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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

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 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

• 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

• ### 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

• 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

• 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