Elementary volume

Result of reconstruction is a 3D matrix of output data assigned with the values of the local density inside elementary volumes. The ways of obtaining the 3D matrix of output data can be various. They are determined by the structure of tomographic system and chosen way of collected data processing. 

The analytical method of jet trajectory study developed by Shepelev allows the derivation of several other useful features and is worth describing. On the schematic of a nonisothermal jet supplied at some angle to the horizon (Fig. 7.25), 5 is the jet s axis, X is the horizontal axis, and Z is the vertical axis. The ordinate of the trajectory of this jet can be described as z = xtga a- Az, where Az is the jet s rise due to buoyancy forces. To evaluate Az, the elementary volume dW with a mass equal to dm dV on the jet s trajectory was considered. The buoyancy force influencing this volume can be described as dP — g(p -Pj). Vertical acceleration of the volume under the consideration is j — dP / dm — -p,)/ g T,-T / T. Vertical... 

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... 

In order to determine the force with which an arbitrary body acts on a particle located around the point p, we mentally divide the volume of the body into many elementary volumes, so their dimensions are much smaller than the corresponding distance from the particle p. It is clear that the magnitude and direction of each force depends on the position of the point q inside a body. Now, applying the principle of superposition, we can find the total force acting on the particle p. Summation of elementary forces gives ... 

In the limit when elementary volumes tend to zero we have... 

It is appropriate to note that to carry out numerical integration the elementary volumes have to satisfy two conditions, namely,... 

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... 

Applying the same approach it is a simple to find an expression for the force of interaction between two arbitrary bodies. It is obvious that the force acting on any elementary volume of a body is the sum of the forces due to other body and the force caused by different elements of the same body. In particular, the resulting force due to body 1 acting on body 2 is... 

Then, we can reduce each elementary volume dV — hih2dl and increase the volume density so that the elementary mass ... 

Suppose that masses are distributed within a plane layer whose thickness is much smaller than the distance from these masses to the observation point. Fig. 1.13a. In other words, the distance between the observation point and any point of the elementary volume is practically the same. Taking into account this fact, we can replace this layer by a plane surface with the same mass, located somewhere at the middle of the layer. Fig. 1.13b. Inasmuch as every elementary volume contains the mass dm — 5(q)hdS,its distribution on the surface can be described by dm — a(q)dS, where... 

FORCES ACTING ON AN ELEMENTARY VOLUME OF THE ROTATING EARTH AND THE GRAVITATIONAL FIELD... 

As is well known, the earth is mainly a fluid the upper crust is an exception, but it is extremely thin layer with respect to the earth s radius. For this reason it is natural to expect that rotation around its axis makes the shape of the earth practically the same as if it was a fluid, and we will follow this conventional point of view. Suppose that during this motion the mutual position of all elementary volumes of the earth remains the same, and correspondingly each of them is involved only in rotation with angular velocity m. This means that the effect of different types of currents inside the earth is neglected and we deal with hydrostatic equilibrium. 

Consider a rotation of the earth around the z-axis in which every particle, elementary volume, of the earth moves along the horizontal circle with the radius r. Our first goal is to find the distribution of forces inside the earth and with this purpose in mind we will derive an equation of motion for an elementary volume of the fluid. Let us introduce a Cartesian system of coordinates with its origin 0, located on the z-axis of rotation. Since this frame of reference is an inertial one, it does not move with the earth, we can write Newton s second law as... 

Fig. 2.1. (a) Elementary volume of a fluid, (b) attraction and gravitational fields on the earth s surface. 

Here F is the pressure, which is uniformly distributed over the face of the cube. This force causes a deformation of the elementary volume and, as a result, it gives rise to a force on the medium in front of the cube. In accordance with the Newton s third law this side is subjected to the force ... 

Multiplying Equations (2.5 and 2.6) by the corresponding unit vectors i, j, and k and adding them, we obtain the equation of motion of an elementary volume inside... 

As follows from Chapter 1, we have formulated an external Dirichlet s boundary value problem, which uniquely defines the attraction field. In this light it is proper to notice the following. In accordance with the theorem of uniqueness its conditions do not require any assumptions about the distribution of density inside of the earth or the mechanism of surface forces between the elementary volumes. In particular, these forces may not satisfy the condition of hydrostatic equilibrium. 

It is important to note that opposite faces of elementary volumes no longer have equal areas. Suppose that the surface on the left has area ai and that the component of the vector A in the direction of h dqi has a value of A at this surface. The outward flow through this surface is — a A. The contribution from the opposite face will be... 

Integral c(x) can be taken with the adequate accuracy by saddle-point technique [1,5]. Change of (13) introduces an essential difference between w(s) and w(x) the last determines the probability w(x)dx of fact that the SAR W trajectory at given values m, N and cr will finished in the elementary volume dx = ] [ dxi lying on the surface of the ellipsoid... 

Theoretical studies are primarily concentrated on the treatment of flame blow-off phenomenon and the prediction of flame spreading rates. Dunskii [12] is apparently the first to put forward the phenomenological theory of flame stabilization. The theory is based on the characteristic residence and combustion times in adjoining elementary volumes of fresh mixture and combustion products in the recirculation zone. It was shown in [13] that the criteria of [1, 2, 5] reduce to Dunskii s criterion. Longwell et al. [14] suggested the theory of bluff-body stabilized flames assuming that the recirculation zone in the wake of the baffle is so intensely mixed that it becomes homogeneous. The combustion is described by a second-order rate equation for the reaction of fuel and air. 

As mentioned in section 12.1, Dunskii [12] was the first who put forward the phenomenological theory of flame stabilization. The theory is based on the characteristic residence time, L, and combustion time, tc, in adjoining elementary volumes of fresh mixture and combustion products in the recirculation zone behind the bluff body. Dunskii s condition for flame blow-off is U/tc = Mi, where Mi is the Mikhelson number close to unity (for example, for cone flame holder the measurements give Mi = 0.45 [36]). Residence time L is taken proportional to the flame holder size, H, and inversely proportional to the approach flow velocity, U, i.e., L = H/U. Combustion time is estimated as tc = at/Si, where... 

The basic equations, as recently slightly updated from further experience [51], are now briefly reviewed. Consider a molecule (A) and let p be the electron density in an elementary volume V] centered at point k), with charge qj = For a... 

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