Stokes assumption

If the dominant relaxation is described by the shear viscosity t] (the Stokes assumption), then... 

Note from Eq. (197) for a rotating sphere that the ratio of inertial to viscous terms is of 0(/ r" ). Thus, Stokes assumption is consistent for alljr when R - -0. In essence, this is the reason why regular perturbation schemes work for the rotating sphere, but are invalid for streaming flow. 

At infinite dilution, the assumption of a constant relaxation time is reasonable and, using Stokes law as well, we have... 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Further support for this approach is provided by modern computer studies of molecular dynamics, which show that much smaller translations than the average inter-nuclear distance play an important role in liquid state atom movement. These observations have conhrmed Swalin s approach to liquid state diffusion as being very similar to the calculation of the Brownian motion of suspended particles in a liquid. The classical analysis for this phenomenon was based on the assumption that the resistance to movement of suspended particles in a liquid could be calculated by using the viscosity as the frictional force in the Stokes equation... 

Equations (4-5) and (4-7) are alternative expressions for the estimation of the diffusion-limited rate constant, but these equations are not equivalent, because Eq. (4-7) includes the assumption that the Stokes-Einstein equation is applicable. Olea and Thomas" measured the kinetics of quenching of pyrene fluorescence in several solvents and also measured diffusion coefficients. The diffusion coefficients did not vary as t) [as predicted by Eq. (4-6)], but roughly as Tf. Thus Eq. (4-7) is not valid, in this system, whereas Eq. (4-5), used with the experimentally measured diffusion coefficients, gave reasonable agreement with measured rate constants. 

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... 

As mentioned, the data obtained by this method are expressed as cumulative size distribution curves. Since the computations assume Stokes law for spherical particles, the plotted curves give the distribution of spherical particles which would behave like the actual sample with respect to this experiment. For this reason, the sizes on the distribution curves should be labelled Stokes Equivalent Diameter . Because of the underlying assumptions and the above interpretation of the results, it is clear that the repeatability of this method has more meaning than accuracy of comparison with results of other methods... 

We refer those who are interested in the details of the Debye Hiickel derivation to the following sources R. A. Robinson and R. H. Stokes. "Electrolyte Solutions", Academic Press, Inc., New York (1955). The Robinson/Stokes reference does an especially good job of summarizing and evaluating the assumptions made in the derivation H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions." Rcinhold Publishing Corporation, New York (1958) K. S. Pitzer. "Thermodynamics." Third Edition, McGraw Hill, Inc., New York (1995). 

A reactive gas is slowly bubbled into a column of liquid. The bubbles are small, approximately spherical, and are well separated from each other. Assume Stokes law and ignore the change in gas density due to elevation. The gas is pure and reacts in the liquid phase with first-order kinetics. Derive an expression for the size of the bubbles as a function of height in the column. Carefully specify any additional assumptions you need to make. 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

Thus, the Stokes-Einstein equation is expected to be valid for colloidal particles and suspensions of large spherical particles. Experimental evidence supports these assumptions [101], and this equation has occasionally been used for much smaller species. 

Henry [ 157] solved the steady-flow continuity and Navier-Stokes equations in spherical geometry, neglecting inertial terms but including pressure and electrical force terms, coupled with Poisson s equation. The electrical force term in Henry s analysis consisted of the sum of the externally applied electric field and the field due to the double layers. His major assumptions are low surface potential (i.e., potentials less than approximately 25 mV) and undistorted double layers. The additional parameter ku appearing in the Henry... 

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

This concept is due to Bjerrum, who in 1926 suggested that in simple electrolytes ions of the opposite charge could associate to form ion-pairs (Szwarc, 1965 Robinson Stokes, 1959). This concept of Bjerrum arose from problems with the Debye-Huckel theory, when the assumption that the electrostatic interaction was small compared with IcTwas not justified. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Equation (7.44) is known as the third approximation of the Debye-Hiickel theory. Numerous attempts have been made to interpret it theoretically, hi these attempts, either individual simplifying assumptions that had been made in deriving the equations are dropped or additional factors are included. The inclusion of ionic solvation proved to be the most important point. In concentrated solutions, solvation leads to binding of a significant fraction of the solvent molecules. Hence, certain parameters may change when solvation is taken into account since solvation diminishes the number of free solvent molecules (not bonded to the ions). The influence of these and some other factors was analyzed in 1948 by Robert A. Robinson and Robert H. Stokes. 

Stokes law is rigorously applicable only for the ideal situation in which uniform and perfectly spherical particles in a very dilute suspension settle without turbulence, interparticle collisions, and without che-mical/physical attraction or affinity for the dispersion medium [79]. Obviously, the equation does not apply precisely to common pharmaceutical suspensions in which the above-mentioned assumptions are most often not completely fulfilled. However, the basic concept of the equation does provide a valid indication of the many important factors controlling the rate of particle sedimentation and, therefore, a guideline for possible adjustments that can be made to a suspension formulation. 

The method preferred in our laboratory for determining the UWL permeability is based on the pH dependence of effective permeabilities of ionizable molecules [Eq. (7.52)]. Nonionizable molecules cannot be directly analyzed this way. However, an approximate method may be devised, based on the assumption that the UWL depends on the aqueous diffusivity of the molecule, and furthermore, that the diffusivity depends on the molecular weight of the molecule. The thickness of the unstirred water layer can be determined from ionizable molecules, and applied to nonionizable substances, using the (symmetric) relationship Pu = Daq/ 2/iaq. Fortunately, empirical methods for estimating values of Daq exist. From the Stokes-Einstein equation, applied to spherical molecules, diffusivity is expected to depend on the inverse square root of the molecular weight. A plot of log Daq versus log MW should be linear, with a slope of —0.5. Figure 7.37 shows such a log-log plot for 55 molecules, with measured diffusivities taken from several... 

The ionic mobility and diffusion coefficient are also affected by the ion hydration. The particle dimensions calculated from these values by using Stokes law (Eq. 2.6.2) do not correspond to the ionic dimensions found, for example, from the crystal structure, and hydration numbers can be calculated from them. In the absence of further assumptions, diffusion measurements again yield only the sum of the hydration numbers of the cation and the anion. 

Robinson and Stokes have demonstrated that, for simplifying assumptions, the following relationship is valid between ar s and /Ts ... 

Loutfy and coworkers [29, 30] assumed a different mechanism of interaction between the molecular rotor molecule and the surrounding solvent. The basic assumption was a proportionality of the diffusion constant D of the rotor in a solvent and the rotational reorientation rate kOI. Deviations from the Debye-Stokes-Einstein hydrodynamic model were observed, and Loutfy and Arnold [29] found that the reorientation rate followed a behavior analogous to the Gierer-Wirtz model [31]. The Gierer-Wirtz model considers molecular free volume and leads to a power-law relationship between the reorientation rate and viscosity. The molecular free volume can be envisioned as the void space between the packed solvent molecules, and Doolittle found an empirical relationship between free volume and viscosity [32] (6),... 

This analysis is based on the assumption that Stokes law applies, i.e., /VRe < 1. This is frequently a bad assumption, because many industrial centrifuges operate under conditions where iVRe > 1. If such is the case, an analytical solution to the problem is still possible by using the Dallavalle equation for CD, rearranged to solve for N t as follows ... 

Two spheres of equal terminal falling velocity settle in water starting from rest at the same horizontal level. How far apart vertically will the particles be when they have both reached 99 per cent of their terminal falling velocities It may be assumed that Stokes law is valid and this assumption should be checked. 

Estimate the terminal falling velocity /infinite dilution. On the assumption that Stokes law is applicable, calculate the particle diameter d. 

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