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MAE 5130: VISCOUS FLOWS

MAE 5130: VISCOUS FLOWS. Momentum Equation: The Navier-Stokes Equations, Part 1 September 7, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk. DEVELOPMENT OF N/S EQUATIONS: ACCELERATION. Momentum equation, Newton’s second law

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MAE 5130: VISCOUS FLOWS

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  1. MAE 5130: VISCOUS FLOWS Momentum Equation: The Navier-Stokes Equations, Part 1 September 7, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

  2. DEVELOPMENT OF N/S EQUATIONS: ACCELERATION • Momentum equation, Newton’s second law • System is fluid particle so convenient to divide by volume, V, of particle so work with density, r • Concerned with: • Body forces • Gravity • Applied electromagnetic potential • Surface forces • Friction (shear, drag) • Pressure • External forces • Eulerian description of acceleration • Substitution in to momentum • Forces are per unit volume • Recall that body forces apply to entire mass of fluid element • Gravitational body force, rg • Now ready to develop detailed expressions for surface forces (and how they related to strain, which are related to velocity derivatives)

  3. SURFACE FORCES • Surface forces are those applied by external stresses on the side of the element • Quantity tij is a tensor (just as strain rate eij) • Pay attention to sign convention for stress components Stress on a face normal to i axis Stress acting in j direction

  4. FORCES ON FRONT FACES • Force in x-direction due to stress • Force in y-direction due to stress • Force in z-direction due to stress • Stress can also be written as symmetric tensor • Written in this way to keep analogy with strain rate tensor, eij, because stress tensor is also symmetric • Symmetry is required to satisfy equilibrium of moments about three axes of element • Rows of tensor correspond to applied force in each coordinate direction

  5. SYMMETRIC STRESS TENSOR • Stress tensor • Viscous Flows, 3rd Edition, by F. White • Stress tensor • Fluid Mechanics, 3rd Edition, by F. White • Recall tij • i: Stress on a face normal to i axis • j: Stress acting in j direction

  6. EXPRESSION OF STRESS FORCES • If element in equilibrium, this forces balanced by equal and opposite force on back face of element • If accelerating, front and back face stresses will be different by differential amounts • Net force in the x-direction • Compare this with conservation of mass derivation • Put force on per unit volume basis (divide by dxdydz) • Force per unit volume in x-direction is equivalent to taking the divergence of the vector (txx, txy, txz), which is the upper row of the stress tensor (shown in previous slide) • Total vector surface force • Divergence of a tensor is a vector • Newton’s second law • All that remains is to express tij in terms of velocity • Assume viscous deformation-rate law between tij and eij

  7. FLUID AT REST: HYDROSTATICS • Newton’s second law of motion • Fluid at rest • Velocity = 0 • Viscous shear stresses = 0 • Normal stresses become equal to the hydrostatic pressure

  8. HYDROSTATICS EXAMPLE • Depths to which submarines can dive are limited by the strengths of their hulls • Collapse depth, popularly called crush depth, is submerged depth at which a submarine's hull will collapse due to surrounding water pressure • Seawolf class submarines estimated to have a collapse depth of 2400 feet (732 m), what is pressure at this depth? • P = rgh = (1025 kg/m3)(9.81 m/s2)(732 m) = 7.36x106 Pa = 73 atmospheres • HY-100 a a yield stress of 100,000 pounds per square inch

  9. TENSOR COMMENT • Tensors are often displayed as a matrix • The transpose of a tensor is obtained by interchanging the two indicies, so the 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

  10. INDEX NOTATION RULES AND COORDINATE ROTATION • Key to classifying scalars, vectors, or tensors is how their components change if the coordinate axes are rotated to point in new directions • A scalar (temperature, density, etc.) is unchanged by rotation – it has the same value in any coordinate system, which is a defining characteristic of a scalar • Vectors and tensors change with rotating coordinate system

  11. INDEX NOTATION, VECTORS, AND TENSORS • Index notation • A free index occurs once and only once in each and every term in an equation • A dummy or summation index occurs twice in a term • Vector has a magnitude and direction that is measured with respect to a chosen coordinate system • Alternative description is to give three scalar components • Not every set of 3 scalar components is a vector • Essential extra property of a vector is its transformation properties as coordinate system is rotated • 3 scalar quantities vi(i=1,2,3) are scalar components of a vector if they transform according to: • A tensor (2nd rank) is defined as a collection of 9 scalar components that change under rotation of axes according to:

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