Viscosity Stokes relation

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. 

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. 

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... 

It was shown in [179] that the effective viscosity is related to the ratio of the velocity of free sedimentation of a single particle according to the Stokes law to the velocity of particles in the suspension, that is, the effective viscosity is related to the correction factor A in the drag force. The expression... 

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 ... 

A different method, often used for calibration of electromagnetic tweezers, determines the viscous drag from the velocity v of the magnetic bead with radius r in a fluid of known viscosity r/ given by Stokes relation F= 6xj/n>. ... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

The relation between the microscopic friction acting on a molecule during its motion in a solvent enviromnent and macroscopic bulk solvent viscosity is a key problem affecting the rates of many reactions in condensed phase. The sequence of steps leading from friction to diflfiision coefficient to viscosity is based on the general validity of the Stokes-Einstein relation and the concept of describing friction by hydrodynamic as opposed to microscopic models involving local solvent structure. In the hydrodynamic limit the effect of solvent friction on, for example, rotational relaxation times of a solute molecule is [ ]... 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

Falling ball viscometers are based on Stokes law, which relates the viscosity of a Newtonian fluid to the velocity of the falling sphere. If a sphere is allowed to fall freely through a fluid, it accelerates until the viscous force is exactly the same as the gravitational force. The Stokes equation relating viscosity to the fall of a soHd body through a Hquid may be written as equation 34, where ris the radius of the sphere and d are the density of the sphere and the hquid, respectively g is the gravitational force and p is the velocity of the sphere. 

Supercritical Mixtures Dehenedetti-Reid showed that conven-tionaf correlations based on the Stokes-Einstein relation (for hquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group D g l/T was constant. Thus, although no general correlation ap es, only one data point is necessaiy to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SFg and COg. 

The identity tensor by is zero for i J and unity for i =J. The coefficient X is a material property related to the bulk viscosity, K = X + 2 l/3. There is considerable uncertainty about the value of K. Traditionally, Stokes hypothesis, K = 0, has been invoked, but the vahdity of this hypothesis is doubtful (Slattery, ibid.). For incompressible flow, the value of bulk viscosity is immaterial as Eq. (6-23) reduces to... 

Using the Stokes-Einstein equation for the viscosity, which is unexpectedly useful for a range of liquids as an approximate relation between diffusion and viscosity, explains a resulting empirical expression for the rate of formation of nuclei of the critical size for metals... 

Very commonly Eq. (4-5) is combined with Eq. (4-6), the Stokes-Einstein equation relating the diffusion coefficient to the viscosity -q. 

Viscosity is a measurement of resistance to flow. Although the unit of absolute viscosity is poise, its measurement is difficult. Instead, kinematic (flowing) viscosity is determined by measuring the time for a given flow through a capillary tube of specific diameter and length. The unit of kinematic viscosity is the stoke. However, in general practice, centistoke is used. Poise is related to stoke by the equation ... 

Elutriation differs from sedimentation in that fluid moves vertically upwards and thereby carries with it all particles whose settling velocity by gravity is less than the fluid velocity. In practice, complications are introduced by such factors as the non-uniformity of the fluid velocity across a section of an elutriating tube, the influence of the walls of the tube, and the effect of eddies in the flow. In consequence, any assumption that the separated particle size corresponds to the mean velocity of fluid flow is only approximately true it also requires an infinite time to effect complete separation. This method is predicated on the assumption that Stokes law relating the free-falling velocity of a spherical particle to its density and diameter, and to the density and viscosity of the medium is valid... 

Since thermal agitation is the common origin of transport properties, it gives rise to several relationships among them, for example, the Nemst-Einstein relation between diffusion and conductivity, or the Stokes-Einstein relation between diffusion and viscosity. Although transport... 

It is important from a practical viewpoint to predict the shear viscosity of mixtures from those of pure melts. For alkali nitrate melts, a linear dependence has been found between the reorientational line width obtained by Raman measurements and the ratio of temperature divided by shear viscosity.For NO3 ions, the depolarized Raman scattering from 1050cm" total stretching vibrational mode (Al) has a contribution to the line width L, which is caused by the reorientational relaxation time of the Csv axis of this ion. The Stokes-Einstein-Debye(SED) relation establishes a relation between the shear viscosity r of a melt and the relaxation time for the reorientation of a particle immersed in it ... 

Viscosity is a useful quantity, in that both rotational and translation mobility of molecules in solution are viscosity dependent and can be related to viscosity through the Stokes-Einstein equation ... 

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