The second velocity field at a later time t 1 t is different from the first and

The second velocity field at a later time t 1 t is

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The second velocity field at a later time t 1 > t 0 is different from the first, and although the fluid parcel started at the same time and place as in the streamline case, the change in the velocity field has caused the location of the fluid parcel to differ from what would have been the case in the instantaneous streamline case. But once the fluid parcel has traveled a portion of its trajectory that portion is fixed ; i.e. , we see in the third velocity field that approximately the first one third of the pathline is the same as that in the previous part of the figure. But the velocity field has changed, so the remainder of the pathline will be different from what it would have been if the velocity field had stayed constant. This is continued for two more steps in the figure, and from this it is easily seen that the pathline obtained during this evolution is quite different in detail from the streamline. We should also note that if we were to now make a streamline at this final time it would also differ from the pathline. In particular, although we have not drawn the figure to display this, the velocity field adjacent to the older part of the pathline is not actually the one shown, in general. (We have frozen the velocities at each new location of the fluid parcel to make them look consistent with the pathline but, in fact, these too are constantly changing.) We emphasize that what the velocity field does after passage of the fluid parcel under consideration is of no consequence to the pathline. This can be seen easily from the mathematical formulation. If we integrate Eq. (2.20) between times t 0 and t 1 , we obtain x 1 = x 0 + integraldisplay t 1 t 0 u ( x ( t ) , y ( t ) , t ) dt
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44 CHAPTER 2. SOME BACKGROUND: BASIC PHYSICS OF FLUIDS t = t location at later time 3 t = t initial location of fluid parcel initial location of fluid parcel t = t 4 final time location at streamline corresponding to original velocity field t = t 1 0 initial location of fluid parcel t = t 2 location at later time initial location of fluid parcel later time location at Figure 2.25: Temporal development of a pathline. as the x coordinate of the fluid parcel at t = t 1 , with a similar expression for the y coordinate. Now we can repeat this process to get to time t = t 2 : x 2 = x 1 + integraldisplay t 2 t 1 u ( x ( t ) , y ( t ) , t ) dt What we see, just as we have indicated schematically in the figure, is that the coordinates of the pathline between times t 1 and t 2 do not explicitly depend on the velocity field at locations visited by the fluid parcel prior to time t 1 . On the other hand, if an instantaneous streamline were constructed at, say t = t 2 , it would depend on all locations within the flow field at that instant. 2.5.3 Streaklines Especially for unsteady flows the streakline is the closest of the three visualization techniques con- sidered here to what is usually produced in a laboratory experiment. We begin with the definition.
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2.6. SUMMARY 45 Definition 2.15 A streakline is the locus of all fluid elements that have previously passed through a given point.
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