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Hydrodynamics

  • The fundamental equation of hydrodynamics of ideal fluid (Eulerian equation):
    $$ \rho \frac{d\vec{v}}{dt} = \vec{f} - \vec{\nabla} p,$$

    where $ \rho $ is fluid density, $ \vec{f} $ is the volume density of mass forces ($\displaystyle \vec{f} = \rho \vec{g} $ in the case of gravity), $ \vec{\nabla} p $ is the pressure gradient.

  • Bernoulli's equation. In the steady flow of an ideal fluid
    $$\frac{\rho v^2}{2} + \rho g h + p = const$$

    along any streamline.

  • Reynolds number defining the flow pattern of a viscous fluid:
    $$Re = \frac{\rho v l}{\eta},$$

    where $ l $ is a characteristic length, $ \eta $ is the fluid viscosity.

  • Poiseuille's law. The volume of liquid flowing through a circular tube (in $ m^3/s $):
    $$Q = \frac{\pi R^4}{8\eta} \frac{p_1 - p_2}{l},$$

    where $ R $ and $ l $ are the tube's radius and length, $ p_1 - p_2 $ is the pressure difference between the ends of the tube.

  • Stokes' law. The friction force on the sphere of radius $ r $ moving through a viscous fluid:
    $$F = 6\pi\eta r v.$$

 

Keywords: 
hydrodynamics, ideal fluid, Euler, Eulerian equation, Bernoulli's equation, steady flow, Reynolds, Reynolds number, viscous, viscosity, Poiseuille, Poiseuille's law, Stokes, Stokes' law,
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