Factor hydrodynamic resistance

An alternative point of view assumes that each repeat unit of the polymer chain offers hydrodynamic resistance to the flow such that f-the friction factor per repeat unit-is applicable to each of the n units. This situation is called the free-draining coil. The free-draining coil is the model upon which the Debye viscosity equation is based in Chap. 2. Accordingly, we use Eq. (2.53) to give the contribution of a single polymer chain to the rate of energy dissipation ... 

Fio. 9. Experimental measurements of CAST soot hydrodynamic resistance factor as a function of Peclet number and aggregate mobility diameter. The continuous lines are plotted using the scaling relation form Eqs. (8) and (9). 

The stabilizing factors for dispersions are the repulsive surface forces, the particle thermal motion, the hydrodynamic resistance of the medium, and the high surface elasticity of fluid particles and films. 

The hydrodynamic resistance of dispersion medium in the gap between particles against flowing out is one of the kinetic stability factors. The decrease in thickness of fluid layer between the particles during coagulation is related to viscous flow of liquid out of a narrow gap between the particles. For solid particles the liquid flow velocity is zero at the interface and highest in the center of a gap. The rate with which the gap between two circular plane-parallel plates of radius r (Fig. VII-7) shrinks, dh/dt, is related to the volume of liquid that flows per second across the side surface of cylindrical gap, dV/dt, via the following relationship ... 

Any mobility of the surface decreases the velocity difference and the viscous stresses. The result is that the hydrodynamic resistance becomes smaller and the floating velocity of a bubble according to (8.6) increases by a factor of 3/2 as compared to Stokes Eq. (8.5). In early experiments, under the condition of Re < 1, it was found (Lebedev 1916) that small bubbles of a diameters less than 0.01 cm behave like rigid spheres since their velocity is described by Stokes formula (8.5). At the same time. Bond (1927) has found that drops of a sufficiently large size fall at velocities described by Eq. (8.6). To overcome contradictions with the Hadamard-Rybczynski theory, Boussinesq (1913) considered the hypothetical influence of the surfaee viscosity and derived the following relation. 

The Reynolds number for a particle Rep of supercritical size, deposited on the surface of a sufficiently large bubble (for which a potential distribution of the liquid velocity field is valid), is much larger than imity. In this case, the hydrodynamic resistance is expressed by a resistance coefficient. In aerosol mechanics a technique is used (Fuks, 1961) in which the non-linearity from the resistance term is displaced by the inertia term. As a result, a factor appears in the Stokes number which, taking into account Eq. (11.20), can be reduced to (l + Rep /b). This allows us to find the upper and the lower limits of the effect by introducing K instead of K " into Eq. (10.47) and the factor X in the third term. 

Shear Factor, F. The shear factor F is a generalized hydrodynamic resistivity of porous media. It appears in the momentum equation 63 and is needed to solve the problem of single-phase flow in porous media. The shear factor F can be related to the pressure drop of a unidirectional flow without bounding wall effects, that is, in a one-dimensional medium through equation 21. In this section, we give a detailed account for the derivation of the expressions for fv and F. 

The role of hydrodynamic interaction in Brownian diffusion was discussed in Section 8.2. Consider now its effect on turbulent coagulation. Formally, it can be taken into account in the same manner as in Brownian motion, by introducing a correction multiplier into the factor of turbulent diffusion (10.57). Another, more correct way (see Section 11.3) is to use the Langevin equation that helped us determine the factor of Brownian diffusion in Section 8.2. As was demonstrated in [60], the factor of turbulent diffusion is inversely proportional to the second power of the hydrodynamic resistance factor ... 

Sometimes instead of the hydrodynamic resistance factor, we use mobility b = h, thus expressing velocity in terms of force ... 

Hence, the factor of Brownian diffusions is inversely proportional to the first power of factor of hydrodynamic resistance of particle. 

In case of turbulent diffusion, the situation is somewhat different. Motion of particles under action of turbulent pulsations is not connected to thermal fluctuations. Therefore B = const and the factor of turbulent diffusion is inversely proportional to the second power of factor of hydrodynamic resistance. 

If the change of hydrodynamic resistance factor with distance is great as compared with particle s displacement in pulsation, then the quantity h figuring in (11.66) depends on displacement x. The same takes place in motion of particles near the wall or at mutual approach of particles. 

At the distances between particles big in comparison with their sizes, each of them is completely entrained by the liquid and h9 = hr, uf = u. Decrease of the distances between particles results in change of factors of hydrodynamic resistance of particles H and hr. The first factor changes slightly, whereas the second one grows and becomes infinite at the particles contact. 

Expression (11.70) for the factor of turbulent diffusion does not take into account motion of the second particle. To take proper account of mutual influence of particles on the velocity of their approach, it should proceed as follows. Let u be the velocity of one particle relative to another, and Ui and Ui - the particles velocities relative to a reference frame whose origin lies between the particles on the line connecting their centers. Then u = ui — U2. Forces Fading on particles, are equal in magnitude and opposite in diredions. Then the fador of hydrodynamic resistance to particles approach can be written as... 

Here h denote the factors of hydrodynamic resistance to motion of each particle (i = 1,2). 

Expression for hydrodynamic resistance factor of drops with mobile interface will be given in section 13.7, in which the coalescence of drops in emulsion will be considered. 

We take the following approximate expression for the hydrodynamic resistance factor... 

For Browrtian diffusion of small particles, the influence of hydrodynamic interaction on the collision frequency was studied in works [28, 29], which also mention the decrease in the collision frequency by a factor of 1.5-2. This decrease is not as large as in the case of turbulent coagulation. There are two reasons why the effect of hydrodynamic interaction on the collision frequency of particles differs so substantially in the cases of turbulent flow and Brownian motion. First, the particle size is different in these two cases (the characteristic size of particles participating in Brownian motion is smaller than that of particles in a turbulent emulsion flow). Second, the hydrodynamic force behaves differently (the factor of Browrtian diffusion is inversely proportional to the first power of the hydrodynamic resistance factor h, and the factor of turbulent diffusion - to the second power of h). 

The assumptions made allow us to consider the coalescence of drops with a mobile surface in the same manner as that of drops with a fully retarded surface. The main difference from the case considered in Section 13.6 is in the form of the hydrodynamic resistance factor. If the drops are placed far apart, the factor of hydrodynamic resistance for the relative motion of drops is determined by the formula (11.71), where each of the factors hi and hi is determined according to Hadamar-Rubczynskis formula... 

Finally, consider the case of interacting drops having viscosities and //2 that are different from the viscosity of the ambient liquid. It should be treated in the same way as the previous case where the drops have equal internal viscosities, except that we must change the expression for the factor of hydrodynamic resistance h. To this end, consider two drops of types 1 and 2 which move with abso-... 

We have introduced new designations JI2 = Fi Fe Fi = FiIFc-The factors of hydrodynamic resistance hy depend on the relative distance s between the drops. Paper [43] solves the problem of slow central motion of two drops in a liquid where the two drops and the liquid all have different viscosities. The factors hy are found in the infinite series form. The appoximate solution of a similar problem is obtained by method of reflection in [46], and hy are found as power series in ratios Ri/r and R2/T which may be considered as asymptotic expressions for factors hy at s 2. An asymptotic expression for h is obtained in [39] for small values of the gap between the drops at s 2 ... 

Note that in calculations of such factors, the hydrodynamic resistance of drop motion, as well as forces of molecular interactions, are not taken into account. Hereinafter, it will be shown that correct allowance for these forces significantly increases the characteristic time of drop coagulation. 

Factor of hydrodynamic resistance at motion of non-hindered (free) particle kg-s-... 

For other applications such as the simultaneous sedimentation at large co (Eq. 17), the overall minimization of the flow rate Q may be required. Resorting again to Eq. 9, we can also reduce the quotient Ar to throttle the flow rate. This geometric factor reflects the ratio of the hydrostatic pressure head ( Ar) to the hydrodynamic resistance ( Uct). So a radially shallow channel with a high tilt angle between the channel axis and the radial direction, i.e., Ar I, diminishes Q (Fig. 1). A further reduction of Q can be accomplished by increasing the flow resistance, e.g., by a narrow, meander-like flow channel with an embedded stationary phase. 

Polyacrylamide El, with the lowest electrochemical degradation factor of 11.2 in Table 3, experiences the smallest reduction of resistance factor in the presence of univalent and divalent electrolytes, from 55.9 in river water to 49.5 in an 80/20 mixture of river and formation waters. These unusually large resistance factors probably resulted from the hydrodynamic resistance of the long linear polymer chain which is a unique characteristic of its gamma radiation manufacturing process. There appears to be some correspondence between the effect of electrolytes on viscosity and screen factor since polymers C and D1 with the lowest electrochemical degradation exhibit the greatest reduction in screen factor on... 

Figure 2.17 fa) The correction factor = F/Fr for hydrodynamic resistance force exerted on a disk interacting with superhydrophobic stripes vs. a dimensionless gap width H/L. Solid curves correspond to a local slip length b/L = 10 (from top to bottom 2 = 0.2, 0.5, and 0.9), dashed curves to b/L = 0.1 (from top to bottom = 0.2 and 0.5), dash-dotted curve to b/L = 0.01 and = 0.5. (b) The plot of f g vs. for a thin gap CW < L) and several superhydrophobic patterns anisotropic stripes (solid line), isotropic textures attaining HS bounds (dashed and dash-dotted lines), and isotropic Schulgasser structure (diamond), all with a local slip b/H = 10 (reprinted with permission from Ref. 14, copyright 2011, lOP Publishing). 

Hydrodynamic principles for gas bearings are similar to those involved with Hquid lubricants except that gas compressibility usually is a significant factor (8,69). With gas employed as a lubricant at high speeds, start—stop wear is minimized by selection of wear-resistant materials for the journal and bearing. This may involve hard coatings such as tungsten carbide or chromium oxide flame plate, or soHd lubricants, eg, PTFE and M0S2. 

The structure of the cake formed and, consequently, its resistance to liquid flow depends on the properties of the solid particles and the liquid phase suspension, as well as on the conditions of filtration. Cake structure is first established by hydrodynamic factors (cake porosity, mean particle size, size distribution, and particle specific surface area and sphericity). It is also strongly influenced by some factors that can conditionally be denoted as physicochemical. These factors are ... 

Due to the combining effects of hydrodynamic and physicochemical factors, the study of cake structure and resistance is extremely complex, and any mathematical description based on theoretical considerations is at best only descriptive. 

Electrostatic repulsion of the anionic carboxylate groups elongates the polymer chain of partially hydrolyzed polyacrylamides increasing the hydrodynamic volume and solution viscosity. The extensional viscosity is responsible for increased resistance to flow at rapid flow rates in high permeability zones (313). The screen factor is primarily a measure of the extensional (elonga-tional) viscosity (314). The solution properties of polyacrylamides have been studied as a function of NaCl concentra-tion and the parameters of the Mark-Houwink-Sakaruda equation calculated... 

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