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# Hydrodynamics

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• 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 ($!\vec{f} = \rho \vec{g}$ 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: $$F = 6\pi\eta r v.$$