Parallel Velocities

If the two velocities vi and vz are parallel to each other, the situation is simplified to a large extent. We can always rotate all three inertial systems simultaneously such that vi and vz are parallel to the x-axes of all three frames of reference. Consequently, all three transformation matrices occurring in Eq. (3.95) are of the simple form given by Eq. (3.66) or Eq. (3.74), respectively. Eq. (3.95) 

For parallel velocities, the rapidities are simply additive, tp = tpi + tp2- Furthermore, according to Eq. (3.96) the Lorentz boosts from IS to IS and IS to IS commute. 

This may be compared with the successive application of three-dimensional rotations, which also only commute if they are applied about the same axis of rotation. In order to derive an explicit expression for the velocity V = V vi,V2), we consider 

For parallel velocities Vi V2, the resulting velocity V with which IS is moving relative to IS is therefore given by 

For general, i.e., non-parallel velocities V and vz we cannot simplify the problem by application of a suitable rotation of the coordinate axes. Now all three Lorentz boost matrices occurring in Eq. (3.95) are of the most general form given by Eq. (3.81) with a yet undetermined resulting velocity V. We thus have to evaluate Eq. (3.95) directly. In order to achieve this task, it is convenient to introduce the abbreviations 

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For flow parallel to an electrode, a maximum in the value of the mass-transfer rate occurs at the leading edge of the electrode. This is not only the case in flow over a flat plate, but also in pipes, annuli, and channels. In all these cases, the parallel velocity component in the mass-transfer boundary layer is practically a linear function of the distance to the electrode. Even though the parallel velocity profile over the hydrodynamic boundary layer (of thickness h) or over the duct diameter (with equivalent diameter de) is parabolic or more complicated, a linear profile within the diffusion layer (of thickness 8d) may be assumed. This is justified by the extreme thinness of the diffusion layer in liquids of high Schmidt number ... 

This formula defines the Einstein addition law for parallel velocities. It shows that, no matter how closely P and (3 approach unity f3" can never exceed unity. In this sense c can be considered to be the ultimate speed allowed by special relativity. 

Problem 10-10. Mixing Layer. Two semi-infinite streams of the same fluid, but with different uniform, parallel velocities, Uand all, are brought into contact. Owing to the action of viscosity, a boundary layer will develop over which the initial discontinuity in velocity will be smoothed out as illustrated in the figure. Derive the boundary-layer equations for... 

These lead to the law for the composition of parallel velocities, and certainly would not account for the result of the Michelson and Morley experiment. To arrive at the required relation one has to apply what is called the Lorentz transformation. According to this... 

Eq. (3.105) constitutes the addition theorem for general relativistic velocities. For parallel velocities V V2 it reduces to Eq. (3.100). Note that for general, i.e., non-parallel velocities V and V2 the result is not symmetric in v and V2- It thus makes a difference whether we first transform with A(z7i) and then with A(z 2) or vice versa. This is most dramatically seen for orthogonal velocities Vi J-V2, for which the resulting velocity V is given by... 

General Lorentz boosts A(z ) are therefore in general not commutative, whereas boosts for parallel velocities commute, cf. Eq. (3.98). 

In Examples 2-1 and 5-11, the assumption of a producing fraeture motivated the use of logarithms as singularities these are responsible for equal and opposite Darcy velocities normal to the fraeture plane and simulate production. In practice, flow also moves parallel to the fracture, toward the penetrating well that taps the fluid. For this motion to be possible, a pressure gradient must exist along the fracture, and the variable Pf(x) considered in Chapter 2 applies. Often the fracture contains solids and debris, and the parallel velocity on one side of the fracture will not be the same as that on the other. For this flow, we specify along z = 0 the discontinuous velocity... 

Permanent translation (fling) is a consequence of permanent fault displacement due to an earthquake it appears in the form of step displacement and one-sided velocity pulse in the strike-parallel direction for strike-slip faults or in the strike-normal direction for dip-slip faults. In the latter case, directivity and permanent translation effects build up in the same direction. Figure 3 illustrates characteristic examples of permanent translation (fling) from the 1999 Izmit earthquake. The fault-parallel velocity and displacement time histories recorded at Yarimca (YPT) and Sakarya (SKR) stations are affected by the permanent displacement along the right-lateral strike-slip North Anatolian Fault. 

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