Navier-Stokes Equations

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Navier-Stokes equations

[nä′vyā ′stōks i‚kwā·zhənz]
(fluid mechanics)
The equations of motion for a viscous fluid which may be written d V/ dt = -(1/ρ)∇ p + F + ν∇2V + (⅓)ν∇(∇·V), where p is the pressure, ρ the density, F the total external force per unit mass, V the fluid velocity, and ν the kinematic viscosity; for an incompressible fluid, the term in ∇·V (divergence) vanishes, and the effects of viscosity then play a role analogous to that of temperature in thermal conduction and to that of density in simple diffusion.

Navier-Stokes Equations


the differential equations that describe the motion of a viscous fluid. These equations are named after L. Navier and G. Stokes. For an incompressible (density ρ = constant) and unheated (temperature T = constant) fluid, the Navier-Stokes equations projected on the axes of a rectangular Cartesian coordinate system (a system of three equations) have the form

Here t is the time; x, y, and z are the coordinates of a particle of fluid; vx, vy, and vz are the projections of the velocity of the particle; X, Y, and Z are the projections of the body force; ρ is the pressure; ν = μ/ρ is the kinematic viscosity coefficient (where μ is the dynamic viscosity coefficient), and

Two other equations are obtained by replacing x with y, y with z, and z with x.

The Navier-Stokes equations are used to determine vx, vy vz., and ρ as functions of x, y, z, and t. In order to close the system, we add to equations (1) a continuity equation, which for an incompressible fluid has the form

In order to integrate equations (1) and (2), we must be given the initial conditions (if the motion is not steady state) and the boundary conditions, which for a viscous fluid are the conditions of adhesion to rigid walls. In the general case of the motion of a compressible and heated fluid, the Navier-Stokes equations also take into account the variability of ρ and the temperature dependence of μ, changing the form of the equations. In this case, the equation of energy balance and the Clapeyron equation are also used.

The Navier-Stokes equations are used in the study of the motions of real liquids and gases; in most such specific problems, only various approximate solutions are sought.


References in periodicals archive ?
In agreement with the previous suppositions, the Navier-Stokes equations in cylindrical coordinates take the form
El analisis de los resultados es basado en tecnicas conocidas, principalmente en el caso de las ecuaciones de Navier-Stokes (con densidad constante), con lo cual, los resultados presentados pueden ser considerados como una generalizacion natural de resultados ya conocidos para ciertos modelos de las ecuaciones de la hidrodinamica.
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2003), it is necessary to perform a spatial average of the Reynolds Averaged Navier-Stokes equations.
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R stands for the residuals of the Navier-Stokes equations, n is the unit normal vector, v and u are the primal and adjoint velocities respectively, p and q are the primal and adjoint pressures respectively and v is the kinematic viscosity.
In this article, the fluid flow is described by the Navier-Stokes equations for a three-dimensional incompressible two-phase flow.
The computation of these parameters is based on Navier-Stokes and conservation equations.
Velocity distribution in the tube side is assumed a parabolic profile considering laminar, steady state, and fully developed conditions [18], For achieving velocity distribution in the shell side for the liquid solution stream, the Navier-Stokes equations have to be solved in conditions of incompressible and Newtonian fluid.
The earlier book concentrated on the analysis of numerical methods applied to model equations, while the new book concentrates on algorithms for the numerical solution of the Euler and Navier-Stokes equations.
2) can be viewed as a generalization of the Navier-Stokes equations with shear-dependent viscosity [?