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The incompressible Navier–Stokes equations with uniform density and viscosity and conservative external field is the fundamental equation of hydraulics. The domain for these equations is commonly a 3 or fewer dimensional Euclidean space , for which an orthogonal coordinate reference frame is usually set to explicit the system of scalar ...
This equation is called the mass continuity equation, or simply the continuity equation. This equation generally accompanies the Navier–Stokes equation. In the case of an incompressible fluid, Dρ / Dt = 0 (the density following the path of a fluid element is constant) and the equation reduces to:
The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case. [ 2 ] [ 4 ] [ 9 ] [ 10 ] They are the leading-order simplification of the full Navier–Stokes equations, valid in the distinguished limit R e → 0. {\displaystyle \mathrm {Re} \to 0.}
In computational fluid dynamics, the projection method, also called Chorin's projection method, is an effective means of numerically solving time-dependent incompressible fluid-flow problems. It was originally introduced by Alexandre Chorin in 1967 [1] [2] as an efficient means of solving the incompressible Navier-Stokes equations.
In fluid mechanics, non-dimensionalization of the Navier–Stokes equations is the conversion of the Navier–Stokes equation to a nondimensional form. This technique can ease the analysis of the problem at hand, and reduce the number of free parameters. Small or large sizes of certain dimensionless parameters indicate the importance of certain ...
Discretization of the Navier–Stokes equations of fluid dynamics is a reformulation of the equations in such a way that they can be applied to computational fluid dynamics. Several methods of discretization can be applied: Finite volume method; Finite elements method; Finite difference method
In mathematics, the Navier–Stokes equations are a system of nonlinear partial differential equations for abstract vector fields of any size. In physics and engineering, they are a system of equations that model the motion of liquids or non-rarefied gases (in which the mean free path is short enough so that it can be thought of as a continuum mean instead of a collection of particles) using ...
Using these properties, the Navier–Stokes equations of motion, expressed in tensor notation, are (for an incompressible Newtonian fluid): = + = + where is a vector representing external forces. Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged, [ b ] to yield: