0. FM_Review_2010 - ENU 4134 Fluid Mechanics Review D....

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Unformatted text preview: ENU 4134 Fluid Mechanics Review D. Schubring September 3, 2010 Continuity t + ~ V = 0 (1) Steady, compressible flow t = 0 (2) ~ V = 0 (3) Incompressible flow t = 0 (4) ~ V = 0 (5) Momentum Equations f x- p x + xx x + yx y + zx z = u t + u u x + v u y + w u z f y- p y + xy x + yy y + zy z = v t + u v x + v v y + w v z f z- p z + xz x + yz y + zz z = w t + u w x + v w y + w w z ~ f- p + ~ ~ = d ~ V dt (6) Equations vs. Unknowns (Scalar) Equations: 3 from momentum, 1 from mass 4 total. (Scalar) Unknowns: 9 components of ~ ~ , 3 components of ~ V , 3 components of ~ f , p 16 total. Differential angular momentum equation shows that ij = ji , so: (Scalar) Unknowns: 6 independent components of ~ ~ , 3 components of ~ V , 3 components of ~ f , p 16 total. Need 9 (scalar) constitutive relations. Constitutive Relation #1 Force f , from gravity ~ f = ~ g (7) Gravity is predominant. This is a vector relation, so 3 down, 6 to go. (Enrichment) But there could be electromagnetic forces in some fluids, so that: ~ f = ~ g + q ~ E + ~ V ~ B (8) Newtonian Fluids For incompressible flow: ij = V i x j + V j x i (9) Equations 4.37 in the text enumerate all terms. This relation has 6 independent relations, so the system can be closed. Navier-Stokes Equation(s) ~ f- p + ~ ~ = d ~ V dt (10) ~ g- p + 2 ~ V = d ~ V dt (11) ~ g- p + 2 ~ V = D ~ V Dt (12) ~ g- p + 2 ~ V = " ~ V t + ~ V ~ V # (13) Simplify and Solve Navier-Stokes...
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0. FM_Review_2010 - ENU 4134 Fluid Mechanics Review D....

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