Chapter%204%20Basic%20Eqns

Chapter%204%20Basic%20Eqns - IV. Basic Equations of Fluid...

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Unformatted text preview: IV. Basic Equations of Fluid Motion General comments: System vs control volume • Laws of physics are often applied to points or systems of fixed mass . • Newton's 2 nd law, F = ma , is applicable to a fixed collection of mass. • System : a fixed collection of mass ; used in Lagrangian approach. • Control Volume : a region in space where fluid flows through it; often used in Eulerian approach; fluid inside c.v. does not remain to be the same fluid. Control volume at time t. Control volume at t+ ∆ t System at t. System at time t+ ∆ t Control volume and system . • We will use Control Volume : to derive basic laws for fluid moion. 4.1. Conservation of Mass v n S u Fig. 4.1 Control volume v for a portion of fluid v n S u Fig. 4.1 Control volume v for a portion of fluid • Total mass of the fluid in v is M = dv v ρ ∫∫∫ Rate of increase of mass = dv v ρ ∫∫∫ . Total mass of the fluid flowing out of v through S per unit time = u nds S ρ ⋅ ∫∫ Ò . • No sources of mass, the conservation of mass requires dv v ρ ∫∫∫ + u nds S ρ ⋅ ∫∫ Ò = 0 (4.1) • Divergence theorem, • Total mass of the fluid in v0 is dv v ρ ∫∫∫ Rate of increase of mass = dv v ρ ∫∫∫ Total mass of the fluid flowing out of v0 through S0 per unit time = u nds S ρ ⋅ ∫∫ Ò • Divergence theorem u nds S ρ ⋅ ∫∫ Ò ⇒ ( ) u dv v ρ ∇ ⋅ ∫∫∫ (4.1) ⇒ [ ( )] u dv v t ρ ρ ∂ + ∇ ⋅ ∂ ∫∫∫ = 0 ⇒ ( ) t u ρ ρ + ∇ ⋅ ∂ = ∂ (4.2) • Since ( ) u u u ρ ρ ρ ⋅ ∇ ∇ ⋅ = + ∇ ⋅ (4.2) ⇒ u u t ρ ρ ρ ∂ + ⋅ ∇ + ∇ ⋅ = ∂ Or D Dt u ρ ρ + ∇ ⋅ = (4.3) • If the flow is incompressible , ρ does not change along its pathline, i.e. D Dt ρ = . (def. of incompressibility) Then (4.3) gives u ∇ ⋅ = . (4.4) (result of incompressibility + continuity eqn.) 4.2 Momentum Equation • Newton's second law: total rate of change of the momentum = the resultant force applied. Considering the fluid in the control volume v 0. f (body force) σ n σ n u C.V. u Fig.4.2 Forces acting on the fluid in a c.v. u nds S ρ ⋅ ∫∫ Ò ⇒ ( ) u dv v ρ ∇ ⋅ ∫∫∫ a. The total fluid momentum in v is udv v ρ ∫∫∫ . b. The total momentum flowing out of v through S per unit time is ( ) u u n ds S ρ ⋅ ∫∫ Ò Hence the net rate of change of the fluid momentum in v is udv v ρ ∫∫∫ + ( ) u u n ds S ρ ⋅ ∫∫ Ò There are both body force and surface force on the fluid. c. The body force is fdv v ρ ∫∫∫ in which f is the force per unit mass. d. The surfaces force is n ds S σ ⋅ ∫∫ Ò based on the discussion in § 2.2. • Thus Newton's 2 nd law results in udv v ρ ∫∫∫ + ( ) u u n ds S ρ ⋅ ∫∫ Ò = fdv v ρ ∫∫∫ + n ds S σ ⋅ ∫∫ Ò (4.5) • Divergence theorem ( ) n d u u s S ρ ⋅ ∫∫ Ò ⇒ ( ) u dv u v ρ ∇ ⋅ ∫∫∫ (you can verify it component by component for u ) n ds S σ ⋅ ∫∫ Ò ⇒ ( ) dv...
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This note was uploaded on 09/09/2010 for the course EGM 6812 taught by Professor Renweimei during the Fall '09 term at University of Florida.

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Chapter%204%20Basic%20Eqns - IV. Basic Equations of Fluid...

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