Chapter05

Mechanics of Fluids

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89 CHAPTER 5 The Differential Forms of the Fundamental Laws 5.1 0 = - + ∂ρ ρ t d V V ndA c v c s . . . . $ . v Using Gauss’ theorem: 0 = - + ∇ ⋅ - = + ∇ ⋅ - ∂ρ ρ ∂ρ ρ t d V V d V t V d V c v c v c v . . . . . . ( ) ( ) . v v v v Since this is true for all arbitrary control volumes (i.e., for all limits of integration), the integrand must be zero: ∂ρ ρ t V + ∇ ⋅ = v v ( ) . 0 This can be written in rectangular coordinates as - = + + ∂ρ ρ ρ ρ t x u y v z w ( ) ( ) ( ). This is Eq. 5.2.2. The other forms of the continuity equation follow. 5.2 & & . m m m t in out element - = ( 29 ρ θ ρ ρ θ v rd dz v r v dr r dr d dz r r r ( ) ( ) - + + + - + ρ ρ ∂θ ρ θ θ θ θ v drdz v v d drdz ( ) + + - + + = + ρ θ ρ ρ θ ρ θ v r dr d dr v z v dz r dr d dr t r dr d drdz z z z 2 2 2 ( ) . Subtract terms and divide by rd drdz q : - - + - - + = + ρ ρ ∂θ ρ ρ ρ θ v r r v r dr r v r z v r dr r t r dr r r r z ( ) ( ) ( ) / / . 1 2 2 Since dr is an infinitesimal, ( )/ ( / )/ . r dr r r dr r + = + = 1 2 1 and Hence, ∂ρ ρ ∂θ ρ ρ ρ θ t r v r v z v r v r z r + + + + = ( ) ( ) ( ) . 1 1 0 This can be put in various forms.
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