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

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
• 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
• 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
• 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
• 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
• 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
• 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 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