $n$ the number density of molecules, $m$ the molecular mass, and ${\cal C}$ the molecular heat capacity. It turns out that this prediction is only approximately true due to additional factors, such as intermolecular forces, that have not been incorporated into our highly simplified analysis. For example, the ratio $\kappa\,m/(\eta\,{\cal C})$, which should be unity according to simple kinetic theory, actually takes the values $2.40$, $2.49$, $1.91$, $1.91$, and $1.90$ for helium, argon, hydrogen, nitrogen, and oxygen, respectively, at standard temperature and pressure.

The prediction that the thermal conductivity and viscosity of an ideal gas are both independent of the gas pressure, at constant temperature, breaks down when the pressure becomes sufficiently low that the mean free path between collisions becomes comparable with the size of the gas's container. Under these circumstances, the thermal conductivity and viscosity both become approximately proportional to the pressure, at constant temperature.