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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

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
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
tensor comment
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
examples of tensor properties
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
comment on notation
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
comments from section 2 4 2
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
formulating the deformation law
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
formulating the deformation law1
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
n s equation for incompressible constant m flow
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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