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Unformatted text preview: ENU 4133 – Chapter 4, Part 1 February 1, 2010 Chapter 4 – Differential Relations for Fluid Flow In scope of Exam #1 and/or HW #4 and #5 I 4.1 – acceleration field of a fluid I 4.2 – differential mass conservation (continuity) I 4.3 – differential momentum conservation I 4.3, 4.6, 4.10 – NavierStokes equation and solutions I 4.5 – differential energy equation Additional Chapter 4 material will be covered for Exam #2. Acceleration Field of a Fluid General velocity field (Cartesian coordinates – Eulerian/fixed frame): ~ V ( ~ r , t ) = ~ iu ( ~ r , t ) + ~ jv ( ~ r , t ) + ~ kw ( ~ r , t ) (1) Acceleration field – differentiate w/r/t t . ~ a = d ~ V dt = ~ i du dt + ~ j dv dt + ~ k dw dt (2) Note: total time derivatives, not partial derivative. Considering u / x component: du dt = ∂ u ∂ t + ∂ u ∂ x dx dt + ∂ u ∂ y dy dt + ∂ u ∂ z dz dt (3) du dt = ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z (4) du dt = ∂ u ∂ t + ~ V · ∇ u (5) Acceleration Field of a Fluid (2) du dt = ∂ u ∂ t + ~ V · ∇ u (6) dv dt = ∂ v ∂ t + ~ V · ∇ v (7) dw dt = ∂ w ∂ t + ~ V · ∇ w (8) ~ a = d ~ V dt = ∂ ~ V ∂ t + ~ V · ∇ ~ V (9) Acceleration composed of local and convective parts. In general, this is termed the substantial or material derivative, e.g. , of B (can be ~ B or ~ ~ B ): DB Dt = ∂ B ∂ t + ~ V · ∇ B (10) Differential Mass Conservation Equation I Take control volume as differential volume dx × dy × dz = dV I Identify generation and destruction terms (hint: very easy) I Identify inflow and outflow terms I Compile terms & do a little math...
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 Spring '11
 Schubring
 Fluid Dynamics, Momentum, ∂t, ∂z

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