Secondary motion

Detailed descriptions of the convective airflow patterns in cast replicas of the human respiratory tract during steady inspiration were given by Olson et al, Their results show that the effect of the larynx is such that flow patterns typical of smooth bifurcating tubes (secondary motions and high shear rates along the inside wall) do not occur until the lobar bronchi are reached. Turbulent eddies produced by flow separation below the larynx do not decay as rapidly as predicted by theory. Indeed, small eddies were observed as far down as the sublobar bronchi with 200-mt/s flows in the trachea. 

Figure 5.14 shows terminal velocities of spheres of various densities in air and water at 20 C calculated from the correlations in Tables 5.2 and 5.3, incorporating corrections for secondary motion, Eq. (5-18), and slip (see Chapter 10). 

Figures 5.22 and 5.23 present the result of combining the equations in Table 5.4 with the correlations of Table 5.3 to predict heat transfer for spheres falling in air at 20 C and mass transfer for spheres in water at 20 C with Sc = 10. The decrease in terminal velocity due to secondary motion has not been taken into account because the transfer rate depends on the overall relative velocity between the sphere and the fluid, not the vertical velocity component alone.

Willmarth et al. (W5) showed that secondary motion of a freely falling disk depends on a dimensionless moment of inertia. 

The upper bound of the region of stable steady motion is shown in Fig. 6.7 as a function of (CdRcx ) and /. For large /, secondary motion starts at Rcy = 100, i.e., (Cd Rcj ) = 23.4. At lower /, steady motion persists to higher Rcy the boundary shows a maximum at Rcy = 172, (CdRcj ) = 32.6 for/ = 8 X 10 Three kinds of secondary motion have been observed (S8), although the distinctions between them are not sharp. Immediately above the transition to unsteady motion, a disk shows regular oscillations about a diameter the amplitude of oscillation and of the associated horizontal motion increases with... 

For spheroids with E = 0.5, Stringham et al. (S8) showed that steady motion with the axis vertical persists over a much wider range of Rcj than for thin disks. Secondary motion started at Rcj = 4 x 10 = 10. On increasing... 

No data are available for heat and mass transfer to or from disks or spheroids in free fall. When there is no secondary motion the correlations given above should apply to oblate spheroids and disks. For larger Re where secondary motion occurs, the equations given below for particles of arbitrary shape in free fall are recommended. 

C3, 12, J2, M3, S8). The amplitude of angular oscillations decreases as E increases, and a very long cylinder falls steadily to high Rey(I2, J2). If Re > 3500 (S8), motion also occurs in a horizontal plane. For relatively low y, the cylinder oscillates about a vertical axis (12, S8), while for dense particles in liquids or particles in gases the cylinder rotates continuously about a vertical axis (C3,12). A cylinder with E = 1 follows a trajectory inclined to the vertical, and tumbles in the direction of horizontal travel (12). For F < 1, the axis oscillates and rotates about a vertical line, so that the secondary motion resembles the final stages of motion of a coin spinning on a flat surface (12). 

Drag Coefficients and Terminal Velocities for Cylinders with Secondary Motion ... 

Since the motion of particles of simple shapes in free fall or rise is poorly understood when secondary motion occurs, it is not surprising that the behavior of particles with more complex or irregular shapes in this range cannot be predicted with certainty. As for the regular shapes, is only weakly dependent on Re, but depends on y (T2). The correlation developed by Wadell (Fig. 6.14) is not recommended since it shows dependence on Re but not on y and has already been shown to be unreliable in the intermediate range. 

Aybers and Tapucu (A4, A5) measured trajectories of air bubbles in water. When surface-active agents continue to accumulate during rise, the terminal velocity may never reach steady state (A4, Bl) and may pass through a maximum (W4). Five types of motion were observed, listed in Table 7.1 with Re based on the maximum instantaneous velocity. Secondary motion of fluid par-... 

The generalized graphical correlation presented in Fig. 2.5 gives one method of estimating terminal velocities of drops and bubbles in infinite liquid media. For more accurate predictions, it is useful to have terminal velocities correlated explicitly in terms of system variables. To obtain such a correlation is especially difficult for the ellipsoidal regime where surface-active contaminants are important and where secondary motion can be marked. 

Bubbles and drops of intermediate size show two types of secondary motion ... 

While other explanations have been proposed [e.g. (B6, El, H6)], secondary motions are most plausibly related to wake shedding. The onset of oscillations coincides with the onset of vortex shedding from the wake (El, E2, S5, W8). For high k or contaminated drops and bubbles, the onset of oscillations... 

In general, oscillations may be oblate-prolate (H8, S5), oblate-spherical, or oblate-less oblate (E2, FI, H8, R3, R4, S5). Correlations of the amplitude of fluctuation have been given (R3, S5), but these are at best approximate since the amplitude varies erratically as noted above. For low M systems, secondary motion may become marked, leading to what has been described as random wobbling (E2, S4, Wl). There appears to have been little systematic work on oscillations of liquid drops in gases. Such oscillations have been observed (FI, M4) and undoubtedly influence drag as noted earlier in this chapter. Measurements (Y3) for 3-6 mm water drops in air show that the amplitude of oscillation increases with while the frequency is initially close to the Lamb value (Eq. 7-30) but decays with distance of fall. 

Secondary motion plays an important role in increasing drag (L7) and in promoting heat and mass transfer from bubbles or drops. The onset of oscillations corresponds approximately to the maximum in Uj d ) and minimum in... 

It has already been noted in Chapters 5 to 7, that y = p /p must be included when secondary motion is superimposed on steady particle translation. 

Free-fall experiments with Re >10 show that a sphere released from rest initially accelerates vertically, and then moves horizontally while its vertical velocity falls sharply (R3, S2, S3, V2). As for steady motion discussed in Chapter 5, secondary motion results from asymmetric shedding of fluid from the wake (S3, V2). Wake-shedding limits applicability of the equations given above. Data on the point at which wake-shedding occurs are scant, but lateral motion has been detected for in the range 4-5 (C7). Deceleration occurs for Re > 0.9 Re. The first asymmetric shedding occurs at much higher Re than in steady motion (Re = 200 see Chapter 5), due to the relatively slow downstream development, as shown in Fig. 11.12. 

Cj drag coefficient for sphere subject to secondary motion drag coefficient for... 

It was proposed that the temperature dependence of polymer 5 arises from the temperature dependence of the kA step. Specifically, it was suggested that the polymer segments to which the radicals are attached are conformationally stressed. There are two possible modes for the newly formed radicals to relax and become separated They can rotate or recoil away from each other (Scheme 9). These secondary motions of the polymer arise from the relaxation of unfavorable bond conformations that are formed during the polymer casting process. The increased thermal energy facilitates the rotation and recoil relaxation processes, which effectively increases the rate constant for diffusion of the radicals out of the cage, kA. This leads to decreased radical-radical recombination and consequently an increase in photodegradation efficiency. 

The data also show that mixing is probably better than expected for diffusion as the only mechanism [33], This is to be expected, since owing to shearing of the layers it is likely that secondary motion is induced. Further, the measurements show that the FTOL and the mixing length are not necessarily equal. 

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