Sec.1_Review_Notes - AE 312 Section 1. REVIEW OF FLUID...

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AE 312 Section 1. REVIEW OF FLUID MECHANIC AND THERMODYNAMIC PRINCIPLES I. Governing Equations Derivation Procedure: (1) Write governing equations in known system form (2) Apply Reynolds' Transport Theorem (3) Obtain corresponding control volume form Definitions System (Closed System): a fixed set of identifiable particles of constant mass; the same set of particles is followed throughout the analysis. (abbreviation: sys ) Control Volume (Open System): a region of space which may or may not be moving (often taken to be fixed in space) and through which fluid may flow. Therefore, the identity of the particles in the control volume may vary from instant to instant. (abbreviation: CV ) Control Surface: the geometric bounding surface of the control volume (abbreviation: CS ) A. Reynolds' Transport Theorem Reynolds' Transport Theorem relates the time rate of change of an extensive property following a system to appropriate terms for a control volume , the latter form being more useful for flow problems. Take: = arbitrary extensive property, i.e., proportional to mass = corresponding specific property, i.e., amount of per unit mass, dm d By considering the situation in which a system and control volume coincide at time "t 0 " and an infinitesimal increment of time t later during which the control volume remains fixed, but the system moves in the general direction of the mean streamlines, Reynolds' Transport Theorem can be derived. [For details, see Munson, Young, Okiishi, and Huebsch, 6 th Edition, Sec. 4.4 or White, 6 th Edition, Sec. 3.2.] Reynolds' Transport Theorem:   CS CV sys A d V V d t Dt D dt d CS through of outflow or efflux of rate net CV in of change of rate time system following of change of rate time
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2 where: Dt D substantial derivative V volume density V velocity A d infinitesimal area vector, normal and outward from control surface Note: In RTT both t and V are measured with respect to the control volume. B. Conservation of Mass (Continuity) Choose: = mass = m 1 dm dm dm d System form: 0 dt dm sys Applying RTT to the left-hand side: 0 A d V V d t CS CV 0 CS through mass of efflux net CV in mass of change of rate time
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3 C. Linear Momentum Equation (Newton's Second Law) Choose: P linear momentum (vector) V dm P d dm d System form: dt P d F F F sys B S (assuming an inertial coordinate system) Applying RTT to the right-hand side: CS CV B S A d V V V d V t F F F CS
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Sec.1_Review_Notes - AE 312 Section 1. REVIEW OF FLUID...

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