![\"\"](\"http:\/\/underthehood.blogwyrm.com\/wp-content\/uploads\/2020\/06\/Linear-Couette-Flow.png\")
<\/figure><\/div>\n\n\n\nThe $x$-component of the fluid\u2019s velocity $u_x$ varies as a function of $y$ and the usual definition of the stress in the problem relates it to the velocity gradient<\/p>\n\n\n\n
\\[ d_{xy} = \\mu \\frac{\\partial u_x(y)}{\\partial y} \\; . \\]<\/p>\n\n\n\n
The physical meaning of the stress is that $d_{xy}$ is the amount of momentum $p_x$ in the $x$-direction transported across any plane in the $y$-direction, which we will call $p_{xy} = – d_{xy}$ (with an appropriate change in sign to account for difference between a force acting on the fluid and the reaction of the fluid itself).<\/p>\n\n\n\n
Again, using the Reif\u2019s argument $1\/6$ of the molecules will be crossing some plane $y=Y$ in the upward direction and $1\/6$ of them will crossing downward. Assuming the molecular mass as $M$ and the number density as $n$, the difference in momentum is<\/p>\n\n\n\n
\\[ p_{xy} = \\frac{1}{6} M n {\\bar V} \\left[ u_x(y – \\lambda) – u_x(y + \\lambda) \\right] = – \\frac{1}{3} M n \\frac{\\partial u_x}{\\partial y} \\lambda \\; . \\]<\/p>\n\n\n\n
Comparing the two expressions gives the viscosity in terms of the microscopic parameters as<\/p>\n\n\n\n
\\[ \\mu = \\frac{1}{3} M n {\\bar V} \\lambda = M n D \\; . \\]<\/p>\n\n\n\n
As with all these types of results, this one should be taken with a grain of salt regarding the numeric factor. Other authors also get this $1\/3$ but they do so using quite different ways of \u2018averaging\u2019 across the populations. Nonetheless, the functional dependence of the viscosity on the microscopic parameters is correct and, in this case, the analysis of this dependence leads to some interesting conclusions.<\/p>\n\n\n\n
The first thing to note is that Reif expresses the mean free path generally as<\/p>\n\n\n\n
\\[ \\lambda = \\frac{1}{\\sqrt{2} \\sigma n } \\; , \\]<\/p>\n\n\n\n
where $\\sigma$ is the generic cross-section, which takes the value $\\pi d^2$ for hard sphere collisions giving the expression derived earlier. This generalization will prove useful in a bit but first let\u2019s look at using this formula for the mean free path explicitly for viscosity. Putting these together gives an expression for the viscosity<\/p>\n\n\n\n
\\[ \\mu = \\frac{1}{3 \\sqrt{2} \\sigma} M {\\bar V} \\; \\]<\/p>\n\n\n\n
that may be quite surprising.\u00a0 The amount of viscosity delivered by a gas is independent of density, at least over a wide range of values for the physical parameters that enter into this theory.\u00a0 The limitations occur in the limits of a very small mean free path, in which the gas molecules are nearly always colliding with each other, or when the mean free path is larger than the physical size of the experiment.\u00a0 Kittel and Kroemer, in their book Thermal Physics<\/em>, cite a quote from Robert Boyle in 1660 as reading<\/p>\n\n\n\n Before pressing on with more physics, it is worth noting that there are two satisfying things about the above quote. The first is that Boyle was scrupulous enough to report the null result of performing this experiment; a sentiment that bucks the trend of modern science where only \u2018breakthroughs\u2019 are reported. Second, it is refreshing to hear the \u2018snark\u2019 that Boyle conveys on his own behalf. <\/p>\n\n\n\n Finally, Reif presents how the viscosity changes as a function of temperature. There is an obvious dependence of speed through the explicit appearance of ${\\bar V} \\propto T^{1\/2}$ in calculating the viscosity. And, if the collision process were accurately modeled in terms of hard spheres this would be all there was to the dependence. However, the scattering cross section is, generically, a function of speed since the underlying forces (i.e., Coulomb scattering) have a stronger influence on the particle when it is moving slowly. This is where the relaxation of the hard sphere scattering assumption becomes relevant as the scattering cross section becomes a function of speed, which, in turn, is a function of temperature in the general case. Thus, in the general case<\/p>\n\n\n\n \\[ \\mu = \\mu \\left[ {\\bar V}(T), \\sigma ( {\\bar V}(T)) \\right] \\; \\]<\/p>\n\n\n\n leading to an overall dependance that Reif cites going as<\/p>\n\n\n\n \\[ \\mu \\propto T^{0.7} \\; . \\] An interesting implication of this functional dependence (with or without the extra, implicit temperature dependence coming through the effective cross section) is that the viscosity of a gas increases with increasing temperature, which is the exact opposite effect as is active in the vast majorities of liquids.<\/p>\n","protected":false},"excerpt":{"rendered":" Last month\u2019s blog presented the prototype \u2018algorithm\u2019 for relating macroscopic transport properties (e.g., the diffusion coefficient) to their microscopic mechanical attributes (e.g., the mean free path). This post extends the… Read more ><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1949","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/1949","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1949"}],"version-history":[{"count":9,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/1949\/revisions"}],"predecessor-version":[{"id":1964,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/1949\/revisions\/1964"}],"wp:attachment":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1949"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1949"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1949"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}