Interaction between moving spheres

Equation 1.61 describes the flow of the liquid around the moving sphere, the medium being at rest at infinity. Since U = t/k, eq 1.61 can be rewritten as 

Then the velocity of the medium at rj, is not simply given by an equation of the type of eq 1.63 because now the fluid is moving with respect to the coordinate system at r as a consequence of the motion of Let us denote by V/(r) the velocity of the fluid at position r and Vr(a) the relative velocity of S with respect to the fluid at Tn- The two spheres are not necessarily of equal radii Kais the radius of Sa and Ht, that of St,. Hence the actions of the spheres on the fluid are different. Different tensors 4 (eq 1.63) must be considered for Sa and St, 

The velocity of the sphere St with respect to the fluid, i.e. the velocity v,(fr), modified by the wake-effect can be calculated with the help of the simple following model the relative velocity of a sphere is supposed to be the difference between the velocity expressed in the coordinate system in which the fluid is at rest at infinity and the velocity that the fluid would have if the sphere were not existent. With regard to the sphere St follows 

This approximation holds the better, the larger the distance between the two spheres. When the distance between the spheres is very large in comparison with their radii, the velocity field of the fluid in the vicinity of point r jin the absence of the sphere Sj, is approximately uniform. 

Then the sphere moves at that position as if the liquid had a constant velocity v/(t) at infinity. According to this model eq 1.62 is modified to 

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Let us assume that sizes of drops are essentially various / , << R. Then, it is possible to consider that the small corpuscle moves simply in a hydrodynamic field big, and at definition of force of resistance of medium to its motion by inhomogeneity of this field it is possible with sufficient accuracy to neglect. If the distance between surfaces of drops several times is more R, it is possible to neglect also forces of hydrodynamic interacting of moving sphere with a motionless flat wall. These assumptions allow to present the equation of motion of a small corpuscle in an aspect ... 

Several expressions of varying forms and complexity have been proposed(35,36) for the prediction of the drag on a sphere moving through a power-law fluid. These are based on a combination of numerical solutions of the equations of motion and extensive experimental results. In the absence of wall effects, dimensional analysis yields the following functional relationship between the variables for the interaction between a single isolated particle and a fluid ... 

The molecular dynamics method is based on the time evolution of the path (p (t), for each particle to feel the attractions and repulsions from all other particles, following Newton s law of motion. The simplest case is a dilute gas following the hard sphere force field, where there is no interaction between molecules except during brief moments of collision. The particles move in straight lines at constant velocities, until collisions take place. For a more advanced model, the force fields between two particles may follow the Lennard-Jones 6-12 potential, or any other potential, which exerts forces between molecules even between collisions. 

Here /ie and are effective masses of electron and hole, respectively. Near to bottom of conductivity band and near to top of valent band where dependence E from k is close to parabolic, electron and hole move under action of a field as particles with effective masses fie — h2l(d2Ec(k)ldk1) and jUh = —h2l( E (k)ldk ) [6]. In particular, in above-considered onedimensional polymer semiconductor /ie — /ih — h2AEQj2PiP2d2 [6]. As a first approximation, it is possible to present nanocrystal as a sphere with radius R, which can be considered as a potential well with infinite walls [6], The value of AE in such nanocrystal is determined by the transition energy between quantum levels of electron and hole, with the account Coulomb interaction between these nanoparticles. 

This process is outer sphere because there is little electronic interaction between the metal centers through the ligand the electron transfer occurs through space as the metal centers move to the correct conformation by rotations about the C—C bond of the... 

Rigid spheres. If the macromolecules are rigid spheres, of diameter d, with no interactions between them and the suspended particles, then the configurational partition function,, can be evaluated from the volume of space in which the centres-of-mass of the two particles can move freely. Stated another way, can be obtained by subtracting from the total volume that volume from which the centres of mass of the particles are excluded due to the physical presence of the other particles. For two spherical particles, each of radius a, geometrical considerations lead to... 

Maxey and Riley [47] derived an equation of motion for a small rigid sphere of radius R in a nonuniform flow. If one considers small bubbles moving in a polar liquid, this equation might be appropriate because surfactants would tend to immobilize the surface of a bubble and make it behave like a rigid sphere. Maxey and Riley assumed that the Reynolds number based on the difference between the sphere velocity and the undisturbed fluid velocity was small compared to unity. In addition, they assumed that the spatial nonuni-formity of the undisturbed flow was sufficiently small that the modified drag due to particle rotation and the Saffman [48] lift force could be neglected. Finally, they ignored interactions between particles. 

Repulsive interactions are important when molecules are close to each other. They result from the overlap of electrons when atoms approach one another. As molecules move very close to each other the potential energy rises steeply, due partly to repulsive interactions between electrons, but also due to forces with a quantum mechanical origin in the Pauli exclusion principle. Repulsive interactions effectively correspond to steric or excluded volume interactions. Because a molecule cannot come into contact with other molecules, it effectively excludes volume to these other molecules. The simplest model for an excluded volume interaction is the hard sphere model. The hard sphere model has direct application to one class of soft materials, namely sterically stabilized colloidal dispersions. These are described in Section 3.6. It is also used as a reference system for modelling the behaviour of simple fluids. The hard sphere potential, V(r), has a particularly simple form ... 

A slight but systematic decrease in the wave number of the complexes bond vibrations, observed when moving from sodium to cesium, corresponds to the increase in the covalency of the inner-sphere bonds. Taking into account that the ionic radii of rubidium and cesium are greater than that of fluorine, it can be assumed that the covalent bond share results not only from the polarization of the complex ion but from that of the outer-sphere cation as well. This mechanism could explain the main differences between fluoride ions and oxides. For instance, melts of alkali metal nitrates display a similar influence of the alkali metal on the vibration frequency, but covalent interactions are affected mostly by the polarization of nitrate ions in the field of the outer-sphere alkali metal cations [359]. 

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