Turbulence under a Magnifying Glass
نویسنده
چکیده
where the vector v(t, x) describes the velocity of the fluid at time t and space point x, positive constant ν is the viscosity, ρ is the constant density, f(t, x) is the (intensive) force and p(t, x) is the pressure. The above equations date back to the works of Navier (1823) and Stokes (1843) and modify the even older Euler equation (1755) by the addition of the dissipative term ν∆v . In most physical applications the dimensionality of the space is 3 or 2, but the equations make sense in general dimension d. The Euler equation without the ν∆v term is really the F = ma (or rather a = 1 m F ) equality for the volume element of the fluid. The ν∆v term represents the friction forces. By taking the L scalar product of the NS equation with v and assuming that the flow velocity vanishes sufficiently fast at large distances (or at the boundary), one deduces the energy balance:
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