Distribution of Mass

The radial distribution function is a useful tool to describe the structure of dendritic stracture. The distribution of sites from the molecular center of mass is given as  

Where N is the total number of molecules, r is the position of the center of mass and a rans over all other sites belonging to the same molecule. Similarly, the distribution from the central site (the core) can be defined as  

Where r. is the position of the core. Another useful function for characterization of internal structure and spatial ordering of sites composing the materials is the atomic radial distribution. 

Where r is the distance between the sites / and j, = NNs is the total number of 

Dispersion agent FIGURE 4 Potential application of dendritic polymers. 

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There exists some radial distance from the axis of rotation at which all of the mass could be concentrated to produce the same moment of inertia that the actual distribution of mass possesses. This distance is defined to be the radius of gyration. According to this definition,... 

Each type of blood cell has its own distribution of mass densities (Fig. 2). Most blood cell separators are based on the formation of blood components into layers by density gradient only. Some cell separators, ie, Haemonetics MGS, apply methods based on a combination of mass density and cell size. 

An uneven distribution of mass about the geometric axis of the system. This distribution causes the center of mass to be different from the center of rotation. 

Experiments by Schmidli et al. (1990) were focused on the distribution of mass on rupture of a vessel containing a superheated liquid below its superheat-temperature limit. Flasks (50-ml and 100-mI capacity) were partially filled with butane or propane. Typically, when predetermined conditions were reached, the flask was broken with a hammer. Expansion of the unignited cloud was measured by introduction of a smoke curtain and use of a high speed video camera. Large droplets were visible, but a portion of the fuel formed a liquid pool beneath the flask. Figure 6.5 shows that, as superheat was increased, the portion of fuel that... 

Theoretical models presented in previous sections give no information on distributions of mass, velocity, or range of fragments, and very little information on the number of fragments to be expected. Apparently, these models are not developed sufficiently to account for these parameters. More information can probably be found in the analysis of results of accidental explosions. It appears, however, that vital information is lacking for most such events. 

Fig. 4-7 Example of a coupled reservoir system where the steady-state distribution of mass is not uniquely determined by the parameters describing the fluxes within the system but also by the initial conditions (see text).

Figure 7-12 depicts the main physical pathways by which aerosol particles are introduced into and removed from the air. Processes that occur within the atmosphere also transform particles as they age and are transported. This form of distribution of mass with size was originally discovered in polluted air in Los Angeles, but it is now known to hold for remote unpolluted locations as well (Whitby and Sverdrup, 1980). In the latter case, the... 

Thus, integration over an arbitrary volume allows us to find the force caused by any distribution of masses. It is essential that the particle p can be located either outside or inside of a body and at any distance from its surface. Equation (1.3) describes the total force that is a result of a superposition of the elementary forces, vectors, at the same point. Correspondingly, this force can cause a translation of the particle only. It is also instructive to consider the force F generated by the particle and acting on an arbitrary body. Each elementary volume is subjected to the force... 

Assuming that the distribution of masses inside the volume V is given, this vector function g p) depends only on the coordinates of the observation point p, and by definition it is a field. It is appropriate to treat the masses in the volume V as sources of the field g p). In other words, these masses generate the field at any point of the space, and this field may be supposed to exist whether a mass is present or absent at this point. When we place an elementary mass at some point p, it becomes subject to a force equal to... 

Fig. 1.3. (a) Linear distribution of masses, (b) surface distribution of masses. 

Inasmuch as Equation (1.6) allows us to solve the forward problem for any distribution of masses, we may say in this sense that the theory of the gravitational method is completely developed. However, in order to understand better the behavior of the field of the earth and sometimes to improve the quality of the... 

Thus, we deal with a system of four differential equations of the first order and, in general, there are four unknown functions g, gy, g, and S. If the distribution of masses is known, then we do not need to use the set (1.31) to find the field. In a fact, this task is solved by integration, using Equation (1.6) ... 

As follows from Equation (1.33) there are relationships between different components of the field, and they indicate that each component of the field contains the same information about a distribution of masses. Note, that Equation (1.33) directly follows from Equation (1.34). By definition, we have ... 

Next, applying the principle of superposition, we arrive at an expression for the potential caused by a volume distribution of masses ... 

Suppose, first, that a distribution of masses is known everywhere. Then, as was shown earlier, the potential of the attraction field is... 

Besides, the potential and its first derivatives are continuous at boundaries where a volume density is discontinuous function. It is obvious that in this case the solution of the forward problem is unique. Now consider a completely different situation, when a density 5 q) is given only inside some volume V surrounded by a surface S, Fig. 1.8a. Inasmuch as the distribution of masses outside V is unknown, it is natural to expect that Poisson s equation does not uniquely define the potential U, and in order to illustrate this fact let us represent its solution as a sum ... 

This means that Poisson s equation defines the potential with an uncertainty of a harmonic function 14. Regardless of a distribution of masses outside the volume the potential C4 remains harmonic function inside V and, correspondingly, there are an infinite number of potentials U which satisfy Equation (1.70), and they can be represented as ... 

This first case vividly illustrates the importance of the boundary condition. Indeed, Poisson s equation or the system of field equations have an infinite number of solutions corresponding to different distributions of masses located outside the volume. Certainly, we can mentally picture unlimited variants of mass distribution and expect an infinite number of different fields within the volume V. In other words, Poisson s equation, or more precisely, the given density inside the volume V, allows us to find the potential due to these masses, while the boundary condition (1.83) is equivalent to knowledge of masses situated outside this volume. It is clear that if masses are absent in the volume V, the potential C7 is a harmonic function and it is uniquely defined by Dirichlet s condition. 

At the same time it is worth to notice that in modern numerical methods of a solution of boundary value problems, based on replacement of differential equations by finite difference, these steps are performed simultaneously. In accordance with the theorem of uniqueness, the field inside the volume V is defined by a distribution of masses inside this volume and boundary conditions, and correspondingly it is natural to derive an equation establishing this link. With this purpose in mind we will again proceed from Gauss s theorem,... 

Now we begin to study the attraction field caused by different distributions of masses, and start from a very simple case. 

The behavior of g as a function of R is shown in Fig. 1.12c, and, of course, it is a continuous function. Now let us mentally decrease the thickness h and increase the volume density so that the mass remains the same. In such a way we arrive at a distribution of masses with a surface density, and this replacement does not change the field outside the shell, but it leads to a discontinuity of the field at the surface masses. It is instructive to demonstrate why the field inside the shell, Relementary surfaces dS and dS2- By definition we have ... 

One more feature of the field behavior is worth noting. Inasmuch as the layer has infinite extension in horizontal planes, the distribution of masses possesses axial symmetry with respect to any line parallel to the z-axis that passes through the observation point. For this reason, it is always possible to find two elementary masses such that the tangential component of the field caused by them is equal to zero. Respectively, the field due to all masses of the layer has only a normal component gz. 

In order to study the attraction of masses of the earth which moves around the axis of rotation, it seems appropriate to use the field g, which depends on the distribution of masses and the angular velocity, as well as coordinates of the point. Besides, it has a physical meaning of the reaction force per unit mass. However, it has one very serious shortcoming, namely, unlike the attraction force it is directed outward. In other words, it differs strongly from the attraction field, in spite of the fact that the contribution of rotation is extremely small. To overcome this problem we introduce the gravitational field g which differs from the reaction field in direction only ... 

Then, taking into account that the distribution of masses inside the spheroid is independent of the azimuth coordinate cp, we have for Laplace s equation... 

This expression for the potential is valid for any distribution of masses, provided that R>Ri- Now we focus our attention on the potential of the normal field caused by regular part of masses. As was assumed before, their density is independent of the longitude, and the equator is a plane of symmetry. For this reason this part of the potential of the earth has a similar behavior and, correspondingly, Equation (2.213) is greatly simplified. Let us rewrite it in the form ... 

It is a simple matter to show that for a regular distribution of masses this term vanishes. 

Let us recall that this equation was derived assuming that the distribution of masses is characterized by a symmetry with respect to the z-axis and the equator. In addition, the center of mass is located at the origin and the total mass M is equal to that of the earth. 

Here the point p belongs to the spherical surface A of radius R. In order to find the upper limit on the left hand side of this equality, let us recall that T is the disturbing potential. In other words, it is caused by the irregular distribution of masses whose sum is equal to zero. This means that its expansion in power series with Legendre s functions does not contain a zero term. The next term is also equal to zero, because the origin coincides with the center of mass. Therefore, the series describing the function T starts from the term, which decreases as r. This means that the product r T O if oo and... 

Here is a point inside the volume V. Hence, it is impossible to distinguish between the field caused by a volume distribution of masses and the field generated by masses on the equipotential surface S, provided that the condition (4.6) is met and the observation point is located outside S. As a rule, a three-dimensional body and... 

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