Stoke’s formula

All other cases are between the extreme limits of Stokes s and Newton s formulas. So we may say, that modeling the free-falling velocity of any single particle by the formula (14.49), the exponent n varies in the region 0.5 s n < 2. In the following we shall assume that k and n are fixed, which means that we consider a certain size-class of particles. 

Similar to scalar field problems, in order to obtain an integral representation for the momentum eqns. (10.63) and (10.64) for the flow field (u, p), Green s formulae for the momentum equations (Theorems (10.2.1) and (10.2.2)) are used together with the fundamental singular solution of Stokes equations, i.e.,... 

Let us, for a moment, consider a single particle in one dimension with a Hamiltonian of the type// = p2/2m + V(x). This is a second-order differential operator, and this means that the general solution to the inhomogeneous Eq. (3.51)—considered as a second-order differential equation—will consist of a linear superposition of two special solutions, where the coefficients will depend on the boundary conditions introduced. As a specific example, one could think of the two solutions to the JWKB problem, their connection formulas, and the Stoke s phenomenon for the coefficients. 

The force due to the shear in the vdscous sublayer on a particle lying on a surface, is analogous to the formula for Stoke s drag ... 

A number of numerical calculations was carried out for various Reynolds numbers up to 2000, and the duct s initial region length was found as a graphical function Lx = f(Re h, A) represented in Fig. 3.11. The dependence Lx vs A significantly differs for small and big Reynolds numbers Re. The value A = 0 corresponds to the case where the EPR is absent. It can be seen that, for A -> 0, all the curves arrive at the constant value c 0.03, but the curves associated with small Reynolds numbers 1, 5, and 10. Hence, it can be concluded that the principal Schlichting s formula (3.45) has been justified for large Re despite it was derived from the boundary-layer approach, rather than from the complete Navier - Stokes equations. 

Substituting v for a potential velocity field, Sutherland s formula is obtained. When we substitute Stokes velocity field into Eq. (10.25) and use Eq. (10.22), we obtain... 

Levin (1961) has shown that inertia deposition of particles below a critical size, which corresponds to a critical Stokes number St = 1/12, is impossible. Regarding a finite size of particles, the collision is characterised by Sutherland s formula (10.11). Comparison of the results obtained from Sutherland s relation and by Levin enables to conclude that in the region of small St < St the approximation of the material point, accepted by Levin and useful at fairly big St, becomes unsuitable for Stsmall Stokes numbers were studied by Dukhin (1982 1983b) for particles of finite size. Under these conditions inertia forces retard microflotation. 

The effect of the specific density of a particle and its radius on the combined effect of centrifugal forces is shown in Fig. 10.16. It can be expected that Sutherland s formula describes the transport stage of the elementary act also at Stokes number close to the critical one. It becomes clear from the results in Fig. 10.16, that it is applicable only at 0, close to 90°. This condition is not fulfilled over a wide range of 0, and Ap. 

The same procedure can be applied to the cuiiiulaiits or moiiieiils f7(r). With Stokes-Einstein s formula, the -average hydrodynamic riidius of the inacromolecules is determined from the z-average diffusion coefficient... 

Applying the divergence theorem to the third term of the r.h.s. and using Stokes power formula (3.18), we obtain the following local Eulerian form of the dissipative energy equation ... 

The Navier-Stokes (NS) equations can be used to describe problems of fluid flow. Since these equations are scale-independent, flow in the microscale structure of a porous medium can also be described by a NS field. If the velocity on a solid surface is assumed to be null, the velocity field of a porous medium problem with a small pore size rapidly decreases (see Sect. 5.3.2). We describe this flow field by omitting the convective term v Vv, which gives rise to the classical Stokes equation We recall that Darcy s theory is usually applied to describe seepage in a porous medium, where the scale of the solid skeleton does not enter the formulation as an explicit parameter. The scale effect of a solid phase is implicitly included in the permeability coefficient, which is specified through experiments. It should be noted that Kozeny-Carman s formula (5.88) involves a parameter of the solid particle however, it is not applicable to a geometrical structure at the local pore scale. 

If a homogenization analysis (HA) is applied to porous media flow, which is described by the Stokes equation, we can immediately obtain Darcy s formula and the seepage equation in a macroscale field while in the microscale field the distributions of velocity and pressure are specified (Sanchez-Palencia 1980). We can also apply HA for a problem with a locally varying viscosity. 

Calculate the uncertainty in the fitted parameters - the slope and the intercept. In cell C25 enter the formula = SQRT(C21 C24 C24/(C21 E19 - C19 C19)). This should yield a value of 0.032. The slope is thus —0.988 0.032. When reporting uncertainties, only one significant figure is meaningful, unless the first digit is 1. Similarly, it is not meaningful to report the fitted parameters to decimal places beyond the uncertainty. Thus the slope is —0.99 0.03. We calculate the uncertainty of the intercept to be 0.049. Thus the intercept is 1.40 0.05. Stokes s law predicts a slope of —1 and an intercept of 1.398 (10 =25). Our fitted parameters and their uncertainties are consistent with Stokes s law. 

In the article on electro-osmosis (q.v.) a similar formula, but with 4 in place of the factor 6, is derived. Since electrophoresis is the reverse of electro-osmosis, the same expression should apply in both cases to the potential at the surface of shear between the two phases. The explanation of the apparent discrepancy is that instead of applying Stokes s law to a small sphere, the derivation of the electro-osmotic effect is based on the model of a parallel plate capacitor, i.e. on a large solid surface whose radius of curvature is negligible (compared with the thickness of the diffuse double layer). Closer analysis of the problem by Henry and Booth has shown that 4 is the correct factor for large particles, independent of their size and shape, but that for most systems, e.g. stable colloidal solutions, the factor varies between 4 and 6, depending on the size of the particle and the thickness of its atmosphere. 

The American Petroleum Institute recommends various minimal horizontal and vertical cross-section areas and depm-to-widm ratios. These minima are based on the formula version of Stokes s law, which relates me rate of rise of an oil droplet through a tank of water to (among omer things) me difference between me liquid densities and the square of me droplet diameter. 

---