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The Navier–Stokes equations are nonlinear partial differential equations in the general case and so remain in almost every real situation. [23] [24] In some cases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear equations.
In addition to the mathematical challenges of solving the Navier–Stokes equations, there are also many practical challenges in applying these equations to real-world situations. For example, the Navier–Stokes equations are often used to model fluid flows that are turbulent, which means that the fluid is highly chaotic and unpredictable.
The Navier–Stokes equations govern the velocity and pressure of a fluid flow. In a turbulent flow, each of these quantities may be decomposed into a mean part and a fluctuating part. Averaging the equations gives the Reynolds-averaged Navier–Stokes (RANS) equations, which govern the mean flow.
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 .
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 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 ...
Navier–Stokes equation and the continuity equation [ edit ] In order to analytically find the stability of fluid flows, it is useful to note that hydrodynamic stability has a lot in common with stability in other fields, such as magnetohydrodynamics , plasma physics and elasticity ; although the physics is different in each case, the ...
In 1845, George Gabriel Stokes published another important set of equations, today known as the Navier-Stokes equations. [1] [11] Claude-Louis Navier developed the equations first using molecular theory, which was further confirmed by Stokes using continuum theory. [1] The Navier-Stokes equations describe the motion of fluids: [1]