Moving coordinates

Here r, 9, 4> are dimensionless co-moving coordinates attached to fundamental observers and R(t) a scale factor with a dimension of length depending only on cosmic time t. k is the curvature constant, which with suitable choice of units takes one of the three values +1 (closed world model with positive curvature), 0 (flat, open model) or —1 (open model with negative curvature). Some consequences of Eq. (4.7) are the relation between redshift and scale factor Eq. (4.2) and the variation of temperature... 

Puff equations with moving coordinate system (Equations 5-54 through 5-56)]... 

If the pulse width is of the order of picoseconds or shorter, the terms with 1 0 have to be taken into account. In the case of plane waves, this effect (usually referred to as the SS effect) leads to a change in the shape of the temporal envelope and to the shift of the pulse peak with respect to the center of the moving coordinate system. Another effect significant for... 

Figure 26. Longitudinal variation of the pulse displacement (2.13) at the waveguide axis in nonlinear waveguide of the structure A with respect to the center of moving coordinate system, ro=70fs (1), 60fs (2), 50fs (3), a=3.0pm (solid lines), a= 2.4pm (dashed line),...

Though the form of the Rankine-Hugoniot equation, Eqs. (1.42)-(1.44), is obtained when a stationary shock wave is created in a moving coordinate system, the same relationship is obtained for a moving shock wave in a stationary coordinate system. In a stationary coordinate system, the velocity of the moving shock wave is Ml and the particle velocity is given by u = u M2. The ratios of temperature, pressure, and density are the same for both moving and stationary coordinates. 

The PFTR was in fact assumed to be in a steady state in which no parameters vary with time (but they obviously vary with position), whereas the batch reactor is assumed to be spatially uniform and vary only with time. In the argument we switched to a moving coordinate system in which we traveled down the reactor with the fluid velocity , and in that case we follow the change in reactant molecules undergoing reaction as they move down the tube. This is identical to the situation in a batch reactor ... 

The primary difference between the two equations is the unsteady term in equation (E2.2.2) and the convective term in equation (E5.3.2). Now, let s convert our coordinate system of Example 2.2 to a moving coordinate system, moving at the bulk velocity, U, which suddenly experiences a pulse in concentration as it moves downstream. This is likened to assuming that we are sitting in a boat, moving at a velocity U, with a concentration measuring device in the water. The measurements would be changing with time, as we moved downstream with the flow, and the pulse in concentration would occur at x = 0. We can therefore convert our variables and boundary conditions as follows ... 

The solution of Example 7.3 will be compared with an analytical solution of a diffusive front moving at velocity U, with D = 1/2U Az. First, we must derive the analytical solution. This problem is similar to Example 2.10, with these exceptions (1) convection must be added through a moving coordinate system, similar to that described in developing equation (2.36), and (2) a diffusion gradient will develop in both the +z-and -z-directions. 

So, now that we have the fixed interface fairly well described, what about the free interface This actually presents another problem for the techniques of Campbell and Hanratty. The z = 0 axis is on the water surface. If the water surface moves up and down, the axis moves with it, and is actually a moving coordinate system. It is difficult to get a water surface, exposed to turbulence, that does not move more than 100 /u.m. 

A common form of analyzing film blowing is by setting-up a coordinate system, , that moves with the moving melt on the inner surface of the bubble, and that is oriented with the film as shown in Fig. 6.21. Using the moving coordinates, we can define the three non-zero terms of the local rate of deformation tensor as... 

Under Lorentz transformation a 3D line element appears to be contracted in the direction of motion. A time interval as measured in two relatively moving coordinate system is likewise, not invariant under Lorentz transformation,... 

The conservation of momentum is one of the basic principles of mechanics and it is to be expected that momentum remains invariant under Lorentz transformation. The fact that the velocity of a moving body is observed to be different in relatively moving coordinate systems therefore implies that,... 

Transformation of the above system into moving coordinates r=t f=z-w rm 1 yields ... 

A moving coordinate system was used to account for the velocity term of the ADE. A problem similar to the one presented in Skaggs and Kabala [58] is used for the QR study. The results are less accurate than that of the TR approach, but it is computationally less expensive. The authors claimed that it is much easier to incorporate heterogeneous parameters in the QR method. However, to date, heterogeneous parameters have not been incorporated either in the QR method or in the Tikhonov method by Skaggs and Kabala. 

This problem is most easily solved with a simple change of variable into a moving coordinate system. If time is measured from the start of ablation then a convenient variable change is... 

The effect of this transformation is to fix the origin at the surface in the moving coordinate system. The derivatives in Eq. (12-61) may now be expressed as... 

Another model worth considering is to assume all the deadwater resides in the stagnant pockets in bubble wakes. Here, a moving coordinate system would be used, taking the bubble swarm velocity to be U. For this model, equation (13) is replaced by the distributed parameter equation ... 

A moving coordinate accounting for the movement of shrinking particles was used to handle the problem of structural changes during drying (Ratti, 1991). 

From this perspective of the moving coordinate z, the two fluids at infinity appear to be stationary. 

This equation can be solved again by transforming the solution domain into a moving coordinate system. The velocity of the coordinate system is slowed by the isotope exchange reaction. The following transformation is used ... 

Let us now derive the simplified form of (25.27) corresponding to a moving coordinate system. Let x , /, and z7 be the coordinates in the new system and u x, u y, and be the wind velocities with respect to this new moving coordinate system. The coordinate system moves horizontally with velocity equal to the wind speed and therefore ux = u y = 0, while u = u-. Therefore the second and third terms on the left-hand side [LHS] of (25.27) are zero in this case. Physically, the air parcel is moving with velocity equal to the windspeed, so there is no exchange of material with its surroundings by advection. The atmospheric diffusion equation then simplifies to... 

Equation 3-19 defines the diffusion coefficient Da, but is inconvenient for many calculations since the flux is defined with respect to a moving coordinate system. For binary systems, JaIS simply related to the mass flux of A with respect to a stationary coordinate system, rt ... 

The application of the equations in preceding sections to moving coordinate systems is quite simple, once we grasp the idea that the reservoir conditions are a function of the frame of reference. This appears startling at first, but it must be so. 

The characteristic scale of the polymerization wave (i.e., the spatial region over which the major variation of the temperature and the species concentrations occurs) is typically much smaller than the length of the tube. Thus, on the scale of the polymerization wave the tube can be considered infinite, — oo < X < 00. It is convenient to introduce a moving coordinate x = x — (fit, y), where (p is the position of a characteristic point of the wave. The specific choice of (p will be described later. Expressed in the moving coordinate system, the mass and energy balance equations become... 

Here, the Laplacian in the moving coordinate system is given by... 

Since the system (3.22), (3.23) is already written in a moving coordinate system, the traveling wave solution is a stationary solution of this problem,... 

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