Tangential motion

Besides the resuspension of particles, the perfect sink model also neglects the effect of deposited particles on incoming particles. To overcome these limitations, recent models [72, 97-99] assume that particles accumulate within a thin adsorption layer adjacent to the collector surface, and replace the perfect sink conditions with the boundary condition that particles cannot penetrate the collector. General continuity equations are formulated both for the mobile phase and for the immobilized particles in which the immobilization reaction term is decomposed in an accumulation and a removal term, respectively. Through such equations, one can keep track of the particles which arrive at the primary minimum distance and account for their normal and tangential motion. These equations were solved both approximately, and by numerical integration of the governing non-stationary transport equations. 

Fig. 13D Motion of the liquid at the surface of an RDE. (o) tangential motion in the plane of the electrode, (b) perpendicular motion towards the electrode.

Electrokinetic phenomena are generally characterized by the tangential motion of liquid with respect to an adjacent charged surface. In the above example the surface was that of a negatively charged clay particle the particle moved with respect to the stationary liquid. The surface may also be that of a droplet as in emulsions. Alternatively, the particles may be stationary with the liquid moving, as for Instance in electro-osmosis. For sand this phenomenon was also discovered by Reuss I... 

Elaborations along these lines are the most fundamental, and in that respect the most satisfactory. They require simulations with large samples under lateral shear, with ions embedded in the water and charges on the surface, and with realistic interactions between the water molecules. Surface roughness on a molecular scale should also be accounted for the presence of material obstacles on a surface may drastically affect the tangential motion of molecules and ions that have to negotiate such asperities. There is still a long way to go. 

Figure 6-7. Three configurations for the shallow-cavity problem (a) Four isothermal solid walls with motion driven by tangential motion of the lower wall (b) the same problem as (a) except, in this case, the upper surface is an interface that may deform because of the flow (c) the configuration is the same as (b), except, in this case, the lower wall is stationary and the motion in the cavity is assumed to be driven by Marangoni stresses caused by nonuniform interface temperature that is due to the fact that the end walls are at different temperatures.

Three examples of shallow-cavity flows that we consider in this section are sketched in Fig. 6-7. At the top is the case in which all four boundaries are solid walls, the fluid is assumed to be isothermal, and the motion is driven by tangential motion of the lower horizontal boundary. In the middle, a generalization of this problem is sketched in which the fluid is still assumed to be isothermal and driven by motion of the lower horizontal boundary, but the upper boundary is an interface with air that can deform in response to the flow within the cavity. Finally, the lower sketch shows the case in which fluid in the shallow cavity is assumed to have an imposed horizontal temperature gradient, produced by holding the end walls at different, constant temperatures, and the motion is then driven by Marangoni stresses on the upper interface. In the latter case, there will also be density gradients that can produce motion that is due to natural convection, but this contribution is neglected here (however, see Problem 6-13.)... 

A y-gradient is thus very effective in withstanding a tangential stress and arresting tangential motion of an interface. Actually, the situation is more complicated. Generally, the surfactant is soluble in at least one of the phases, and exchange between interface and bulk will thus occur. Moreover, Eq. (10.17) is not always fully correct. See further Section 10.8.3. 

The concept of adhesive interaction of contacting surfaces is already familiar to us from previous discussion of the adhesive mechanism of friction (Chapters 8 and 12). If the two bodies participating in the adhesive junction are in motion relative to each other, in particular tangential motion, the junction is ruptured shortly after it is established. Rupture of the junction at a location other than the original interface results in transfer of material from one body to the other. According to the broad definition of wear of Section 13.1, each body has been worn—one by loss of material, the other by gain—but there has been no net loss or gain in the system as a whole. 

Let us now explain how the centrifugal force varies with increasing St. The description of the tangential motion of a particle as inertia-free (cf. Section 10.14) assumes small particles and very small St numbers (less than the critical value). When St grows, the tangential velocity Vq... 

Handles and Baron (H5) proposed another model for the more practical range of Reynolds numbers (about 1000). They assumed that the tangential motion caused by circulation is combined with an assumed random radial motion caused by internal vibration, and determined the eddy diffusivity subsequently used in solving the appropriate Fourier-Poisson equations. They postulated radial stream lines, as shown in Fig. 11, rather than those... 

Most theories for polymer adsorption kinetics start from (combinations of) the models discussed earlier. Other theories, often proposed for (bio)polymer adsorption, are based on the random sequential adsorption (RSA) model. According to this model, the adsorbate molecules arrive randomly at the interface and they stick where they hit. It implies that both desorption and tangential motion of the adsorbate at the interface are absent. Because the center of a newly arriving spherical molecule cannot be accommodated within the shaded areas enclosed by the dashed circles shown in Figure 15.6, only the unshaded fraction < ) of the surface is available for adsorption. It is obvious that ( ) is a function of 0, the fraction of the surface that is covered by the adsorbate. For sphere-like molecules, 0 = niUi (R being the radius of the molecule). The following expressions for the available surface function < )(0) can be derived from the RSA theory ... 

Electrokinetic effects arise when one of the two phases is caused to move tangentially past the second phase. Tangential motion can be caused by Electric field ... 

A classification of adhesion as a function of the changes in the interaction between contiguous bodies in the course of detachment was given by Deryagin [1] on the basis of the analogy between adhesion and friction. (Friction prevents the tangential motion of particles and adhesion prevents the motion of particles in a direction perpendicular to the surface on which they have been deposited.)... 

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