Spherical cap

Figure 1.10 The formation of a spherical-cap nucleus of radius r on a substrate upon which the nucleus has a wetting angle 6...

The bubbles shapes in gas purging vary from small spherical bubbles, of radius less than one centimen e, to larger spherical-cap bubbles. The mass transfer coefficient to these larger bubbles may be calculated according to the equation... 

If we know the contact angle we can work out r quite easily. We assume that the nucleus is a spherical cap of radius r and use standard mathematical formulae for the area of the solid-liquid interface, the area of the catalyst-solid interface and the volume of the nucleus. For 0 0 90° these are ... 

If the nucleus wets the catalyst well, with 6= 10°, say, then eqn. (7.15) tells us that het IS.lrt,. In other words, if we arrange our 10 atoms as a spherical cap on a good catalyst we get a much bigger crystal radius than if we arrange them as a sphere. And, as Fig. 7.4 explains, this means that heterogeneous nucleation always "wins" over... 

Fig. 7.4. Heterogeneous nucleation takes place at higher temperatures because the maximum random fluctuation of 10 atoms gives a bigger crystal radius if the atoms are arranged as a spherical cap.

In the JKR experiments, a macroscopic spherical cap of a soft, elastic material is in contact with a planar surface. In these experiments, the contact radius is measured as a function of the applied load (a versus P) using an optical microscope, and the interfacial adhesion (W) is determined using Eqs. 11 and 16. In their original work, Johnson et al. [6] measured a versus P between a rubber-rubber interface, and the interface between crosslinked silicone rubber sphere and poly(methyl methacrylate) flat. The apparatus used for these measurements was fairly simple. The contact radius was measured using a simple optical microscope. This type of measurement is particularly suitable for soft elastic materials. 

Brown [46] continued the contact mechanics work on elastomers and interfacial chains in his studies on the effect of interfacial chains on friction. In these studies. Brown used a crosslinked PDMS spherical cap in contact with a layer of PDMS-PS block copolymer. The thickness, and hence the area density, of the PDMS-PS layer was varied. The thickness was varied from 1.2 nm (X = 0.007 chains per nm-) to 9.2 nm (X = 0.055 chains per nm-). It was found that the PDMS layer thickness was less than about 2.4 nm, the frictional force between the PDMS network and the flat surface layer was high, and it was also higher than the frictional force between the PDMS network and bare PS. When the PDMS layer thicknesses was 5.6 nm and above, the frictional force decreased dramatically well below the friction between PDMS and PS. Based on these data Brown [46] concluded that ... 

However, for the case of the rigid particle indenting a compliant substrate, the change in area, which arises from the stretching of the surface of the substrate, is given by the expansion in size of a spherical cap. 

This view is supported by our observation of hemi-spherically capped single-wall and multi-wall tubes on the same samples. It suggests that the Cf,o-derived tube could be the core of possible multilayer concentric graphitic tubes. After the single-shell tube has been... 

Observations of bubbles emerging through the bed surface show that bubble shape is markedly dependent on liquid velocity. This indicates the existence of a relationship between bed viscosity and liquid velocity. A bed near incipient fluidization is characterized by a high viscosity, and an emerging bubble is of nearly spherical shape, whereas a fluidized bed of high porosity is characterized by a viscosity not very much higher than that of water, so that an emerging bubble is of spherical cap shape. 

Dispersed bubbles are observed (Fig. 5.6a) when the gas flow rate is very small such as [/gs = 0.0083 m/s. Two kinds of bubbles are observed one type is finely dispersed with a size smaller than the tube diameter, and the other type has a length of near to or a little larger than the mbe diameter with spherical cap and tail. The distance between two consecutive bubbles may be longer than ten times the tube diameter. This flow pattern is also considered as a dispersed bubbly flow. Often in air-water flow two kinds of bubbles appear together as pairs of bubbles in which the small-sized bubbles follow the larger ones. 

In quiescent liquids and in bubble columns, buoyancy-driven coalescence is more important. Large fluid particles with a freely moving surface will also have a low-pressure region at the edge of the particle where the velocity is maximum. This low-pressure region will not only allow the bubble to stretch out and form a spherical cap but also allow other bubbles to move into that area and coalesce. Figure 15.14 shows an example of this phenomenon. 

It is well known that when liquid droplets form on a flat substrate they adopt spherical cap shapes (neglecting gravity effects) with a contact angle 6. This angle depends solely on the interfacial energies as described by the Young s equation ... 

In many cases the potential P(z) is small compared with the surface energy of the liquid and the droplet shape is very close to a spherical cap. If the height e and the radius of curvature R at the top of the droplets can be measured, an effective contact angle can be defined through the expression ... 

Figure 6 shows droplets of KOH solution on mica produced by similar methods. In both cases the drop profiles are very close to a spherical cap. In Figure 7 we have plotted the effective contact angle as a function of droplet height. The deviation from the macroscopic contact angle with decreasing droplet volume can clearly be seen. 

A very different behavior is observed when condensation occurs on a mica surface that has been exposed to air for a few hours (we will refer to it as contaminated mica ). In this case glycerol forms droplets in the shape of spherical caps, indicating that it does not completely wet the surface. This behavior is similar to that of water, which we present in detail later. The contact angle of water on mica surfaces increased from 0° on the freshly cleaved surface to a small value between 2 and 3° on the contaminated mica [51]. 

In region III near the tube center, viscous stresses scale by the tube radius and for small capillary numbers do not significantly distort the bubble shape from a spherical segment. Thus, even though surfactant collects near the front stagnation point (and depletes near the rear stagnation point), the bubble ends are treated as spherical caps at the equilibrium tension, aQ. Region... 

We utilize the ad hoc procedure of Bretherton (15) which has been formally justified by Park and Homsy (20). Figure 3 displays the meridional circle characterizing the spherical cap of the... 

Figure 3. Schematic of matching to the spherical cap at the bubble front. The radius of the flow-altered sphere is For a static bubble, the bubble radius is R. ...

Problem. Let a polymer fiber contain rod-shaped structural entities in an amorphous matrix with some preferential orientation. Let us assume that the rods are crystalline. Our interest is to study the crystalline structure of the rods. Instead of sharp hkl reflections we observe that each reflection is smeared over a spherical cap in solid angle. Thus the observed intensity is suitably expressed in polar coordinates... 

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