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Expressing the Navier–Stokes vector equation in Cartesian coordinates is quite straightforward and not much influenced by the number of dimensions of the euclidean space employed, and this is the case also for the first-order terms (like the variation and convection ones) also in non-cartesian orthogonal coordinate systems.
Oseen looked for a solution for the Navier–Stokes equations in cylindrical coordinates (,,) with velocity components (,,) of the form =, = (,), = where is the circulation of the vortex core.
The cross differentiated Navier–Stokes equation becomes two 0 = 0 equations and one meaningful equation. The remaining component ψ 3 = ψ is called the stream function. The equation for ψ can simplify since a variety of quantities will now equal zero, for example:
Projected streamlines of the Sullivan vortex on the axial -plane; is the origin.. In fluid dynamics, the Sullivan vortex is an exact solution of the Navier–Stokes equations describing a two-celled vortex in an axially strained flow, that was discovered by Roger D. Sullivan in 1959.
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 ...
The Burgers vortex is a stable solution of the Navier–Stokes equations. [4] One of the important property of the Burgers vortex that was shown by Jan Burgers is that the total viscous dissipation rate per unit axial length is independent of the viscosity, indicating that dissipation by the Burgers vortex is non-zero even in the limit . For ...
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]
The Batchelor vortex is an approximate solution to the Navier–Stokes equations obtained using a boundary layer approximation. The physical reasoning behind this approximation is the assumption that the axial gradient of the flow field of interest is of much smaller magnitude than the radial gradient.