Chapt03 - Table tennis ball suspended by an air jet The...

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128 Table tennis ball suspended by an air jet. The control volume momentum principle, studied in this chapter, requires a force to change the direction of a flow. The jet flow deflects around the ball, and the force is the ball’s weight. (Courtesy of Paul Silverman/Fundamental Photographs)
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3.1 Basic Physical Laws of Fluid Mechanics Motivation. In analyzing fluid motion, we might take one of two paths: (1) seeking to describe the detailed flow pattern at every point ( x , y , z ) in the field or (2) working with a finite region, making a balance of flow in versus flow out, and determining gross flow effects such as the force or torque on a body or the total energy exchange. The second is the “control-volume” method and is the subject of this chapter. The first is the “differential” approach and is developed in Chap. 4. We first develop the concept of the control volume, in nearly the same manner as one does in a thermodynamics course, and we find the rate of change of an arbitrary gross fluid property, a result called the Reynolds transport theorem . We then apply this theorem, in sequence, to mass, linear momentum, angular momentum, and energy, thus deriving the four basic control-volume relations of fluid mechanics. There are many applications, of course. The chapter then ends with a special case of frictionless, shaft- work-free momentum and energy: the Bernoulli equation . The Bernoulli equation is a wonderful, historic relation, but it is extremely restrictive and should always be viewed with skepticism and care in applying it to a real (viscous) fluid motion. It is time now to really get serious about flow problems. The fluid-statics applications of Chap. 2 were more like fun than work, at least in my opinion. Statics problems ba- sically require only the density of the fluid and knowledge of the position of the free surface, but most flow problems require the analysis of an arbitrary state of variable fluid motion defined by the geometry, the boundary conditions, and the laws of me- chanics. This chapter and the next two outline the three basic approaches to the analy- sis of arbitrary flow problems: 1. Control-volume, or large-scale, analysis (Chap. 3) 2. Differential, or small-scale, analysis (Chap. 4) 3. Experimental, or dimensional, analysis (Chap. 5) The three approaches are roughly equal in importance, but control-volume analysis is “more equal,” being the single most valuable tool to the engineer for flow analysis. It gives “engineering” answers, sometimes gross and crude but always useful. In princi- 129 Chapter 3 Integral Relations for a Control Volume
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Systems versus Control Volumes ple, the differential approach of Chap. 4 can be used for any problem, but in practice the lack of mathematical tools and the inability of the digital computer to model small- scale processes make the differential approach rather limited. Similarly, although the dimensional analysis of Chap. 5 can be applied to any problem, the lack of time and money and generality often makes experimentation a limited approach. But a control- volume analysis takes about half an hour and gives useful results. Thus, in a trio of ap-
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