The understanding of the organization of a flow cannot be based solely on knowledge of its local properties, velocity, pressure, density, temperature, etc. at each point in space (*x*, *y*, *z*) and every time *t.* We often need a more “dynamic” representation by using the motion of fluid particles, or at least the direction of the displacement. Such a visualization of the flow is a valuable aid in the design of vehicles or systems usinsg fluids. It plays a vital role in the study of separated three-dimensional flows whose understanding is based on the topological analysis of the velocity and skin friction fields. To assist this representation, we use the concepts of trajectory and streamlines, and to a lesser extent, streak lines.

We consider an *unsteady* flow in the three-dimensional space whose velocity field (*x*, *y*, *z*, *t*) is known at any point (*x*, *y*, *z*) and at any time *t.* Given a fluid element (*E*_{1}) passing through the point *P*_{0} at time *t*_{0} (see Figure 8.1a), at a later time *t* = *t*_{0} + Δ*t*, that same element can be found at a point *P*_{1}. The path followed by (*E*_{1}) on its displacement from *P*_{0} to *P*_{1} is called a *trajectory.*

We consider another fluid element (*E*_{2}) passing through the same point *P*_{0} at a later time *t*_{1}. At time *t* = *t*_{0} + Δ*t*, (*E*_{2}) has no reason to be in ...

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