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Unformatted text preview: Fluid Dynamics IB Dr Natalia Berloff § 1.3 Material derivative Consider a field F ( x , t ). Rate of change with time seen by an observer moving with fluid, DF Dt , is found by using the chain rule for differentiation (Remember: the value of x for a given fluid particle varies with time) Material derivative: DF Dt ≡ ∂F ∂t + u · ∇ F Other names: Lagrangian, total, convected, or substantial derivative. § 1.4 Conservation of Mass We assume that fluid is neither created nor destroyed , i.e., that fluid mass is con- served. Look at what this means mathematically, for a general flow field. Consider an arbitrary finite volume V fixed in space, bounded by surface S , with outw’d normal n . The total mass of fluid contained within V : Mass inside changes because it flows accross the boundary surface. Volume out (per unit of time) Mass out of V through S (per unit of time): This must be equal to the rate of change of the total mass occupying V , R V ρdV , implying that d dt Z V ρdV =- Z S ρ u · n dS . Now the volume V is fixed in space, so Use divergence theorem: Equation holds for arbitrary V , hence, assuming that integrands are continuous functions, ∂ρ ∂t + ∇ · ( ρ u ) = 0 , (1) the equation of mass conservation ....
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This note was uploaded on 10/01/2011 for the course ME 509 taught by Professor Wereley during the Spring '11 term at Purdue.
- Spring '11
- Fluid Dynamics