Chapter4 - 04-R3868 7/20/06 10:12 AM Page 89 CHAPTER F L U...

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FLUID KINEMATICS F luid kinematics deals with describing the motion of Fuids without neces- sarily considering the forces and moments that cause the motion. In this chapter, we introduce several kinematic concepts related to Fowing Fuids. We discuss the material derivative and its role in transforming the con- servation equations from the Lagrangian description of Fuid Fow (following a Fuid particle )tothe Eulerian description of Fuid Fow (pertaining to a Fow ±eld ). We then discuss various ways to visualize Fow ±elds— streamlines , streaklines , pathlines , timelines , and various surface Fow visualization meth- ods. The concepts of vorticity , rotationality , and irrotationality in Fuid Fows are then discussed. ²inally, 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 Fuid Fow into and out of a control volume . 89 CHAPTER 4 OBJECTIVES When you fnish reading this chapter, you should be able to n Understand the role oF the material derivative in transForming between Lagrangian and Eulerian descriptions n Distinguish between various types oF ±ow visualizations n Distinguish between rotational and irrotational regions oF ±ow based on the ±ow property vorticity n Understand the useFulness oF the Reynolds transport theorem Satellite image of a hurricane near the ²lorida 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 n LAGRANGIAN AND EULERIAN DESCRIPTIONS The subject called kinematics concerns the study of motion . In Fuid dynam- ics, fuid kinematics is the study of how Fuids Fow and how to describe Fuid motion. ±rom a fundamental point of view, there are two distinct ways to de- scribe motion. The ²rst and most familiar method is the one you learned in high school physics—to follow the path of individual objects. ±or 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 (±ig. 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 ,...,andthe velocity vector of each object, V A , V B , . . . , as functions of time (±ig. 4–2). When this method is applied to a Fowing Fuid, we call it the Lagrangian description of Fuid 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 ²xed identity. The Lagrangian description re- quires us to track the position and velocity of each individual Fuid parcel,
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This note was uploaded on 10/16/2010 for the course GEEN 3311 taught by Professor Dr.ra’fatal-waked during the Fall '10 term at Prince Mohammad Bin Fahd University, Dhahran.

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Chapter4 - 04-R3868 7/20/06 10:12 AM Page 89 CHAPTER F L U...

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