Effects of Inertia

Forces in microsystems decrease as size is scaled down. Different forces scale down into the microdomain differently, depending on the dimensions [5]. The forces were found to scale in one of four ways following the applicable physical laws. If the scale size is decreased by a factor of 10, the forces may decrease according to the various physical laws by 10, 100, 1,000, or 10,000 times, respectively. For example, at the same downsizing rate, surface tension decreases only by a factor of 10, while gravity decreases by a factor of 1,000. 

Photograph of prototype endoscope with tunable-focus liquid microlens at distal end. An image acquisition fiber bundle and 12 optical fibers are bound together and attached to the back side of the microlens by adapters. Source Zeng, X. et al. 2011. Journal of Mkroelectromechaniad Systems, 20, 583-593. With permission.) 

The scaling of surface tension is advantageous and may dominate gravity when the feature size of the system is smaller than a specific value. Normally such feature value for most liquids is close to 1 mm (10 3 mL) [6,7], as shown by an example in Chapter 2. Therefore, for most microlenses whose sizes are smaller than or near 1 mm, the effect of gravity may be neglected. 

Theisen et al. presented a dynamic model to describe the oscillations resulting from the competition between liquid inertia and capillarity [8]. They demonstrated that gravity was weaker than surface tension but exerted a non-negligible effect. It was found that gravity could distort the shapes of 

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Of course, the limiting case, (4-3), contains no influence of inertia. To determine the effects of inertia for very small, but nonzero, values of Rm we look for an approximate... 

The effect of inertia is to slightly increase the drag relative to Stokes drag. 

The characteristic balance of elasticity and viscosity can be shifted in polymers through either the temperature or the frequency of the deforming force one works on the effects of structure, the other on the effects of inertia (mass).2... 

Negative Effect of Inertia Forces on Flotation of Small Particles. Generalisation of Sutherland s Formula. Extension of Limits of Applicability of Microflotation Theory... 

The less Stokes number is, the smaller is the effect of inertia forces on the particle trajectory since the viscous resistance of the medium inhibits displacement of the particle from the respective liquid stream line. 

There is a point of inflection on each liquid stream line, which divides the line into two parts (Fig. 10.12). One of these branches is called the near and the other the distant part. On the distant part of the stream line, inertia forces shift the direction of particle motion to the bubble surface promoting deposition, a positive effect of inertia forces. 

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. 

It is worth noticing that negative effects of inertia forces appear at subcritical values of Stokes numbers when a positive effect is practically absent (cf Section 10.1). The inertia-free approach of a particle and a bubble is caused by the radial particle velocity when its centre is located at a distance from the bubble surface approximately equal to a. When the particle radius tends to zero, this velocity also tends to zero and deposition depends on the finite size of the particle. 

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.

Considering the negative effects of inertia forces, most important is the calculation of the point of tangency 0, where the radial component of the particle velocity at the time of contact with bubble surface becomes zero,... 

While neglecting the finite size of a particle at subcritical Stokes numbers excludes inertia forces at all, the situation changes with the consideration of a finite particle size. Inertia forces become essential at subcritical but not too small Stokes numbers. This effect can turn out to be negative. Thus, a critical Stokes number separates the regions of positive and negative effects of inertia forces on particle deposition. 

At — <0.1 the negative effect of inertia becomes insignificant at any size of bubbles and P... 

This theory enables to characterise quantitatively the conditions under which flotation proceeds inertia-free. It is simultaneously a generalisation of the theory of small particle flotation which not only allows to describe inertia-free flotation, but also flotation complicated by a negative effect of inertia forces. If flotation proceeds practically inertia-free at ap<3-10pm, the negative effect of inertia its quantitatively described over the range of bubble size from 10 to 30 pm. 

Unfortunately, up to now all quantitative investigations of collision efficiency have been performed under conditions which do not allow to check the validity of the present theory. In experiments by Anfhms Kitchener (1977), bubble surfaces were retarded to a large degree which strongly decreases the negative effect of inertia forces. 

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