Drag coefficient deformed

Drops accelerated by an air stream may split, as described in Chapter 12. For drops which do not split, measured drag coefficients are larger than for rigid spheres under steady-state conditions (R2). The difference is probably associated more with shape deformations than with the history and added mass effects discussed above. For micron-size drops where there is no significant deformation, trajectories may be calculated using steady-state drag coefficients (SI). 

Wall thickness Channel width Acoustic velocity Friction coefficient Conductance Capillary number Discharge coefficient Drag coefficient Diameter Diameter Dean number Deformation rate tensor components Elastic modulus Energy dissipation rate Eotvos number Fanning friction factor Vortex shedding frequency Force... 

Up — Uc represents the resultant slip velocity between the particulate and continuous phase. Some other commonly used drag coefficient correlations are listed in Appendix 4.2. For fluid particles such as gas bubbles or liquid drops, the drag coefficient may be different than that predicted by the standard drag curve, due to internal circulation and deformation. For example, Johansen and Boysen (1988) proposed the following equation to calculate Cd, which is valid for ellipsoidal bubbles in the range 500 < Re < 5000 ... 

Kariyasaki [70] studied bubbles, drops, and solid particles in linear shear flow experimentally, and showed that the lift force on a deformable particle is opposite to that on a rigid sphere. For particle Reynolds numbers between 10 and 8 the drag coefficient could be estimated by Stokes law. The terminal velocity was determined to be equal to that of a particle moving in a quiescent... 

FIGURE 5.49 Deformed fluid particle (the inset) moving tangentially to an immobile solid snrface plot of the dimensionless drag coefficient, fy, vs. the dimensionless film thickness, hlR for three valnes of the dimensionless film radius, R/R (see Equation 5.303). 

Here, as usual, h and R denote the film thickness and radius, and R is the curvature radius of the spherical part of the particle surface. The dependence of the dimensionless drag coefficient, fy, on the distance h for different values of the ratio R/R is illustrated in Figure 5.49. The increase of R/R and the decrease of hlR may lead to an increase of the drag force,/j, by an order of magnitude. That is the reason the film between a deformed particle and a wall can be responsible for the major part of the energy dissipation. Moreover, the formation of doublets and flocks of droplets separated by liquid films seems to be of major importance for the rheological behavior of emulsions. 

Moore32 generalized the boundary-layer analysis for a spherical bubble to include the deformation to an oblate ellipsoidal shape. His analysis is not reproduced here, but it is worth recording the final result for the drag coefficient, which takes the form... 

Abstract In this chapter the basic physics and methods of calculation of the effective drag forces acting on drops in isolated-drop and multidrop configurations relevant to sprays are provided. The effect of various physical phenomena such as drop deformation, nonuniformity of the incoming flow, drop-drop interactions, drop-gas interactions, and evaporation on the drag coefficient on the drop, with special focus on the underlying physics, is highlighted. 

Keywords Drag coefficient Drag of deformed drops ing droplets Flow past a droplet Interacting drops... 

One of the earliest works on the flow past a slightly deforming viscous drop in a flow with a large density ratio is that of Hadamard [7], which considers Stokes flows. They offered the following expression for the drag coefficient ... 

Even further increases in the Weber number can lead to a more pronounced deformation in the drop. This is shown in Fig. 4.7 for We = 100. For this case, there is no returning point and the drop continuously deforms and spreads out into a sheet-like shape. A drop with this much deformation ultimately breaks up into smaller pieces by either particles getting pinched off its tip or the whole flat drop breaking into several pieces. Panel (a) in Fig. 4.7 clearly shows how the enhanced vortex in the leeward side of the jet helps stretching the drop into a thin shape which in turn expands the vortex itself. As expected, the continuous deformation leads to a smooth increase in the drag coefficient. 

Fig. 4.6 Streamlines for various stages of deformation in (a), drop deformation in (b), and transient drag coefficient compared to that of a solid sphere in (c) for a viscous drop decelerating in an initially stationary gas. Ohi = 0.01, We = 10, Re = 150, P /Pg = 50. Reprinted with permission fi-om Wadha et al., Phys. Fluids, 19,113301, 2007. Copyright 2007, American Institute of Physics...

The distortion parameter y as defined in Fig. 4.12a can be used to modify the drag coefficient to account for the drop deformation. This can be done using the relation... 

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