Handout_7 - Classical Dynamics and Fluids P 125 Classical...

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Classical Dynamics and Fluids P 125 F LOW OF I DEAL F LUIDS We shall look frst at the Fow o± an ideal Fuid, which is incompressible and has no viscosity a . This is sometimes a good model ±or the Fow o± liquids and gases. A±ter looking at the basic equations o± Fuid Fow, we will derive the important Bernoulli equation that relates the pressure to the Fow velocity. ²or the case o± irrotational fow , we will use velocity potential theory to fnd the Fow patterns around obstacles in the Fuid, and then use Bernoulli’s equation to fnd the pressure. We shall then look at the e±±ects o± viscosity and its e±±ects in simple laminar Fows when viscosity dominates. In real Fows a very important consideration is the ratio the inertial stresses to the viscous stresses. This Reynolds number determines whether Fows are laminar or turbulent. We will then look very brieFy at the qualitative e±±ects o± compressibility on Fuid Fow (e.g. shock ±ronts). a Feynman calls this the “fow o± dry water”, in contrast to the “fow o± wet water” when viscosity is included Classical Dynamics and Fluids P 126 F LUID DYNAMICS When the mean ±ree path λ o± particles in a liquid, gas or plasma is small compared to the scales o± interest, we can treat the system as a hydrodynamical fuid . In an ideal fuid di±±erences in normal stresses decay so ±ast that the only stresses present can be an isotropic pressure τ 1 = τ 2 = τ 3 = - P . A Fuid is composed o± fuid elements , which are regions bigger than λ , large enough to have well-defned values o± macroscopic properties. ²or compressible Fuids these properties are the density ρ ( x ,t ) , the velocity v ( x ,t ) ; the pressure P ( x ,t ) ; the energy density u ( x ,t ) and anything else that is relevant, e.g. magnetic feld. A Fuid element is acted on by gravity (and other body ±orces), by pressure ±orces and other stresses on the sur±ace. The Fuid elements stay more or less well-defned (on scales ± λ ), but the sur±ace o± an element is moving at the local Fuid velocity, so elements distort as they move around. The distortions can become very large — in the presence o± turbulent motions the Fuid elements can be dispersed There are still some microscopic physical processes occurring on scales o± λ that determine important properties o± a Fuid: heat conduction; viscosity . The macroscopic properties can also change discontinuously on scale o± λ at shock Fronts .
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Classical Dynamics and Fluids P 127 C ONSERVATION OF M ASS We shall derive the equations of compressible Fuid dynamics and restrict them to the incompressible case afterwards.
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This note was uploaded on 05/09/2009 for the course DAMTP NST 1B Phy taught by Professor Sfgull during the Spring '07 term at Cambridge.

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Handout_7 - Classical Dynamics and Fluids P 125 Classical...

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