3controlvolume-print

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Unformatted text preview: • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Chap 3 Fluid Flow Concepts and Basic Control Vol- ume Equation • Introduce concepts needed for analysis of fluid motion. • Derive basic equations that enable us to predict fluid behavior. In- cluding – Continuity equation – Equation of momentum – First and second laws of thermodynamics. Control volume approach is used. Basically this chapter deals with • One-dimensional flow theory • Incompressible cases • Viscous effects do no predominate. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit 3.1 Flow Concepts and Kinematics 3.1.1 Analysis Approach ? Lagrangian approach ? Eulerian approach 3.1.1.1 Lagrangian Ap- proach In solid mechanics, we use Lagrangian approach for given masses. We study each object or mass point or system. We can use same approach to describe fluid, that is, fluid is considered as many mass points (a fluid parcel is considered as a fluid mass point in macroscopic motion). We follow fluid mass points for studying their mechanical and physical states. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit How to identify mass? ? Specify a initial time t = 0 ¶ ? Write down the initial position coordinates for each fluid mass point A = ( a 1 ,a 2 ,a 3 ) ¶ ? Identify each fluid mass point with the initial position coordinates. (similar to student ID no.) In Lagrangian approach, we care about motion, variation and physical states for each mass point, say, after a period of time, at t , x = x ( A ,t ) , T = T ( A ,t ) , p = ( A ,t ) A ,t are Lagrangian variables. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Property of position function Position function x = x ( A ,t ) is the trace of fluid mass point, has the following properties: ? Initial time t = 0 , x = A , that is x ( A , 0) = A ? At any time, position variable x and initial position variable A are one to one corresponding continuous functions. According to implicit function theorem, one can show that there exists an inverse function for x = x ( A ,t ) A = A ( x ,t ) • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit One-to-one correspondence: According to x , we can find A (that is we can find where you come from, by checking your ID); On the other hand, according to A , we can find x (we know where you go, by put a GPS sensor on you). Comments: Lagrangian approach is natural extension of solid body kinematics, it is easy to understand. If what we care are just some particular fluid parcels, say, particles in two phase flow, it works well. But in fluid, the deformation is large, when many fluid parcels are under consideration, it is hard to follow fluid parcels (fluid mass points). • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit It is easy to follow particle in figure (a) as it is so neatly ordered....
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This note was uploaded on 04/18/2009 for the course CE 309 taught by Professor Lee during the Spring '07 term at USC.

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