# Navier-Stokes Equations

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## Navier-Stokes equations

[nä′vyā ′stōks i‚kwā·zhənz]*d*V/

*dt*= -(1/ρ)∇

*p*+ F + ν∇

^{2}V + (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; *v _{x}*,

*v*, and

_{y}*v*are the projections of the velocity of the particle;

_{z}*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 *v _{x}*,

*v*., and

_{y}v_{z}*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.

S. M. TARG