Integral Eulerian

Following Legendre, the gamma function, or the second Eulerian integral, is usually symbolised by r(w). By definition, therefore,... 

The first member of (10) is sometimes called the first Eulerian integral, or beta fumtion. It is written B(m, n). The beta function is here expressed in terms of the gamma function. Substitute x = ay/h in the second member of (10), and we get... 

With 14 5-10 and 1000, the hydrodynamic relaxation time can be interpreted as an Eulerian integral time scale... 

The solutions discussed so far, both Eulerian and Lagrangian, are known as block implicit solutions. Eulerian integrations into the steady state may, however, be further simplified by uncoupling the separate equations, so that each dependent variable is considered separately during each time step, under... 

Each of the Hamiltonians Hi is easily integrated in terms of rotations about one or the other of the Eulerian axes. 

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As shown above in (6.162), the Lagrangian fluid-particle PDF can be related to the Eulerian velocity, composition PDF by integrating over all initial conditions. As shown below in (6.168), for the Lagrangian notional-particle PDF, the same transformation introduces a weighting factor which involves the PDF of the initial positions y) and the PDF of the current position /x.(x t). If we let V denote a closed volume containing a fixed mass of fluid, then, by definition, x, y e V. The first condition needed to reproduce the Eulerian PDF is that the initial locations be uniform ... 

We recognize in the integral a form that is close to the Eulerian gamma function (4.1.22), which becomes more visible by preparing for variable change as... 

Quoting Cornille [87, p. 168] The calculation concerning the electromagnetic conservation laws given in most textbooks, for example, in Jackson [7, p. 239] is not correct, as noted by Selak et al. [88], because it is not permissible to substitute a convective time derivative for an Eulerian time derivative even when we have a constant volume of integration. ... 

The selection rules for pure-rotation transitions are found by evaluation of the nine integrals IXOa,/ZOc these involve the nine direction cosines6 cos(XOa),..., cos(ZOc). The volume element in Eulerian angles can be shown to be... 

Sunderland and Grosh (Sll) use an explicit numerical scheme for solving the Landau problem. An Eulerian coordinate system is used with the origin at the melt interface. As noted by Landau, the numerical integration is simplified by appropriate choice of the ratio of the space and time intervals. Extension to time-dependent heat flux by either a numerical or a graphical technique is indicated. 

Equation (1) contains the integral form of the general balance relation. In this form it is a Eulerian result. If we take the volume in question to be the entire volume of the pipe located between two planes located at points 1 and 2 separated by some finite distance, as shown in Fig. 1, Eq. (1) can be written in the following a verage or macroscopic form,... 

If the N solutes are in the infinite dilution condition, we can simplify the integral in the previous equation considering that there is no interaction among the solute molecules. Flence for the great majority of the solutes rototranslational configurations, the integral on the other coordinates and solvent momenta is a constant (note that in the solute mass tensor no elements depend on the center of mass position and the Eulerian angles). Therefore... 

In these equations = (x,y, z,(p,, is the energy of interaction of the molecule with the adsorbent surface, the integration extends over the spatial variables (x,y) which specify the point on the adsorbent surface above which the molecule is situated, the variable z which represents the distance between the molecule and the adsorbent surface, and the Eulerian angles (y , describe the orientation of the molecule with respect to the surface A is the surface area over which (x,y) integration is performed. Calculation of the integrals in equation (1) allows the thermodynamic quantities for the adsorption of a quasi-rigid molecule of the species i to be expressed as follows ... 

Now the adsorption potential is independent of the coordinates (x,y) parallel to the adsorbent surface, and one Eulerian angle (say, tp) can be chosen in such a way that the rotation of the molecule through it does not affect the z coordinates of the atoms of the molecule. Thus the integrations over (x,y,cp) can be performed analytically, and instead of equation (1) one obtains ... 

To establish the integral form of the basic conservation laws for mass, momentum and energy, the fundamental approach is to start out from a system analysis and then transform the balance equations into a control volume analysis by use of the transport theorem. However, to achieve a more compact presentation of this theory it is customary to start out from a generic Eulerian form of the governing equations. That is, the material control volume analysis is disregarded. 

The transport theorem (1.10) enables the time rate of change of the material volume integral (1.12) to be transformed into the Eulerian formulation. 

The integral Lagrangian-Eulerian conversion relation yields ... 

It is seen that the governing system equation (3.51) is transformed into a generic Eulerian control volume formulation (3.81) consisting of a volume integral determining the microscopic transport equations for the bulk phases and a surface integral determining the jump balance at the interface. 

It follows from the continuum assumption that the integrands in (3.415) are continuous and differentiable functions, so the integral theorems of Leibnitz and Gauss (see app. A) can be applied transforming the system description into an Eulerian control volume formulation. The governing mixture... 

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