1. 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. 2 GOAL: INCOMPRESSIBLE, CONSTNAT  N/S EQUATION Start with Newton’s 2 nd 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. 3 TENSOR COMMENT Tensors are often displayed as a matrix The transpose of a tensor is obtained by interchanging the two indicies –Transpose of T ij is T ji Tensor Q ij is symmetric if Q ij = Q ji Tensor is antisymmetric if it is equal to the negative of its transpose, R ij = -R ji Any arbitrary tensor T ij may be decomposed into sum of a symmetric tensor and antisymmetric tensor

4. 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 I 3 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. 5 COMMENT ON NOTATION Recall that in White’s nomenclature: –x 1, y 1, and z 1 are principal axes –x, y, and z are arbitrary axes With respect to principal axes x-axis has directional cosines: l 1, m 1, and n 1 y-axis has directional cosines: l 2, m 2, and n 2 z-axis has directional cosines: l 3, m 3, and n 3 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. 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 1.Fluid is continuous, and its stress tensor  ij is at most a linear function of strain rates  ij 2.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 3.When strain rates are zero (for example if fluid is at rest, V=0), deformation law must reduce to hydrostatic pressure condition,  ij = -p  ij Begin derivation of deformation law with element aligned with principal axes –White notation for principal axes: x 1, y 1, z 1 –Axes where shear stresses and shear strain rates are zero

7. 7 FORMULATING THE DEFORMATION LAW Using the principal axes the deformation law could involve 3 linear coefficients Isotropic condition requires that  22 =  33 (cross-flow terms) be equal -p is added to satisfy hydrostatic condition Re-write with gradient of velocity Try to write  22 and  33 terms

8. 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 2 nd Law Note that shear stresses are expressed as velocity derivatives as desired

9. 9 THE NAVIER-STOKES EQUATIONS

10. 10 N/S EQUATION FOR INCOMPRESSIBLE, CONSTANT  FLOW Start with Newton’s 2 nd 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