Moving frame of reference

The reaction introduces a spatial shift into the solution (tkv), which leads to an asymmetric behavior of the response function in the moving frame of reference. The slight asymmetry of the lower solid line in Fig. 4.31 is only visible in a magnified view (Fig. 4.32). 

In the following, a Thiele modulus of = 0.1 was chosen, in which case d is well approximated by d = 0.2. In Figure 3.94, Eq. (3.13) is presented in the moving frame of reference xt = 0 for three different times. At t = 0.5 s nearly the entire original present component A has reacted irreversibly to component B and the concentration of A is close to zero everywhere. 

Now, we describe the Kirkendall effect [9], The flux, as well as the diffusion coefficient, has to be chosen relative to a frame of reference. In Figure 5.4, the laboratory frame of reference, X, which is the observer frame of reference, and the moving frame of reference, x, which moves with the inert markers, are shown. 

To end this section and the review, we mention briefly the first results from the simulation on laboratory-frame cross-correlation of the type (v(f)J (0)). Here v is the molecular center-of-mass linear velocity and J is the molecular angular momentum in the usual laboratory frame of reference. For chiral molecules the center-of-mass linear velocity v seems to be correlated directly in the laboratory frame with the molecule s own angular momentum J at different points r in the time evolution of the molectilar ensemble. This is true in both the presence and absence of an external electric field. These results illustrate the first direct observation of elements of (v(r)J (0)) in the laboratory frame of reference. The racemic modification of physical and molecular dynamical properties depends, therefore, on the theorem (v(r)J (0)) 0 in both static and moving frames of reference. An external electric field enhances considerably the magnitude of the cross-correlations. 

Galilean invariance (Rothman Zalesky, 1997) is a fundamental tenet of Newtonian mechanics. It is invariance under the transformation x = x - wt, where w is the constant velocity of a moving frame of reference, and embodies the concept that only the relative velocities and positions of two bodies determine their interaction. Galilean invariance is lost in lattice gas simulations because every particle has only one possible speed. This loss is an artifact that can be eliminated for incompressible fluids by re-scaling the velocity. According to Boghosian (1993), more sophisticated lattice gas models overcome this problem. Appropriate application of lattice gas models also requires certain restrictions on the mean free path of a particle (Rothman, 1988). 

This is messy problem analytically. It is fairly easy if you take the viewpoint of someone riding on the jump (the lagrangian viewpoint) and solve by trial and error for the jump velocity that satisfies the hydraulic jump equation in the moving frame of reference. 

The familiar Galilean law of relative motion dictates that a stationary observer measures the position of an object in relative motion, at constant speed V, to change by an amount vt during time t. In the moving frame of reference, where the position P remains constant, the relative motion is described correctly by ... 

The revolutionary feature of the Lorentz equations is that in order to perform a coordinate transformation between relatively moving frames of reference a complex time coordinate must be taken into account. This transformation takes the form of a complex rotation in a four-dimensional pseudo-Euclidean... 

The special theory. For Galileo and Newton, all uniformly moving frames of reference (Galilean frames) are equivalent for describing the dynamics of moving bodies. There is no experiment in dynamics that can distinguish between a stationary laboratory and a laboratory that is moving at uniform velocity. Einstein s special theory of relativity takes this notion of equivalent frames one step further he required all physical phenomena, not only those of dynamics, to be independent of the uniform motion of the laboratory. 

Fig. 8.5 The fountain flow velocity vector field in the moving frame of reference (Reproduced from Zheng (1991))...

The moving mesh method implies that the entire domain is translated as the train moves. This means that all results are presented for locations with constant relative distances from the train, but in a moving frame of reference. The results are presented below. 

Kristensen, L. and Frandsen, S., "Model for Power Spectra of the Blade of a Wind Turbine Measured from the Moving Frame of Reference," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 10, 1982, pp. 249-262. 

The important point now is, that Re[Q ] has a quadratic maximum for some value k. By adjusting v ( choosing the speed of the moving frame of reference properly, we may set this maximum to zero, such that Re[f2n] -(k-kQ). Quite obviously this value of v Vq corresponds to the speed of the fastest propagating mode. This is now what we conjecture to be the operating point of "marginal stability" [6], where the spectrum has its extreme value at Re = 0. 

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