Mae 5130 viscous flows
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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

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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


    The navier stokes equations

    THE NAVIER-STOKES EQUATIONS


    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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