But their solutions consist merely the fluid particle spatial location at each

# But their solutions consist merely the fluid particle

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they result from direct application of Newton’s second law. But their solutions consist merely of the fluid particle spatial location at each instant of time, as depicted in Fig. 3.1. This figure shows two different fluid particles and their particle paths for a short period of time. Notice that it is the location of the fluid parcel at each time that is given, and this can be obtained directly by X z y 2 (1) 2 X x particle path #1 (2) particle path #2 X (0) 1 X 1 1 (2) X (1) X 2 (0) Figure 3.1: Fluid particles and trajectories in Lagrangian view of fluid motion. solving the corresponding equations. The notation X (0) 1 represents particle #1 at time t = 0, with X denoting the position vector ( x, y, z ) T . There are several important features of this representation requiring some explanation. First, it can be seen that the fluid parcel does not necessarily retain its size and shape during its motion. Later, when we derive the equation for mass conservation we will require that the mass of the fluid element remain fixed; hence, if the density is changing, which might well be the case, the volume must also change. Second, we can think of changes in shape as having arisen due to interactions with neighboring fluid elements (not shown, but recall Fig. 2.9); we will treat this in more detail when we derive the momentum equations. We next observe that although the velocity is not directly calculated, it is easily deduced since, e.g. , dx dt = u . Thus, if a sequence of locations of the fluid parcel is known for a period of time, it is easy to calculate its velocity (and acceleration) during this same period. Furthermore, we can think about obtaining values of any other fluid property ( e.g. , temperature or pressure) at this sequence of locations by simply “measuring” them as we ride through the fluid on the fluid parcel. Finally, it must be emphasized that in order to obtain a complete description of the flow field using this approach it is necessary to track a very large number of fluid parcels. From a practical standpoint,
3.1. LAGRANGIAN & EULERIAN SYSTEMS; THE SUBSTANTIAL DERIVATIVE 49 either experimentally or computationally, this can present a significant burden. Furthermore, it is rather typical in engineering applications to need to know fluid properties and behavior at specific points in a flow field. In the context of a Lagrangian description it is difficult to specify, a priori , which fluid parcel to follow in order to obtain the desired information at some later time. 3.1.2 The Eulerian viewpoint An alternative to the Lagrangian representation is the Eulerian view of a flowing fluid. As noted above, this corresponds to a coordinate system fixed in space, and within which fluid properties are monitored as functions of time as the flow passes fixed spatial locations. Figure 3.2 is a simple representation of this situation. It is evident that in this case we need not be explicitly typical measurement locations x z y Figure 3.2: Eulerian view of fluid motion.

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