Stokes relation

For more complicated physical situations, the results again deviate from the usual Navier-Stokes relations, and may be useful in investigating rapidly varying phenomena that cannot be explained in the usual (hydrodynamic) fashion. 

When solvated ions migrate within the electrolyte, the drag force applied by the surrounding solvent molecules is measured by solvent viscosity rj. Thus, in a solvent of lower viscosity, the solvated ions would move more easily in response to an applied electric field, as expressed by the Einstein—Stokes relation (eq 3). Solvents of low viscosity have always been considered the ideal candidates for electrolyte application however, their actual use was restricted because most of these solvents have low dielectric constants (Tables 1 and 2) and cannot dissociate ions effectively enough to prevent ion pairing. 

The relation between friction and viscosity goes beyond the Stokes relation. The Navier-Stokes hydrodynamics has been generalized by Zwanzig and Bixon [23] to include the viscoelastic response of the medium. This generalization provides an elegant expression for the frequency-dependent friction which depends among other things on the frequency-dependent bulk and shear viscosities and sound velocity. 

In this section the studies of the relation between the friction (Q on a tagged solute and the viscosity (rjs) of the medium is presented for neat liquids in the normal regime. The well-known Stokes relation is often used to connect the friction (Q on a spherical molecule with the viscosity ( /,) of the medium and is given by... 

Mode coupling theory provides the following rationale for the known validity of the Stokes relation between the zero frequency friction and the viscosity. According to MCT, both these quantities are primarily determined by the static and dynamic structure factors of the solvent. Hence both vary similarly with density and temperature. This calls into question the justification of the use of the generalized hydrodynamics for molecular processes. The question gathers further relevance from the fact that the time (t) correlation function determining friction (the force-force) and that determining viscosity (the stress-stress) are microscopically different. 

The validity of the Stokes relation has also been investigated from the microscopic point of view, and the following surprising result is obtained. Indi-... 

As emphasized before, the hydrodynamic derivation (based on the contribution of the current mode alone [75]) of the relation between the friction and the viscosity has no validity in the case of neat liquids (where the tagged molecule is one of the solvent molecules). On the other hand, the experiments [76], the computer simulations [77], and the MCT calculations presented here all show that the ratio of friction to viscosity at high density almost always lies between 4n and 67c even for a neat liquid. Therefore, it is imperative to analyze the cause of apparent validity of the Stokes relation in greater depth. 

We may find the effects of hydrodynamical interactions by using the Stokes relation ... 

For a derivation of this equation Debye s monograph may be consulted. It will be observed that equation (27) differs from (22) in that the term fP/ikT is multiplied by the factor 1/(1 + foot), in which i is the operator / — 1, random distribution after the removal of the impressed field. By assuming that the molecules are spheres of radius r, and that Stokes relation for the rotation of a sphere in a medium of viscosity rj is valid, Debye obtains the equation... 

Relative Viscosity of Suspensions One of the most interesting derivations of the T vs. (() dependence (covering the full range of concentration) was published by Simha [1952]. He considered the effects of concentration on the hydrodynamic interactions between suspended particles of finite size. (Note that previously the particles were simply considered point centers of force that decayed with cube of the distance.) Simha adopted a cage model, placing each solid, spherical particle of radius a inside a spherical enclosure of radius b. At distances x < b, the presence of other particles does not influence flow around the central sphere and the Stokes relation is satisfied. This assumption leads to a modified Einstein [1906, 1911] relation ... 

One attempt to improve the agreement with experimental data is to use Hammond-Stokes relation in which the product D,Ti2 is plotted against the molar volume ratio of solvent to solute Vr. The slope is influenced by the following few factors, namely,... 

In addition to fister settling, a destabilised stem will generally possess enhanced fihration characteristics. These inqrrovements follow fiom the increased possibility of particulate colfisions, with the production of clusters or floes of partides. The clusters have a greater effective diameter. Thus fiom the Stokes relation h ... 

Stokes relations. It is interesting—and not unexpected— to observe that the reflection matrix R x) and the transmission matrix T x) are related to one another in simple ways. We refer to these relations as Stokes relations due to Stokes early discoveries of such relations between reflection and transmission coefficients for light rays impinging on slabs see [20]. 

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