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Chapter4 - 04-R3868 10:12 AM Page 89 CHAPTER F L U I D K I...

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FLUID KINEMATICS F luid kinematics deals with describing the motion of fluids without neces- sarily considering the forces and moments that cause the motion. In this chapter, we introduce several kinematic concepts related to flowing fluids. We discuss the material derivative and its role in transforming the con- servation equations from the Lagrangian description of fluid flow (following a fluid particle ) to the Eulerian description of fluid flow (pertaining to a flow field ). We then discuss various ways to visualize flow fields— streamlines , streaklines , pathlines , timelines , and various surface flow visualization meth- ods. The concepts of vorticity , rotationality , and irrotationality in fluid flows are then discussed. Finally, we discuss the Reynolds transport theorem ( RTT ), emphasizing its role in transforming the equations of motion from those fol- lowing a system to those pertaining to fluid flow into and out of a control volume . 89 CHAPTER 4 OBJECTIVES When you finish reading this chapter, you should be able to Understand the role of the material derivative in transforming between Lagrangian and Eulerian descriptions Distinguish between various types of flow visualizations Distinguish between rotational and irrotational regions of flow based on the flow property vorticity Understand the usefulness of the Reynolds transport theorem Satellite image of a hurricane near the Florida coast; water droplets move with the air, enabling us to visualize the counterclockwise swirling motion. However, the major portion of the hurricane is actually irrotational , while only the core (the eye of the storm) is rotational .
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4–1 LAGRANGIAN AND EULERIAN DESCRIPTIONS The subject called kinematics concerns the study of motion . In fluid dynam- ics, fluid kinematics is the study of how fluids flow and how to describe fluid motion. From a fundamental point of view, there are two distinct ways to de- scribe motion. The first and most familiar method is the one you learned in high school physics—to follow the path of individual objects. For example, we have all seen physics experiments in which a ball on a pool table or a puck on an air hockey table collides with another ball or puck or with the wall (Fig. 4–1). Newton’s laws are used to describe the motion of such objects, and we can accurately predict where they go and how momentum and kinetic energy are exchanged from one object to another. The kinematics of such ex- periments involves keeping track of the position vector of each object, x A , x B , . . . , and the velocity vector of each object, V A , V B , . . . , as functions of time (Fig. 4–2). When this method is applied to a flowing fluid, we call it the Lagrangian description of fluid motion after the Italian mathematician Joseph Louis Lagrange (1736–1813). Lagrangian analysis is analogous to the (closed) system analysis that you learned in your thermodynamics class; namely, we follow a mass of fixed identity. The Lagrangian description re-
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