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This chapter contains many important topics about perfect and viscous fluids. First, we formulate the equations of statics of fluids stating Stevino’s and Archimedes’ laws. Then, on dynamics of perfect fluids, we prove Kelvin’s, Lagrange’s and Bernoulli’s theorems with some applications. After defining the vortex tubes, we formulate Helmotz’s theorems. There follows an analysis of plane flows by complex potentials. After showing important examples of complex potentials (vortex, source, doublet, etc.), we prove D’Alembert’s paradox and Blausius’ formula. The analysis of Kutta–Joukowski’s mapping leads to the determination of wing contour. After proving D’Alembert’s paradox, we are compelled to introduce the viscosity by Newtonian fluids. After a dimensionless formulation of Navier–Stokes equations, we show that viscous flows lead us to boundary layers. The chapter contains an extensive analysis of these narrow regions by Prandtl’s approach. Then, we apply the general theory of ordinary waves to perfect fluids in order to prove the existence of longitudinal waves in compressible fluids. Finally, we apply the Rankine–Hugoniot conditions to analyze the behavior of shock waves in perfect fluids. After sections containing exercises, we state the programs, Potential, Wing and Joukowski written with MathematicaⓇ .
KeywordsComplex Potentials Perfect Fluid Stepka Wing Program Complex Velocity
- S. Wolfram, Mathematica ®;. A System for Doing Mathematics by Computer (Addison-Wesley Redwood City, California, 1991)Google Scholar