Nonspherical drop

The process just described, by which a nonspherical drop would be driven toward the spherical equilibrium shape by pressure gradients associated with variations in interface curvature, is but one example of a large number of situations in which fluid motions are actually caused by pressure gradients that are produced by variations in the curvature of a fluid interface. 

Many empirical relations for steady-state velocity of deformed drops and bubbles of various shapes, including shapes more complicated than the ellipsoidal shape, are presented in [94]. Laminar flow past nonspherical drops was studied numerically in [98, 517]. 

Numerous workers (B16, F2, F4, H17, J4, K4, K7, L8, M2, S5, and others) estimated the external heat-transfer coefficient in the continuous phase by assuming a velocity profile in the boundary layer and ambient fluid. Except for very low Reynolds numbers, the exact boundary layer solutions only apply to the front part of the drop, up to the separation point. Fortunately, simple assumptions sometimes suffice for extending the derivation to the entire drop, and the relationships obtained are in agreement with experimental data. The limitations of the analytical solutions, as well as their application to nonspherical drops, is concisely demonstrated in Lochiel and Calderbank s (LI8) recent study on mass transfer around axisymmetric bodies of revolutions. 

Drops larger than about 1 mm in diameter are significantly nonspherical the mean height to width ratio is approximated (P5) by ... 

For viscous energy loss, from Kozeny s equation, the pressure drop is proportional to the square of the specific surface area of solids So- For kinetic energy loss, from Burke and Plummer s relation, the pressure drop is proportional to So- So is related to the particle diameter by Eq. (5.351) for spherical particles for nonspherical particles, the dynamic diameter (see 1.2) may be used for the particle diameter. The general form of the pressure drop can be expressed as... 

One point that has not been emphasized is that all of the preceding analysis and discussion pertains only to the steady-state problem. From this type of analysis, we cannot deduce anything about the stability of the spherical (Hadamard Rybczynski) shape. In particular, if a drop or bubble is initially nonspherical or is perturbed to a nonspherical shape, we cannot ascertain whether the drop will evolve toward a steady, spherical shape. The answer to this question requires additional analysis that is not given here. The result of this analysis26 is that the spherical shape is stable to infinitesimal perturbations of shape for all finite capillary numbers but is unstable in the limit Ca = oo (y = 0). In the latter case, a drop that is initially elongated in the direction of motion is predicted to develop a tail. A drop that is initially flattened in the direction of motion, on the other hand, is predicted to develop an indentation at the rear. Further analysis is required to determine whether the magnitude of the shape perturbation is a factor in the stability of the spherical shape for arbitrary, finite Ca.21 Again, the details are not presented here. The result is that finite deformation can lead to instability even for finite Ca. Once unstable, the drop behavior for finite Ca is qualitatively similar to that predicted for infinitesimal perturbations of shape at Ca = oo that is, oblate drops form an indentation at the rear, and prolate drops form a tail. 

The dependence (4.12.3) can also be used to estimate the intensity of transient mass transfer for nonspherical particles, drops, and bubbles at Pe 1. In this case, all dimensionless variables r, Sh, Shst, and Pe must be defined on the basis of the same characteristic length a. Under this condition, the expression (4.12.3) provides valid asymptotic results for small as well as large times. Equation (4.12.3) can be rewritten as follows ... 

Formula (5.3.8) and Eq. (5.3.9) can be used for the calculation of the mean Sherwood number for nonspherical particles, drops, and bubbles at high Peclet numbers. 

The usage of the flow equations can be summarized as follows. For the case of a one-dimensional single fluid flow, either equation 106 or 108 can be used to predict the normalized pressure drop factor in a porous medium. The determined normalized pressure drop factor is related to the pressure drop by equation 11. For the simple case of packed spherical beads, ds and e are known a priori. The Reynolds number is evaluated using equation 93. For random packs of nonspherical particles, the particle s sphericity needs to be known. Equation 73 can be used to estimate ds. For the case of consolidated porous medium, one can estimate ds from the knowledge of the intrinsic permeability using equation 14. 

Equafion 7.14 can be used for nonspherical particles if the diameter of the sphere wifh fhe same specific surface as the particles is used. If the pressure drop through the bed is a significant proportion of the total pressure and the fluid is compressible, the velocity of the fluid will increase as it passes through the bed. The top of the bed will thus tend to fluidize at a lower flow rate of fluid than the bottom. 

At low Reynolds number (Re 1), the experimental values of X for both drops and gas bubbles are included in Fig. 6. The correspondence between predictions and experiments is seen to be about as good as can be expected in this kind of work. Indeed, the results do lend some support to the hypothesis of the importance of the viscous parameters in this regime. In view of the idealizations inherent in theoretical treatments coupled with the uncertainties arising from the inadequacy of the power law, nonspherical shapes, and possible viscoelastic and wall effects, the match seen in Fig. 6 is not bad at all. 

One encounters the following difficulties in the interpretation of the data from the experiments with interfacial dilatation. As discussed in Ref. 58, the shear viscosity, T jh, does not influence the total stress, 67, only for interfacial flow of perfect spherical symmetry. If the latter requirement is not fulfilled by a given experimental technique, its output data will be influenced by a mixture of dissipative effects (not only -r d but also -qsh and tr). The apparent interfacial viscosity thus determined is not a real interfacial property insofar as it depends on the specific method of measurement. For example, the apparent interfacial viscosity measured by the capillary-wave methods [189-196] depends on the frequency the apparent interfacial viscosity measured by the Langmuir trough method [197,198] is a sum of the dilatational and shear viscosities ("q + -q h) for the methods employing nonspherical droplet deformation, like the spinning-drop method [199-201], the apparent surface viscosity is a complex function of the dilatational and shear interfacial viscosities. 

It is easy to see that the size of the onions decreases dramatically if the innermost shell is not filled, which amounts to dropping the first term in the series (12). To generalize the theory, one has to takep to be variable and express the total bending energy in terms of the radii of the outermost and innermost onion skins. We refrain from this exercise in view of many other uncertainties. For instance, the nonspherical shapes of the outermost skins, because of their additional bending energies, should cause the phase transition from the onion phase to the planar multilayer system to be discontinuous. 

In principle, some types of nonspherical particles could be packed more tightly than spheres, although they would start to interact at lower concentrations. In reality, higher viscosities are normally found with nonspherical particles. The concentration law is approximately exponential at low to moderate concentrations, but equations similar to eq. 10.5.1 can still be used as well. The empirical value of 4>m can be much smaller than that for spherical particles (e.g., 0.44 for rough crystals with aspect ratios close to unity Kitano et al., 1981). If fibers are used, this value drops even further, down to 0.18 for an aspect ratio of 27 (see also Metzner, 1985). The decrease with aspect ratio seems to be roughly linear. Homogeneous suspensions of fibers with large aspect ratios are difficult to prepare and handle. As in dilute systems, the type of flow will determine the extent of the shape effect. Extensional flows are discussed below. 

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