# Fluid mechanics: Some elementary concepts

#### arildno

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In this thread, I will explore some elementary concepts in fluid mechanics, starting with the distinction between a particle description and a field description (Lagrangian vs. Eulerian formalism).
I will continue by distinguishing between the concepts of particle trajectories, streamlines, material curves and streak lines.
Furthermore, I'll look into the basic derivations we may make with these concepts in inviscid flow, namely Bernoulli's equation, Crocco's theorem, Kelvin's theorem, Helmholtz' theorem for vortex lines,and the helicity theorem.
Lastly, I'll look into the assumptions behind irrotational, i.e, potential flow, the generalized potential flow known as the Clebsch flow, and the elegant principle known as Luke's principle of variation.

Quite a bit more to follow in a while..

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#### Divisionbyzer0

I am interested seeing some of this stuff if you're still interested in going through it.

#### Clausius2

Gold Member
arildno said:
In this thread, I will explore some elementary concepts in fluid mechanics, starting with the distinction between a particle description and a field description (Lagrangian vs. Eulerian formalism).
I will continue by distinguishing between the concepts of particle trajectories, streamlines, material curves and streak lines.
Furthermore, I'll look into the basic derivations we may make with these concepts in inviscid flow, namely Bernoulli's equation, Crocco's theorem, Kelvin's theorem, Helmholtz' theorem for vortex lines,and the helicity theorem.
Lastly, I'll look into the assumptions behind irrotational, i.e, potential flow, the generalized potential flow known as the Clebsch flow, and the elegant principle known as Luke's principle of variation.
Quite a bit more to follow in a while..

If you finally do this stuff, honestly I think your point of view should be a more "engineering oriented" one. I have been studying fluid mechanics during 3 years when undergrad and now doing the Ph.D. in aerospace engineering, and I have never heard about Luke's principle nor Clebsch flow. I'm pretty sure they exist and they are important, but I think they are not the kind of stuff you should include in a tutorial for giving the big picture of this science.

To my understanding a better guideline could be something like:

i) Fluid Kinematics: Eulerian description, Pathlines, Streamlines, Streaklines.
ii) Integral Balance Equations: Continuity, Momentum, Total Energy, Kinetic Energy, Enthalpy and Entropy. Incompressibility assumption. Vorticity Equation.
iii) Differential forms. Reynolds Transport Theorem.
iv) Non dimensional Relevant Parameters.
v) Flow at low Reynolds. Poiseuille and Couette Flows. Stokes Equations. Oseen's approximation. Hydrodynamic Lubrication.
vi) Flow at high Reynolds. Boundary Layer. Blausius Similarity Solution. Bernoulli Equation.
vii) Compressible Flow. Rankine-Hugoniot equations. Shock Waves and Expansion Fans.
viii) Classical Hydrodynamics. Surface Gravity Waves. Shallow Water equations. Flow over obstacles.
viii) Stratified Flow. Bousinessq approximation. Buoyancy frequency. Internal Waves.
ix) Turbulent Flow. RANS equations. Reynolds stress tensor. Auschtag Viscosity coefficient. Closure problem. Boussinesq approximation. Round Turbulent Jet.

Last edited:

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#### bilgealp

... I have never heard about Luke's principle nor Clebsch flow. I'm pretty sure they exist and they are important, but I think they are not the kind of stuff you should include in a tutorial for giving the big picture of this science.
Luke's variational principle is quite elegant and worth to mention especially when potential theory is on the scene. Even, variational calculus itself is a must for anyone interested in fluid mechanics. Finite element people are quite good at dealing with the variational form of the partial differential equations.

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