{"id":709,"date":"2007-05-23T16:25:19","date_gmt":"2007-05-23T21:25:19","guid":{"rendered":"http:\/\/softbeam.net\/hobby\/?p=709"},"modified":"2007-05-23T16:25:19","modified_gmt":"2007-05-23T21:25:19","slug":"stokes-einstein-relation-from-wikimedia","status":"publish","type":"post","link":"https:\/\/softbeam.net\/hobby\/?p=709","title":{"rendered":"Stokes-Einstein Relation from wikimedia"},"content":{"rendered":"

<\/a>In physics<\/a> (namely, in kinetic theory<\/a>) the Einstein relation<\/strong> is a previously unexpected connection revealed by Einstein<\/a> in his 1905<\/a> paper on Brownian motion<\/a>:<\/p>\n

\n
\"D<\/dd>\n<\/dl>\n

linking D<\/em>, the Diffusion constant<\/a>, and \u03bc<\/em>, the mobility<\/a> of the particles; where k<\/em>B<\/em><\/sub><\/span><\/em> is Boltzmann’s constant<\/a>, and T<\/em> is the absolute temperature<\/a>.<\/p>\n

The mobility \u03bc<\/em> is the ratio of the particle’s terminal drift velocity to an applied force, \u03bc = vd<\/sub> \/ F<\/em>.<\/p>\n

This equation is an early example of a fluctuation-dissipation relation<\/a>.<\/p>\n

<\/span><\/span>In the limit of low Reynolds number<\/a>, the mobility \u03bc<\/em> is the inverse of the drag coefficient \u03b3<\/em>. For spherical particles of radius r<\/em>, Stokes law<\/a> gives<\/p>\n

\n
\"\\gamma<\/dd>\n<\/dl>\n

where \u03b7<\/em> is the viscosity<\/a> of the medium. Thus the Einstein relation becomes<\/p>\n

\n
\"D=\\frac{k_B<\/dd>\n<\/dl>\n

This equation is also known as the Stokes-Einstein Relation<\/strong>. We can use this to estimate the Diffusion coefficient<\/a> of a globular protein<\/a> in aqueous solution: For a 100 kDalton<\/a> protein, we obtain D<\/em> ~10-10<\/sup> m\u00b2 s-1<\/sup>, assuming a “standard” protein density of ~1.2 103<\/sup> kg m-3<\/sup>.<\/p>\n

Also:<\/p>\n

The word “viscosity” derives from the Latin<\/a> word “viscum<\/span>” for mistletoe<\/a>. A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds.<\/p>\n

Isaac Newton<\/a> postulated that, for straight, parallel<\/a> and uniform flow, the shear stress, \u03c4, between layers is proportional to the velocity<\/a> gradient<\/a>, \u2202u<\/em>\/\u2202y<\/em>, in the direction perpendicular<\/a> to the layers, in other words, the relative motion of the layers.<\/p>\n

\n
\"\\tau=\\eta.<\/dd>\n<\/dl>\n

Here, the constant \u03b7 is known as the coefficient of viscosity<\/em>, the viscosity<\/em>, or the dynamic viscosity<\/em>. Many fluids<\/a>, such as water<\/a> and most gases<\/a>, satisfy Newton’s criterion and are known as Newtonian fluids<\/a>. Non-Newtonian fluids<\/a> exhibit a more complicated relationship between shear stress<\/a> and velocity gradient<\/a> than simple linearity.<\/p>\n

The relationship between the shear stress and the velocity gradient can also be obtained by considering two plates closely spaced apart at a distance y<\/em>, and separated by a homogeneous<\/a> substance. Assuming that the plates are very large, with a large area A<\/em>, such that edge effects may be ignored, and that the lower plate is fixed, let a force F<\/em> be applied to the upper plate. If this force causes the substance between the plates to undergo shear flow (as opposed to just shearing<\/a> elastically<\/a> until the shear stress<\/a> in the substance balances the applied force), the substance is called a fluid<\/a>. The applied force is proportional to the area and velocity of the plate and inversely proportional to the distance between the plates. Combining these three relations results in the equation F = \u03b7(Au\/y)<\/em>, where \u03b7 is the proportionality factor called the absolute viscosity<\/em> (with units Pa\u00b7s = kg\/(m\u00b7s) or slugs\/(ft\u00b7s)). The absolute viscosity is also known as the dynamic viscosity<\/em>, and is often shortened to simply viscosity<\/em>. The equation can be expressed in terms of shear stress; \u03c4 = F\/A = \u03b7(u\/y)<\/em>. The rate of shear deformation is u<\/em> \/ y<\/em><\/span> and can be also written as a shear velocity, du\/dy<\/em>. Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.<\/p>\n

In many situations, we are concerned with the ratio of the viscous force to the inertial<\/a> force, the latter characterised by the fluid<\/a> density<\/a> \u03c1. This ratio is characterised by the kinematic viscosity<\/em>, defined as follows:<\/p>\n

\n
\"\\nu.<\/dd>\n<\/dl>\n

James Clerk Maxwell<\/a> called viscosity fugitive elasticity<\/em> because of the analogy that elastic deformation opposes shear stress in solids<\/a>, while in viscous fluids<\/a>, shear stress is opposed by rate<\/em> of deformation.<\/p>\n

Units<\/strong>
\n<\/a><\/p>\n

<\/span>Viscosity (dynamic\/absolute viscosity): \u03b7<\/span> or \u03bc<\/span><\/span><\/h4>\n

The IUPAC<\/a> symbol for viscosity is the Greek symbol eta (\u03b7<\/span>), and dynamic viscosity is also commonly referred to using the Greek symbol mu (\u03bc<\/span>). The SI<\/a> physical unit<\/a> of dynamic viscosity is the pascal<\/a>–second<\/a> (Pa\u00b7s), which is identical to 1 kg<\/a>\u00b7m\u22121<\/sup>\u00b7s\u22121<\/sup>. If a fluid<\/a> with a viscosity of one Pa\u00b7s is placed between two plates, and one plate is pushed sideways with a shear stress<\/a> of one pascal<\/a>, it moves a distance equal to the thickness of the layer between the plates in one second<\/a>. The name poiseuille<\/a> (Pl) was proposed for this unit (after Jean Louis Marie Poiseuille<\/a> who formulated Poiseuille’s law<\/a> of viscous flow), but not accepted internationally. Care must be taken in not confusing the poiseuille with the poise<\/a> named after the same person!<\/p>\n

The cgs<\/a> physical unit<\/a> for dynamic viscosity is the poise<\/em>[1]<\/a><\/sup> (P; IPA<\/a>: [pwaz]<\/span>)) named after Jean Louis Marie Poiseuille<\/a>. It is more commonly expressed, particularly in ASTM<\/a> standards, as centipoise<\/em> (cP). The centipoise is commonly used because water has a viscosity of 1.0020 cP (at 20 \u00b0C; the closeness to one is a convenient coincidence).<\/p>\n

\n
1 P = 1 g\u00b7cm\u22121<\/sup>\u00b7s\u22121<\/sup><\/dd>\n<\/dl>\n

The relation between Poise and Pascal-second is:<\/p>\n

\n
10 P = 1 kg\u00b7m\u22121<\/sup>\u00b7s\u22121<\/sup> = 1 Pa\u00b7s<\/dd>\n
1 cP = 0.001 Pa\u00b7s = 1 mPa\u00b7s<\/dd>\n<\/dl>\n

<\/a><\/p>\n

<\/span>Kinematic viscosity: \u03bd<\/span><\/span><\/h4>\n

Kinematic viscosity (Greek symbol: \u03bd<\/span>) has SI units (m2<\/sup>\u00b7s\u22121<\/sup>). The cgs physical unit for kinematic viscosity is the stokes<\/em> (abbreviated S or St), named after George Gabriel Stokes<\/a>. It is sometimes expressed in terms of centistokes<\/em> (cS or cSt). In U.S. usage, stoke<\/em> is sometimes used as the singular form.<\/p>\n

\n
1 stokes = 100 centistokes = 1 cm2<\/sup>\u00b7s\u22121<\/sup> = 0.0001 m2<\/sup>\u00b7s\u22121<\/sup>.<\/dd>\n
1 centistokes = 1 mm\u00b2\/s<\/dd>\n<\/dl>\n

<\/a><\/p>\n

<\/span>Dynamic versus kinematic viscosity<\/span><\/h4>\n

Conversion between kinematic and dynamic viscosity, is given by \u03bd\u03c1 = \u03b7<\/span>. Note that the parameters must be given in SI units not in P, cP or St.<\/p>\n

For example, if \u03bd =<\/span> 1 St (=0.0001 m2<\/sup>\u00b7s-1<\/sup>) and \u03c1 =<\/span> 1000 kg m-3<\/sup> then \u03b7 = \u03bd\u03c1 =<\/span> 0.1 kg\u00b7m\u22121<\/sup>\u00b7s\u22121<\/sup> = 0.1 Pa\u00b7s<\/p>\n","protected":false},"excerpt":{"rendered":"

In physics (namely, in kinetic theory) the Einstein relation is a previously unexpected connection revealed by Einstein in his 1905 paper on Brownian motion: linking D, the Diffusion constant, and \u03bc, the mobility of the particles; where kB is Boltzmann’s constant, and T is the absolute temperature. The mobility \u03bc is the ratio of the … Continue reading “Stokes-Einstein Relation from wikimedia”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-709","post","type-post","status-publish","format-standard","hentry","category-physical"],"_links":{"self":[{"href":"https:\/\/softbeam.net\/hobby\/index.php?rest_route=\/wp\/v2\/posts\/709","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/softbeam.net\/hobby\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/softbeam.net\/hobby\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/softbeam.net\/hobby\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/softbeam.net\/hobby\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=709"}],"version-history":[{"count":0,"href":"https:\/\/softbeam.net\/hobby\/index.php?rest_route=\/wp\/v2\/posts\/709\/revisions"}],"wp:attachment":[{"href":"https:\/\/softbeam.net\/hobby\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=709"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/softbeam.net\/hobby\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=709"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/softbeam.net\/hobby\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=709"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}