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This article summarizes equations in the theory of fluid mechanics. Definitions. Flux F through a surface, dS is the differential vector area element, ...
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them. [ 1 ] : 3 It has applications in a wide range of disciplines, including mechanical , aerospace , civil , chemical , and biomedical engineering , as well as geophysics , oceanography , meteorology , astrophysics ...
The Navier–Stokes equations assume that the fluid being studied is a continuum (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at relativistic velocities. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the ...
The vorticity equation of fluid dynamics describes the evolution of the vorticity ω of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity). The governing equation is:
The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum – a continuous substance rather than discrete particles. Another necessary assumption is that all the fields of interest including pressure , flow velocity , density , and temperature are at least weakly differentiable .
For constant fluid density, the incompressible equations can be written as a quasilinear advection equation for the fluid velocity together with an elliptic Poisson's equation for the pressure. On the other hand, the compressible Euler equations form a quasilinear hyperbolic system of conservation equations .
The equation of motion for Stokes flow can be obtained by linearizing the steady state Navier–Stokes equations.The inertial forces are assumed to be negligible in comparison to the viscous forces, and eliminating the inertial terms of the momentum balance in the Navier–Stokes equations reduces it to the momentum balance in the Stokes equations: [1]
In order to derive Torricelli's formula the first point with no index is taken at the liquid's surface, and the second just outside the opening. Since the liquid is assumed to be incompressible, ρ 1 {\displaystyle \rho _{1}} is equal to ρ 2 {\displaystyle \rho _{2}} and; both can be represented by one symbol ρ {\displaystyle \rho } .