fluids_ch2

fluids_ch2 - In general, flows can be descibed as laminar...

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Chapter 2 Hydrodynamics Irrotational Flows and Flux In general, flows can be descibed as laminar flows in which the fluid acts as though it is comprised of orderly layers or plates. Laminar flows can be represented visually using streamlines. On the other hand, turbulent flows are very chaotic and cannot by visually represented by streamlines. In this chapter we investigate the geometrical properties of lanminar irrotaional flows, orderly flows that have no circulation. We will cover steady state flows in which the flows or fields remain steady and constant in time and may be represented by streamlines. The particles that compose the system may move around and accelerate in moving from one part of a system to another, but the field patterns themselves remain constant in time. At the heart of classical field theory lies two very powerful theorems: the Uniqueness theorem and Helmholtz's theorem . We will see that two properties called flux and circulation uniquely determine any vector field. 2.1 Characterizing Vector Fields. One of the incredible things about vector fields is that all the different types of field patterns that one can dream up follow from two very simple field patterns. The first type is known as a diverging field where the field can be thought of as diverging from a source of field, or conversely, converging to a sink or drain of field. source Figure 2.1 A source produces a diverging field. Classical Fluids, Chapter 2 -2 1-
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In Figure 2.1 the field lines for a single source is shown. This is the classic diverging field pattern. The second type of pattern is known as a circulating or solenoidal field. A whirlpool is the classic example, Figure 2.2 A whirlpool is an example of a circulating field. and is shown in Figure 2.2. Complicated fields can be produced from a combination of circulating and diverging fields. For instance, your bath tub drain is a sink that a produces a diverging flow, but the residual angular momentum from filling the tub causes the water to swirl around the drain as it moves in towards the drain forming a more complicated whirlpool. There are some subtleties. Just because a field curves, does not mean that the field has a circulating component. Figure 2.3 A purely diverging field produced by two sources and one sink. Classical Fluids, Chapter 2 -2 2-
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Figure 2.3 shows a purely diverging field produced by sources and sinks. What distinguishes this field from a circulating field is that although the field swirls on one side of a given source it tends to swirl the opposite way on the other side. A circulating field will cause the field to swirl in the same direction on both sides with one exception we will discuss later, however, the defining distinction that labels this field as purely diverging is that as the fields leave (or enter) the sources (sinks) the lines are perpendicular to the boundaries of the sources. In short note that for diverging flows, the field lines end on sources. Anywhere field lines end, the flow will be diverging. In
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This note was uploaded on 10/23/2009 for the course PHY 9B taught by Professor Cheng during the Fall '08 term at UC Davis.

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fluids_ch2 - In general, flows can be descibed as laminar...

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