In Fig 214b is displayed a 2 D flow The physical situation is that of a box of

In fig 214b is displayed a 2 d flow the physical

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In Fig. 2.14(b) is displayed a 2-D flow. The physical situation is that of a box of fluid that is infinite only in the z direction, the top of the box is moving in the x direction, and it is transparent so that we can view the flow field. There are two key observations to be made. First, each of the planes of velocity vectors indicate motion that varies with both the x and y directions; hence, the flow is at least two dimensional. But we also see that the two planes of vectors are exactly alike—they do not change in the z direction. Thus, the flow is 2D. Finally, the 3-D case is presented in Fig. 2.14(c) which represents a box containing fluid with the upper (solid) surface moved diagonally with respect to the x and ( ) z directions. We again show two planes of velocity vectors; but unlike the previous case, they differ significantly from one another, indicating a z dependence of the u and v components as well as the obvious x and y dependence. In addition, it should be clear that for this case the z component of velocity, w , is nonzero and also varies with x , y and z . 2.4.3 Uniform and non-uniform flows We often encounter situations in which a significant simplification can be had if we are able to make an assumption of uniform flow . We begin by giving a precise definition of this useful concept, and we then provide some examples of uniform and non-uniform flows. Definition 2.10 A uniform flow is one in which all velocity vectors are identical (in both direction and magnitude) at every point of the flow for any given instant of time. Flows for which this is not true are said to be nonuniform . This definition can be expressed by the following mathematical formulation: U s ∂u s ∂v s ∂w s = 0 0 0 . (2.12) Here, U is the velocity vector, and s is an arbitrary vector indicating the direction with respect to which differentiation will be performed. For example, s might be in any one of the coordinate directions, or in any other direction. But no matter what the direction is, the derivative with respect to that direction must be everywhere zero for the flow to be uniform. It should also be observed that the above definition implies that a uniform flow must be of zero dimension—it is everywhere constant, and thus does not depend on any spatial coordinates. From this we see that none of the examples in Fig. 2.14 correspond to a uniform flow. Figure 2.15 provides some examples of uniform and non-uniform flow fields. Part (a) of the figure clearly is in accord with the definition. All velocity vectors have the same length and the same direction. Part (b) of Fig. 2.15 contains a particularly important case that we will often encounter later in the course. From the definition we see that this is a non-uniform flow; the velocity vectors have different magnitudes as we move in the flow direction. On the other hand,
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32 CHAPTER 2. SOME BACKGROUND: BASIC PHYSICS OF FLUIDS (a) (b) (c) Figure 2.15: Uniform and non-uniform flows; (a) uniform flow, (b) non-uniform, but “locally uni- form” flow, (c) non-uniform flow. at any given x location they all have the same magnitude. This is usually termed
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