Conservation%20principles%20%26%20Reynolds%20Transport%20Thm

Conservation%20principles%20%26%20Reynolds%20Transport%20Thm...

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Unformatted text preview: Supplemental Notes Elementary Derivation of Governing Equations in Fluid Mechanics To understand the logical development of the various forms of equations governing the motion of fluid flows from the basic conservation laws, the following chart is arranged as a road map for readers to maintain a clear view in the big picture. A road map for the development of governing equations for fluid flows Conservation Laws Mass conservation; no mass is destroyed/created Momentum conservation; Newtons 2 nd law Energy Conservation; 1 st law of thermodynamics Gauss theorem Fluid Flows in control volume Integral form Differential form Navier- Stokes Eq. Constitutive relation Reynolds Ttransport Theorem Mass Conservation Momentum Conservation Energy Conservation Energy Eq. Bernoulli Eq . System of fixed mass Continuity Eq. A. Basic Conservation Principles in Integral Form Learning Objectives: 1. Become familiar with the equations governing the transport of mass, momentum and energy in the integral form 2. Understand the physical meaning of various terms in the conservation laws. Required background: 1. Vector calculus, field theory 2. Elementary thermodynamics 3. Elementary dynamics 1. Control Volume and System Cryogenic fluids can be either in gaseous or liquid form. The flow may appear either in single-phase or two-phase form. A good understanding of the two-phase cryogenic flow must be built on the base that of single-phase flows. In this chapter, we focus on the equations governing the motion of single-phase cryogenic flow. In mechanics, the laws of physics are often applied to points or systems of fixed mass. For example, the Newton's second law, F = ma , is applicable to a fixed collection of mass. We call a fixed collection of mass as the system . In studying fluid flows, the idea of fixed mass system is difficult to implement. A region of fixed mass, such as these enclosed by the blue dash lines on the left of Fig. 2.1 will be deformed due to the nonuniform flow velocity. It is rather difficult to track the system of the fixed mass in the flow field without knowing the velocity field first. To alleviate this problem, a control volume is used in studying the fluid flow. A control volume ( c . v .) is a region in space where fluid flows through it; the fluid inside c . v . does not remain the same. The surface enclosing the c . v . is called control surface ( c . s .). Control volume at time t . Control volume at t + t . System at t . System at time t + t. Fig. 2.1 Control volume and system. In Fig. 2.1, the c . v . at time t is chosen. And we chose the system at time t to coincide with the c . v . At a later instant t + t , the fluid in the system has moved to the downstream region, as shown on the right of Fig. 2.1, so that the region of the system at t + t is different from the system at t . On the other hand, the c . v . at t + t remains in the same place....
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