Path line

A path line is the path traced by a single particle of fluid over a period of time. 

A streamline shows the direction of a number of particles of fluid at the same instant in time. Flow cannot take place across a streamline. Path lines and streamlines will be identical for steady flow. 

Point 2 is found from point 1 by tracing a path (line) of constant entropy on the diagram. The method is illustrated in Example 3.10. 

Figure 10. Gradient paths for an idealized mountain. Two gradient paths (lines of steepest ascent) are shown (a), together with an arbitrary path (b) that is not a line of steepest ascent but represents an easier route up the mountain. The gradient paths cross the contours at right angles.

In Fig. 18, flow path lines are shown in a perspective view of the 3D WS. By displaying the path lines in a perspective view, the 3D structure of the field, and of the path lines, becomes more apparent. To create a better view of the flow field, some particles were removed. For Fig. 18, the particles were released in the bottom plane of the geometry, and the flow paths are calculated from the release point. From the path line plot, we see that the diverging flow around the particle-wall contact points is part of a larger undulating flow through the pores in the near-wall bed structure. Another flow feature is the wake flow behind the middle particle in the bottom near-wall layer. It can also be seen that the fluid is transported radially toward the wall in this wake flow. 

Fig. 25. Wall-segment geometry for 1-hole particles orthographic projections showing (a) flow path lines for particles released from vertical planes close to the tube wall (b) flow path lines for particles released from the bottom horizontal plane.

In Fig. 25b, the simulated marker particles were released from the bottom surface, which generates path lines that show more detail of the flow inside the WS, at lower radial coordinate values. The path lines reinforce the trends seen in Fig. 25a, and it is also possible to see some evidence of flow through the center voids of the particles. Most evident is the mix of spiraling and axial flow between the center front and center right particles. 

It is of interest to determine the extent to which there is flow through the interior holes of the particles, as the reaction activity is proportional to geometric surface area under these conditions. So, it is important to know whether the extra surface area provided by the holes is accessible to the flow. It is not easy to see this internal flow from the path lines in Fig. 25, although there appears to be flow through the center particle. To determine this more clearly we constructed a surface that passed through the midpoint of the center particle, perpendicular to its axis, for each of the particle geometries. This is shown as the dark square in Fig. 26, which illustrates the results for the 4-hole particle. 

In unsteady flows, the streamline pattern changes from instant to instant. In "steadyr state flux the streamlines are constant in time and also represent the path lines, the trajectories of the fluid particles (Ref 1, p 18)... 

Fig. 32. Path lines (spaced at intervals of equal flux) of the wall flow in different asymmetric shape cells in clean (left) and loaded (right) states. From the top, geometries a, b and c of Fig. 30 are shown.

A wet gas exists solely as a gas in the reservoir throughout the reduction in reservoir pressure. The pressure path, line 12, does not enter the phase envelope. Thus, no liquid is formed in the reservoir. However, separator conditions lie within the phase envelope, causing some liquid to be formed at the surface. 

Spiral vortex. So far the discussion has been confined to the rotation of all particles in concentric circles. Suppose there is now superimposed a flow with a velocity having radial components, either outward or inward. If the height of the walls of the open vessel were less than that of a liquid surface spread out by some means of centrifugal force, and if liquid were supplied to the center at the proper rate by some means, then it is obvious that liquid would flow outward, over the vessel walls. If, on the other hand, liquid flowed into the tank over the rim from some source at a higher elevation and were drawn out at the center, the flow would be inward. The combination of this approximately radial flow with the circular flow will result in path lines that are some form of spirals. 

For a forced vortex with spiral flow, energy is put into the fluid in the case of a pump and extracted from it in the case of a turbine. In the limiting case of zero flow, when all path lines become concentric circles, energy input from some external source is still necessary for any real fluid in order to maintain the rotation. Thus a forced vortex is characterized by a transfer of mechanical energy from an external source and a consequent variation of H as a function of the radius from the axis of rotation. 

Spiral vortex. If a radial flow is superimposed upon the concentric flow previously described, the path lines will then be spirals. If the flow goes out through a circular hole in the bottom of a shallow vessel, the surface of liquid takes the form of an empty hole, with an air core sucked down the hole. If an outlet symmetrical with the axis is provided, as in a pump impeller, we might have a flow either radially inward or radially outward. If the two plates are a constant distance B apart, the radial flow with a velocity Vr is then across a series of concentric cylindrical surfaces whose area is 0.2nrB. Thus Q = 2nrBVr is a constant, from which it is seen that rVr is a constant. Thus the radial velocity varies in the same way with r that the circumferential velocity did in the preceding discussion. Hence the pressure variation with the radial velocity is just the same as for pure rotation. Therefore the pressure gradient of flow applies exactly to the case of spiral flow, as well as to pure rotation. 

In Fig. 10.3 it is seen that, if both Vr and Vu vary inversely to the radius r, the angle a is constant. Hence for such a case the path line is the equiangular or logarithmic spiral. For a case where B varies, Vr will not vary in the same way as Vu, and hence the angle a is not a constant. 

The plotting of such spiral path lines may be of practical value in those cases where it is desired to place some object in the stream for structural reasons but where it is essential that the interference with the stream flow be minimized. If the structure is shaped so as to conform to these streamlines as nearly as possible, it will offer a minimum disturbance to the flow. 

Figure 10 shows the path lines in the bottom screw zone for the original screw reactor and the redesigned reactor. The path lines clearly show... 

Fig. 10. Path lines of bottom part (A) original reactor (B) redesigned reactor.

Figure 12. Lagrangian path lines at various stages of a Rayleigh-Taylor collapse for the case of two inviscid, incompressible fluids having a density ratio of 2 1. A free surface is present above the dense fluid and the interface between the fluids is indicated for each stage. The simulation shows how later evolution of the fluid flow is dominated by the strength and dynamics of the vortex pair created during the...

Such analysis can reveal bond paths (lines of maximum ED finking two atoms), exact location of the bond critical points, bond elhpticity (deviation from cylindrical symmetry), and so on. The real atomic boundaries can be found and the effective atomic charges integrated. 

Another technique used in combat is the explosive line charge. The line charge is a cord or rope of explosives that is fired across a suspected minefield. The explosives are set off to detonate or disable nearby mines and thus clear a path. Line charges have been used since World War II and are still being improved today. 

Fig. 2 Path lines (a, c, e) and streamlines (b, d, f) for different Re numbers of 12 (a, b), 80 (c, d) and 240 (e, f). The swirling of the fluid flow obtained at higher Re number results in better dispersion of the fluid within the channel volume and hence an improvement in the mixing quality (Reprinted from [61]. Copyright (2008) with permission from Elsevier)...

Pollock, D. W. (1988) Semianalytical computation of path lines for finite difference models. Ground Water 26(6), 743-750. 

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