One-body forces

It is to be noted that the force F,- determined by Eq. (1.7) is actually a function not only of the coordinates of the segment i but also those of all the other segments. Thus, it may be explicitly written as Ff ( f Ri) to show its many body character. Here Rj represents the coordinates of the segments other than i. On the other hand, as one can see from Eq. (1.6) the viscosity coefficient is determined directly not by such many body forces but the one body force which is obtained from the many body forces by a suitable averaging processes. 

From these observations it becomes evident that one needs to introduce a self consistent equation for a one body force (or energy dissipation). We shall now consider various methods so far developed. 

Our theory may be imderstood better if compared with the KR theory. Their theory has been developed along the observations discussed in Section 1. We note that Ff of Eq. (1.4) which depends on all the segments is replaced in their theory by a one body force determined by the ordering number of a segment irrespective of its location. For this reason it was necessary to replace the Oseen hydrodynamical interaction tensor by its average. 

A one-body force is independent of the locations of other particles. Electrostatic forces on a charged particle due to an external electric field are one-body forces. Near the surface of the earth the force of gravity on a particle is also a one-body force. This gravitational force on an object corresponds to a potential energy... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

We should mention here one of the important limitations of the singlet level theory, regardless of the closure applied. This approach may not be used when the interaction potential between a pair of fluid molecules depends on their location with respect to the surface. Several experiments and theoretical studies have pointed out the importance of surface-mediated [1,87] three-body forces between fluid particles for fluid properties at a solid surface. It is known that the depth of the van der Waals potential is significantly lower for a pair of particles located in the first adsorbed layer. In... 

Force. The action of one body on another. This action will cause a change in the motion of the first body unless counteracted by an additional force or forces. A force may be produced either by actual contact or remotely (gravitation, electrostatics, magnetism, etc.). Force is a vector quantity. 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

The phenomenon of attraction of masses is one of the most amazing features of nature, and it plays a fundamental role in the gravitational method. Everything that we are going to derive is based on the fact that each body attracts other. Clearly this indicates that a body generates a force, and this attraction is observed for extremely small particles, as well as very large ones, like planets. It is a universal phenomenon. At the same time, the Newtonian theory of attraction does not attempt to explain the mechanism of transmission of a force from one body to another. In the 17th century Newton discovered this phenomenon, and, moreover, he was able to describe the role of masses and distance between them that allows us to calculate the force of interaction of two particles. To formulate this law of attraction we suppose that particles occupy elementary volumes AF( ) and AF(p), and their position is characterized by points q and p, respectively, see Fig. 1.1a. It is important to emphasize that dimensions of these volumes are much smaller than the distance Lgp between points q and p. This is the most essential feature of elementary volumes or particles, and it explains why the points q and p can be chosen anywhere inside these bodies. Then, in accordance with Newton s law of attraction the particle around point q acts on the particle around point p with the force d ip) equal to... 

Fet us now confront the EOS predicted by the phenomenological TBF and the microscopic one. In both cases the BHF approximation has been adopted with same two-body force (Argonne uis). In the left panel of Fig. 4 we display the equation of state both for symmetric matter (lower curves) and pure neutron matter (upper curves). We show results obtained for several cases, i.e., i) only two-body forces are included (dotted lines), ii) TBF implemented within the phenomenological Urbana IX model (dashed lines), and iii) TBF treated within... 

A related, relatively unexplored topic is the importance of many-body forces in the simulations of interfacial systems. The development of water-polarizable models has reached some level of maturity, but one needs to explore how these models must be modified to take into account the interactions with the metal surface atoms and the polarizable nature of the metal itself... 

We consider a system made up of N fermions (nucleons or electrons) interacting through two-body forces and with an external field, with respective potentials >(r,r ) and V(r). We assume there exists a possibility of decomposing the one-particle density matrix into an averaged part and an oscillating part, i.e. ... 

Abstract. The physical nature of nonadditivity in many-particle systems and the methods of calculations of many-body forces are discussed. The special attention is devoted to the electron correlation contributions to many-body forces and their role in the Be r and Li r cluster formation. The procedure is described for founding a model potential for metal clusters with parameters fitted to ab initio energetic surfaces. The proposed potential comprises two-body, three-body, and four body interation energies each one consisting of exchange and dispersion terms. Such kind of ab initio model potentials can be used in the molecular dynamics simulation studies and in the cinalysis of binding in small metal clusters. 

As was shown in previous section, the many-body forces play a crucial role in metal cluster stability. So, a model potential must include many-body terms, at least 3- and, sometimes, 4-body ones. For clusters of larger size, the fitted parameters in these terms will include ( absorb ) many-body effects of higher orders. 

For steady, one-dimensional flow without body forces, with local mean velocity u(x) in a channel of constant cross-sectional area A, the energy conservation equation becomes, approximately (13) ... 

The application of force to a stationary or moving system can be described in static, kinematic, or dynamic terms that define the mechanical similarity of processing equipment and the solids or liquids within their confines. Static similarity relates the deformation under constant stress of one body... 

Energy may also be generated by mechanical action that is, the application of physical force by one body upon anodier. Examples of this are the energy created by the friction of one matter upon... 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

The application of force to a stationary or moving system can be described in static, kinematic, or dynamic terms that define the mechanical similarity of processing equipment and the solids or liquids within their confines. Static similarity relates the deformation under constant stress of one body or structure to that of another it exists when geometric similarity is maintained even as elastic or plastic deformation of stressed structural components occurs [53], In contrast, kinematic similarity encompasses the additional dimension of time, while dynamic similarity involves the forces (e.g., pressure, gravitational, centrifugal) that accelerate or retard moving masses in dynamic systems. The inclusion of tune as another dimension necessitates the consideration of corresponding times, t and t, for which the time scale ratio t, defined as t = t It, is a constant. 

It is useful to think of two different kinds of forces, one that acts over the volume of a fluid element and the other that acts on the element s surface. The most common body force is exerted by the effect of gravity. If an element of fluid is less dense than its surroundings (e.g., because it is warmer), then a volumetric force tends to accelerate it upward—hot air rises. Other fields (e.g., electric and magnetic) can exert volumetric body forces on certain fluids (e.g., ionized gases) that are susceptible to such fields. Here we are concerned mostly with the effect of gravity on variable-density flows,... 

The convective terms are the ones most responsible for nonlinearity in the fluid-flow conservation equations. As such they are often troublesome both theoretically and practically. There are a few situations of interest where the convective terms are negligible, but they are rare. As a means of exploring the characteristics of the equations, however, it is interesting to consider how the equations would behave if these terms were eliminated. For the purpose of the exercise, assume further that the flow is incompressible, single species, constant property, and without body forces or viscous dissipation. In this case the governing... 

Weight The gravitational force of attraction between two bodies (where one body is usually Earth). 

---