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MAE 5130: VISCOUS FLOWS

MAE 5130: VISCOUS FLOWS. Momentum Equation: The Navier-Stokes Equations, Part 2 September 9, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk. GOAL: INCOMPRESSIBLE, CONSTNAT m N/S EQUATION. Start with Newton’s 2 nd Law for a fixed mass

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MAE 5130: VISCOUS FLOWS

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  1. MAE 5130: VISCOUS FLOWS Momentum Equation: The Navier-Stokes Equations, Part 2 September 9, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

  2. GOAL: INCOMPRESSIBLE, CONSTNAT m N/S EQUATION • Start with Newton’s 2nd Law for a fixed mass • Divide by volume • Introduce acceleration in Eulerian terms • Ignore external forces • Only body force considered is gravity • Express all surface forces that can act on an element • 3 on each surface (1 normal, 2 perpendicular) • Results in a tensor with 9 components • Due to moment equilibrium only 6 components are independent • Employ Stokes’ postulates to develop a general deformation law between stress and strain rate • White Equation 2-29a and 2-29b • Assume incompressible flow and constant viscosity

  3. TENSOR COMMENT • Tensors are often displayed as a matrix • The transpose of a tensor is obtained by interchanging the two indicies • Transpose of Tij is Tji • Tensor Qij is symmetric if Qij = Qji • Tensor is antisymmetric if it is equal to the negative of its transpose, Rij = -Rji • Any arbitrary tensor Tij may be decomposed into sum of a symmetric tensor and antisymmetric tensor

  4. EXAMPLES OF TENSOR PROPERTIES • Although component magnitudes vary with change of axes x, y, and z, the stress and strain-rate tensor follow the transformation laws of symmetric tensors • 3 invariants are particularly useful • I3 is the determinant • Another property of symmetric tensors is that there exists one and only one set of axes for which the off-diagonal terms (the shear-strain rates in this example) vanish. • These are called the principal axes • Invariants for principal axes

  5. COMMENT ON NOTATION • Recall that in White’s nomenclature: • x1, y1, and z1 are principal axes • x, y, and z are arbitrary axes • With respect to principal axes • x-axis has directional cosines: l1, m1, and n1 • y-axis has directional cosines: l2, m2, and n2 • z-axis has directional cosines: l3, m3, and n3 • Using tensor transformation from principal to arbitrary axes we arrived at general expressions for diagonal and off-diagonal terms for shear stress and strain in arbitrary orientation

  6. COMMENTS FROM SECTION 2-4.2 • Simplest assumption for variation between viscous stress and strain rate is a linear law • Satisfied for all gases and most common liquids Stokes’ 3 postulates • Fluid is continuous, and its stress tensor tij is at most a linear function of strain rates eij • Fluid is isotropic • Properties are independent of directions (no preferred direction) • Deformation law is independent of coordinate system choice • Also implies that principal stress axes be identical with principal strain-rate axes • When strain rates are zero (for example if fluid is at rest, V=0), deformation law must reduce to hydrostatic pressure condition, tij = -pdij • Begin derivation of deformation law with element aligned with principal axes • White notation for principal axes: x1, y1, z1 • Axes where shear stresses and shear strain rates are zero

  7. FORMULATING THE DEFORMATION LAW • Using the principal axes the deformation law could involve 3 linear coefficients • Isotropic condition requires that e22 = e33 (cross-flow terms) be equal • -p is added to satisfy hydrostatic condition • Re-write with gradient of velocity • Try to write t22 and t33 terms

  8. FORMULATING THE DEFORMATION LAW • Examples of general deformation law • Comparing with shear flow between parallel plates • Often called the ‘second coefficient of viscosity’ or coefficient of bulk viscosity or Lamé’s constant (linear elasticity) • Only associated with volume expansion through divergence of velocity field • Now substitute into Newton’s 2nd Law • Note that shear stresses are expressed as velocity derivatives as desired

  9. THE NAVIER-STOKES EQUATIONS

  10. N/S EQUATION FOR INCOMPRESSIBLE, CONSTANT m FLOW • Start with Newton’s 2nd Law for a fixed mass • Divide by volume • Introduce acceleration in Eulerian terms • Ignore external forces • Only body force considered is gravity • Express all surface forces that can act on an element • 3 on each surface (1 normal, 2 perpendicular) • Results in a tensor with 9 components • Due to moment equilibrium (no angular rotation of element) 6 components are independent) • Employ a Stokes’ postulates to develop a general deformation law between stress and strain rate • White Equation 2-29a and 2-29b • Assume incompressible flow and constant viscosity

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