Grazing trajectory

Fig. 10.2, as indicated by a dashed line). Otherwise the particle is carried off by the flow. From Fig. 10.2 it is evident that the calculation is essentially reduced to the so-called "grazing trajectory" (continuous curve) and, correspondingly, the target distance. A similar approach has long been used in the science of aerosols (Langmuir and Blodgett, 1945). 

Figure 10.2 Continuous lines illustrate the concept of the grazing trajectory of particles, dashed lines indicate the trajectories of the particle at b < b and b > b. ...

Displacement of particles along the near part is similar to displacement along the circumference so that inertia forces appear as centrifugal forces inhibiting deposition. The primary effect of inertia forces on the near part of the trajectory at St < St was determined by Dukhin (1983). A displacement of the particle trajectory with respect to stream-line 1 which is the grazing trajectory is shown in Fig. 10.14. After displacement, particles move along the stream-line 1 away from the bubble and do not touch its surface. 

If we take into account the negative effect of inertia forces on particle capture, it turns out that the grazing trajectory (Fig. 10.13) corresponds to values of b smaller than those in Sutherland s theory and the point of tangency moves from the equator towards the front pole. 

Fig. 10.13. Diagram of grazing trajectories of particles taking into account inertia forces and SHRI (near hydrodynamic interaction) 1 - grazing trajectory in terms of Sutherland 2 - liquid stream-line coinciding with grazing trajectory 2 - trajectory branching out from stream-line 2 under the effect of inertia force 2" - trajectory branching out from trajectory 2 under the effect of SHRI 0( collision angle 6 - angle characterising the boundary of the part of trajectory controlled by SHRI.

The collision angle (Fig. 10.14) characterises the position of the point of tangency of the grazing trajectory and results from the following equation,... 

Thus we can define the conditions for which the methods used for the calculation of the grazing trajectory hold. 

Unlike the collision effectiveness which is expressed by Langmuir s formula (cf Section 10.1), the capture efficiency Eg is determined by the grazing trajectory which characterises the possibility of particle attachment. Results obtained above make it possible to estimate this quantity since collisions on the main part of the surface at > ocr do not result in capture. All of the simplifications used cause an increase of Eg. The cross-section of the stream tube from... 

Fig. 11.7. Illustration of the mechanism of deposition prevention in the zone 0Q5r<0< t due to the joint action of inertia reflection of a particle tfom the bubble surface and centrifugal forces 1 -grazing trajectory at a single reflection 2 - impossibility of deposition at > oc.

A particle is attached when h(0) = h at any 0. However, condition (11.106) separates the grazing trajectory. The result for a potential flow (Dukhin Rulyov 1977) is... 

Comparing the conditions (11.108) and (11.103) we can conclude that the final coordinates of the grazing trajectory are given by... 

The attachment can happen due to the instability of the wetting film confined between the surfaces of particle and bubble and its break at some critical thickness h . If h essentially exceeds the effective radius of attraction forces between surfaces, the transport of the particle to distance h, does not act as attraction forces. This means that the grazing trajectory is not... 

A different situation arises when h = 0 and fixation of particles happens due to surface attraction forces. These forces can exceed the gravitation force so fixation of particles at the lower pole of the bubble becomes possible. The appearance even of a small rear stagnant cap in the neighbourhood of the lower pole results in a change of the course of the grazing trajectory. 

FIGURE 17.28 Schematic of the flow around a falling drop. The dashed lines are the trajectories of small drops considered as mass points. Trajectory a is a grazing trajectory, while b is a collision trajectory. 

It is intriguing to think about isotope effects in the time-dependent picture of a grazing trajectory. In substituting a heavier isotope, the potential surface remains the same, but the vibrational wavepacket will contract in certain dimensions, making (j) a smaller "target" for <()(t), which is itself smaller and spreading slower. 

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