# 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.

S. M. TARG

References in periodicals archive ?
Models use the Reynolds Averaged Navier Stokes equations (and sometimes the filtered Navier Stokes equations) to calculate the blood flow using a computer model.
Mattingly and Sinai [5] attempted to show that smooth solutions to 3D Navier Stokes equations exist for all initial conditions u(x, 0) = [u.
Computation of the Eulerian velocity field have been made with Charles Pierce's code [2] and incompressible Navier Stokes equations are solved numerically using an implicit filtering.
The details of Reynolds Averaged Navier Stokes equations as well as the SST turbulence model are not given here since they are well documented in the literature.
2002) studied the forces acting on cuttings layers based on continuity and Navier Stokes equations.
Mihail, Numerical solution of the steady incompressible Navier Stokes equations for the flow past a sphere by a multigrid defect correction technique, Internal J.
Exact solution of the Navier Stokes equations for the fully developed, pulsating flow in a rectangular Duct with a constant cross-sectional velocity.
The results were connected with the role of viscous heating in micro-channel flows, it's occurrence in the Navier Stokes equations and also there was made an experimental validation for verifying it's presence in practice.
As such, the bulk of the book consists of detailed mathematics (Lagrange multipliers, Lyaponov functions, Navier Stokes equations, Jacobian matrices, and covector fields with their associated Hamiltonians) that most non-mathematical biologists would find daunting.
RIVIERE, A discontinuous subgrid eddy viscosity method for the time-dependent Navier Stokes equations, SIAM J.

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