1. 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. 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,  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,  g Now ready to develop detailed expressions for surface forces (and how they related to strain, which are related to velocity derivatives)

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

4. 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. 5 SYMMETRIC STRESS TENSOR Stress tensor –Viscous Flows, 3 rd Edition, by F. White Stress tensor –Fluid Mechanics, 3 rd Edition, by F. White Recall  ij –i: Stress on a face normal to i axis –j: Stress acting in j direction

6. 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 (  xx,  xy,  xz ), 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  ij in terms of velocity –Assume viscous deformation-rate law between  ij and  ij

7. 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. 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 =  gh = (1025 kg/m 3 )(9.81 m/s 2 )(732 m) = 7.36x10 6 Pa = 73 atmospheres HY-100 a a yield stress of 100,000 pounds per square inch

9. 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 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

10. 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. 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 v i (i=1,2,3) are scalar components of a vector if they transform according to: A tensor (2 nd rank) is defined as a collection of 9 scalar components that change under rotation of axes according to: