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## Vectors

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**Vectors**n v • What is the projection of the vector (1, 3, 2) onto the plane described by ? Louisiana Tech University Ruston, LA 71272**Cross Product**In Cartesian Coordiantes: • Used for: • Moments • Vorticity Louisiana Tech University Ruston, LA 71272**Forces**• Body Forces • Pressure • Normal Stresses • Shear Stresses Louisiana Tech University Ruston, LA 71272**What is a Fluid?**Solid: Stress is proportional to strain (like a spring). Fluid: Stress is proportional to strain rate. Louisiana Tech University Ruston, LA 71272**The Stress Tensor (Fluids)**One dimensional For fluids: Three-Dimensional, where u is velocity Louisiana Tech University Ruston, LA 71272**The Stress Tensor (Solids)**One dimensional For solids: Three-Dimensional, where u is displacement Louisiana Tech University Ruston, LA 71272**Non-Newtonian Fluid**(One dimensional) For fluids: m is a function of strain rate Louisiana Tech University Ruston, LA 71272**Apparent Viscosity of Blood**Non-Newtonian Region Rouleau Formation meff 3.5 g/(cm-s) 1 dyne/cm2 Louisiana Tech University Ruston, LA 71272**The Sturm Liouville Problem**Be able to reduce the Sturm Liouville problem to special cases. Or equivalently: Louisiana Tech University Ruston, LA 71272**Orthogonality**From Bessel’s equation, we have w(x)= x, and the derivative is zero at x = 0, so it follows immediately that: Provided that lm and ln are values of l for which the Bessel function is zero at x = 1. Louisiana Tech University Ruston, LA 71272**Lagrangian vs. Eulerian**Consider the flow configuration below: • The velocity at the left must be smaller than the velocity in the middle. • What is the relationship? • If the flow is steady, is v(t) at any point in the flow a function of time? Louisiana Tech University Ruston, LA 71272**Lagrangian vs. Eulerian**Conceptually, how are these two viewpoints different? Why can’t you just use dv/dt to get acceleration in an Eulerian reference frame? Give an example of an Eulerian measurement. Be able to describe both viewpoints mathematically. Louisiana Tech University Ruston, LA 71272**Exercise**Consider the flow configuration below: Assume that along the red line: a. What is the velocity at x = -1 and x = 0? b. Does fluid need to accelerate as it goes to x = 0? c. How would you calculate the acceleration at x = -0.5? d. How would you calculate acceleration for the more general case v = f(x)? e. Can you say that acceleration is a = dv/dt? Louisiana Tech University Ruston, LA 71272**Streamlines, Streaklines, Pathlines and Flowlines**Be able to draw each of these for a given (simple) flow. Understand why they are different when flow is unsteady. Be able to provide an example in which they are different. If shown a picture of lines, be able to say which type of line it is. Be able to write down a differential equation for each type of line. So: Louisiana Tech University Ruston, LA 71272**Flow Lines**Louisiana Tech University Ruston, LA 71272**Flow Lines**Louisiana Tech University Ruston, LA 71272**Conservation Laws: Mathematically**All three conservation laws can be expressed mathematically as follows: Production of the entity (e.g. mass, momentum, energy) Increase of “entity per unit volume” Flux of “entity per unit volume” out of the surface of the volume (n is the outward normal) is some entity. It could be mass, energy or momentum. is some property per unit volume. It could be density, or specific energy, or momentum per unit volume. Louisiana Tech University Ruston, LA 71272**Reynolds Transport Theorem: Momentum**If we are concerned with the entity “mass,” then the “property” is mass per unit volume, i.e. density. Increase of momentum within the volume. Production of momentum within the volume Flux of momentum through the surface of the volume Momentum can be produced by: External Forces. Louisiana Tech University Ruston, LA 71272**Mass Conservation in an Alveolus**Density remains constant, but mass increases because the control volume (the alveolus) increases in size. Thus, the limits of the integration change with time. Term 1: There is no production of mass. Term 2: Density is constant, but the control volume is growing in time, so this term is positive. Term 3: Flow of air is into the alveolus at the inlet, so this term is negative and cancels Term 2. Control Volume (CV) Control Surface CS Louisiana Tech University Ruston, LA 71272**Vector and Tensor Analysis**• In the material derivative in Gibbs notation, we introduced some new mathematical operators • Gradient • In Cartesian, cylindrical, and spherical coordinates respectively: What is this operation? Louisiana Tech University Ruston, LA 71272**Coordinate Systems**Cartesian Cylindrical Spherical What is an area element in each system? What is a volume element in each system? Louisiana Tech University Ruston, LA 71272**Momentum Equation**Navier-Stokes General Form (varying density and viscosity) Louisiana Tech University Ruston, LA 71272**Momentum Equation**• Be able to translate each vector term into its components. E.g. • Be able to state which terms are zero, given symmetry conditions. E.g. what does “no velocity in the radial direction” mean mathematically. What does “no changes in velocity with respect to the q direction” mean mathematically? Louisiana Tech University Ruston, LA 71272**Continuity Equation**Compressible Incompressible Louisiana Tech University Ruston, LA 71272**Last Word of Advice**• Always check your units. Louisiana Tech University Ruston, LA 71272