Quasi-elastic collision

At a quasi-elastic impact the film thinning velocity is much smaller than the particle velocity. Thus dh/dt is very much less than unity and can be neglected in deriving Eq. (11.5). It is easy to obtain the well-known formula of Evans (1954) for the film deformation time from Eq.(11.6), 

It was shown by Dimitrov Ivanov (1978) that a dimple is not formed at the impact of a particle on liquid surface if the following condition is fiilfilled. 

After substitution into (11.9) with regard to (11.5), we obtain 

As mentioned above, h corresponds to the thickness of a film at which the liquid surface can no more be in a quasi-stationary state and its velocity starts to approach the particle velocity. Therefore, the inequality should be fulfilled at h h , 

Using (11.17), one can show that the second term in the root in Eq. (11.13) is much smaller than unity. Moreover, a quasi-elastic impact is obtained in the case A 0.01. Therefore Eq. (11.13), after some simplifications, takes the form 

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Figure 4.11 depicts the space-time plot of a head-on quasi-elastic collision with positive phase shifts between two waves of unequal velocity. The slower wave experiences the larger change in wave velocity. Both waves travel with higher velocities after the collision. This situation is reminiscent... 

It follows from Eq. (11.19) that a quasi-elastic collision (A<0.01) is only possible over a rather narrow range of values of Up and p which are of interest for flotation. In most cases an inelastic... 

The above problem has been addressed in (Li et al, 2003), where we have considered a quasi-one dimensional billiard model which consists of two parallel lines and a series of triangular scatterers (see Fig.3). In this geometry, no particle can move between the two reservoirs without suffering elastic collisions with the triangles. Therefore this model is... 

Estimates of the rotational diffusivity may be made from MD calculations by fitting an exponential function to Legendre polynomials that express the decorrelation of a unit vector that is fixed in the methane coordinate frame (11). The rotational diffusivity was found to increase with concentration (as a result of sorbate-sorbate collisions which act to decorrelate the molecular orientation). The values are of the same order as those for liquid methane and are 2 orders of magnitude larger than those found by Jobic et al. (73) from a quasi-elastic neutron scattering study of methane in NaZSM-5. 

Wave Collisions and Patterns 4.5.3.1 Quasi-Elastic and Inelastic Collisions... 

DAL influences practically all stages of the elementary flotation act. Buoyancy velocity of bubbles of definite size with retarded and non-retarded surfaces can differ from each another by a factor of about 2 (see Fig. 8.2). According to the theory of quasi-elastic (Section 10.1) and inelastic (Section 10.2) collisions, a smaller film thickness h corresponding to the beginning... 

To evaluate quasi-elastic energy transfer from an electron gas to neutral molecules, the rotational excitation can be combined with the elastic collisions. The process is then characterized by a gas-kinetic rate coefficient cro( e) 3 10 cm /s (where (ne> is the average thermal velocity of electrons), and each collision is considered as a loss of about e ( ) of electron energy. 

Since similar approach vas used in [37] for Brownian diffusions, it should be noted the principal difference of turbulent diffusion from Brownian one. In the process of Brownian diffusion, the particles perform random thermal motion due to collisions with molecules of ambient liquid. In [37] the appropriate force acting on the considered particle, is taken into account as quasi-elastic force proportional to the particle s displacement Fcontr = —c(x. As a result, the form of the equation (11.60) changes, there appears a term proportional to x, and from the condition of thermodynamic equUibrium of the system it follows that... 

Resonances are common and unique features of elastic and inelastic collisions, photodissociation, unimolecular decay, autoionization problems, and related topics. Their general behavior and formal description are rather universal and identical for nuclear, electronic, atomic, or molecular scattering. Truhlar (1984) contains many examples of resonances in various fields of atomic and molecular physics. Resonances are particularly interesting if more than one degree of freedom is involved they reflect the quasi-bound states of the Hamiltonian and reveal a great deal of information about the multi-dimensional PES, the internal energy transfer, and the decay mechanism. A quantitative analysis based on time-dependent perturbation theory follows in the next section. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

The basic theories of elastic deformations associated with various contact forces under static contact conditions have been introduced in the last section. Assuming that an impact process of two solids can be regarded as quasi-static, the theories given in 2.3 are used directly to link the dynamic deformations of the colliding solids with the impact forces. In this section, the collisions of elastic spheres are described. 

Collisions between particles with smooth surfaces may be reasonably approximated as elastic impact of frictionless spheres. Assume that the deformation process during a collision is quasi-static so that the Hertzian contact theory can be applied to establish the relations among impact velocities, material properties, impact duration, elastic deformation, and impact force. 

Two approaches can be used for calculating interparticle and particle surface collision heat transfer (Amritkar et al., 2014). The first approach is based on the quasi-steady state solution of the coUisional heat transfer between two spheres (Vargas and McCarthy, 2002). The other approach is based on the analytical solution of the one-dimensional unsteady heat conduction between two semi-infinite objects. This approach was proposed by Sun and Chen (1988) based on the analysis of the elastic deformation of the spheres in contact. 

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