Translational Peclet number

As with spherical particles the Peclet number is of great importance in describing the transitions in rheological behaviour. In order for the applied flow field to overcome the diffusive motion and shear thinning to be observed a Peclet number exceeding unity is required. However, we can define both rotational and translational Peclet numbers, depending upon which of the diffusive modes we consider most important to the flow we initiate. The most rapid diffusion is the rotational component and it is this that must be overcome in order to initiate flow. We can define this in terms of a diffusive timescale relative to the applied shear rate. The characteristic Maxwell time for rotary diffusion is... 

To determine the critical shear stress, %, we note that the translational Peclet number Pet[= 6iTr)ga yJkBT] - 8, allowii the calculation of the critical shear stress ... 

When y is given in dimensionless form as the translational Peclet number, the dimensionless critical shear rate Peclet number, y, is 8, as given in the following equation ... 

In the present section, however, we follow the lead of Taylor s original work and pursue an approach that is only designed to yield an approximation to the solution for large times and for large values of the Peclet number, Pe >0(1). It was stated earlier that the midpoint of the temperature pulse moves with the mean velocity of the fluid for large times where the Taylor analysis is applicable. Although this is a result that we must verify as part of our theoretical work, we shall assume for the time being that it is true. In this case, it is clearly more sensible to develop our theory based on a coordinate frame that also translates with... 

In previous sections of this chapter, we have considered forced convection heat transfer at low Peclet number from particles in a uniform streaming flow. The results are applicable if the density of the particle is different from that of the fluid, so that the particle is subject to gravitational and/or inertial forces that give it a translational motion relative to the fluid. In this case, we have seen that the relationship between Nu and Pe takes a common form,... 

Mass Transfer in Translational Flow at Low Peclet Numbers... 

Peclet numbers, the problem of mass exchange between a particle of arbitrary shape and a uniform translational flow were studied by the method of matched asymptotic expansions in [62]. The following expression was obtained for the mean Sherwood number up to first-order infinitesimals with respect to Pe ... 

Following [270], we first consider steady-state diffusion to the surface of a solid spherical particle in a translational Stokes flow (Re - 0) at high Peclet numbers. In the dimensionless variables, the mathematical statement of the corresponding problem for the concentration distribution is given by Eq. (4.4.3) with the boundary conditions (4.4.4) and (4.4.5), where the stream function is determined by (4.4.2). 

In this section, some interpolation formulas are presented (see [367, 368]) for the calculation of the mean Sherwood number for spherical particles, drops, and bubbles of radius a in a translational flow with velocity U at various Peclet numbers Pe = aU /D and Reynolds numbers Re = aU-Jv. We denote the mean Sherwood number by Shb for a gas bubble and by Shp for a solid sphere. 

Spherical particle as Re —> 0, 0 < Pe < oo. The problem of mass transfer to a solid spherical particle in a translational Stokes flow (Re -f 0) was studied in the entire range of Peclet numbers by finite-difference methods in [1, 60, 281], To find the mean Sherwood number for a spherical particle, it is convenient to use the following approximate formula [94] ... 

Spherical bubble at any Peclet numbers for Re > 35. For a spherical bubble in a translational flow at moderate and high Reynolds numbers and high Peclet numbers, the mean Sherwood number can be calculated by the formula [94]... 

Mass exchange between a spherical particle and the translational-shear flow (4.9.1) at high Peclet numbers was studied in [175]. For the mean Sherwood number depending on the parameters... 

The mean Sherwood number for the translational-shear flow (4.9.1) past a spherical drop under limiting resistance of the continuous phase at high Peclet numbers can be calculated by the formulas [164]... 

To construct approximate formulas for the Sherwood number in the case of a translational shear flow past particles and bubbles in the entire range of Peclet numbers, one can use formulas (4.7.9) and (4.7.10) where Shpoo and Shboo must be replaced by the right-hand sides of Eqs. (4.9.3), (4.9.4) and (4.9.5), (4.9.6), respectively, with 0 = 0. 

At low Peclet numbers, for the translational Stokes flow past an arbitrarily shaped body of revolution, formula (4.10.8) coincides with the exact asymptotic expression in the first three terms of the expansion [358], Since (4.10.8) holds identically for a spherical particle at all Peclet numbers, one can expect that for particles whose shape is nearly spherical, the approximate formula (4.10.8) will give good results for low as well as moderate or high Peclet numbers. 

Diffusion to an elliptic cylinder in a translational flow at high Peclet numbers was considered in [166]. 

Order of magnitude of dimensionless (related to the radius of drops or solid particles) characteristic sizes of regions of diffusion wake in translational Stokes flow at high Peclet number... 

The approximate expression (5.1.5) was tested at moderate Peclet numbers Pe = 10, 20, 50 (the corresponding values of Re are Re = 10, 20, and 0.5) for a translational flow past a solid sphere by comparison with the results of a numerical solution for a first-order surface chemical reaction. For the data taken from [2, 68], the error of Eq. (5.1.5) in these cases does not exceed 1.5%. 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

The dependence of the auxiliary Sherwood number Sho on the Peclet number Pe for a translational Stokes flow past a spherical particle or a drop is determined by the right-hand sides of (4.6.8) and (4.6.17). In the case of a linear shear Stokes flow, the values of Sho are shown in the fourth column in Table 4.4. 

In the case of mass exchange between a bubble and a translational Stokes flow of a quasi-Newtonian power-law fluid (n is close to unity), one can use the following simple approximate formula for calculating the mean Sherwood number at high Peclet numbers ... 

The hydrodynamic forces are always proportional to the viscosity of the medium. Therefore, suspension viscosities are scaled with the viscosity of the suspending medium, meaning that relative viscosities are used. As for dilute systems, the balance between Brownian motion and flow can be expressed by a Peclet number. Here the translational diffusivity D, has to be used, but that does not change the functionality (for spheres. Dr is proportional to D,). A dimensionless number is obtained by taking the ratio of the time scales for diffusion (D,) and convective motion (y). This is again a Peclet number ... 

Circular cylinder. The mass exchange between a circular cylinder of radius a and a uniform translational flow whose direction is perpendicular to the generatrix of the cylinder was considered in [186,218] for low Peclet and Reynolds numbers Pe = Sc Re and Re = aU-Jv. For the mean Sherwood number (per unit length of the cylinder) determined with respect to the radius, the following two-term expansions were obtained ... 

Particles, Drops, and Bubbles in Translational Flow. Various Peclet and Reynolds Numbers... 

Consider convection diffusion toward a spherical particle of radius R, which undergoes translational motion with constant velocity U in a binary infinite diluted solution [3], Assume the particle is small enough so that the Reynolds number is Re = UR/v 1. Then the flow in the vicinity of the particle will be Stoke-sean and there will be no viscous boundary layer at the particle surface. The Peclet diffusion number is equal to Peo = Re Sc. Since for infinite diluted solutions, Sc 10 and the flow can be described as Stokesian for the Re up to Re 0.5, it is perfectly safe to assume Pec 1. Thus, a thin diffusion boundary layer exists at the surface. Assume that a fast heterogeneous reaction happens at the particle surface, i.e. the particle is dissolving in the liquid. The equation of convective diffusion in the boundary diffusion layer, in a spherical system of coordinates r, 6, (p, subject to the condition that concentration does not depend on the azimuthal angle [Pg.128]

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