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Navier-Stokes equations

In fluid dynamics, the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. For example: they model weather or the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena.

The Navier-Stokes equations are

$\rho\frac{Du_i}{Dt}=\rho F_i-\frac{\partial P}{\partial x_i}+\frac{\partial}{\partial x_j}\left[ 2\mu\left(e_{ij}-\Delta\delta_{ij}/3\right)\right]$

for momentum conservation and

$\frac{\partial\rho u_i}{\partial x_i}=0$

for conservation of mass. Here $ρ$ is the density, $u i$ ( $i = 1,2,3$ ) the three components of velocity, $F i$ body forces (such as gravity), $P$ the pressure, $μ$ the dynamic viscosity, of the fluid at a point; $e_{ij}=\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$ ; $Δ = e i i$ is the divergence, and $δ i j$ is the Kronecker delta. $D / D t$ is the substantive derivative.

If $μ$ is uniform over the fluid, the momentum equation above simplifies to

$\rho\frac{Du_i}{Dt}=\rho F_i-\frac{\partial P}{\partial x_i} +\mu \left( \frac{\partial^2u_i}{\partial x_i\partial x_j}+ \frac{1}{3}\frac{\partial\Delta}{\partial x_i}\right)$

(if $μ = 0$ the resulting equations are known as the Euler equations; there, the emphasis is on compressible flow and shock waves).

If now in addition $ρ$ is assumed to be constant:

$\rho \left({\partial v_x \over \partial t}+ v_x {\partial v_x \over \partial x}+ v_y {\partial v_x \over \partial y}+ v_z {\partial v_x \over \partial z}\right)= \mu \left[{\partial v_x^2 \over \partial x^2}+{\partial v_x^2 \over \partial y^2}+{\partial v_x^2 \over \partial z^2}\right]-{\partial P \over \partial x} +\rho g_x$

$\rho \left({\partial v_y \over \partial t}+ v_x {\partial v_y \over \partial x}+ v_y {\partial v_y \over \partial y}+ v_z {\partial v_y \over \partial z}\right)= \mu \left[{\partial v_y^2 \over \partial x^2}+{\partial v_y^2 \over \partial y^2}+{\partial v_y^2 \over \partial z^2}\right]-{\partial P \over \partial y} +\rho g_y$

$\rho \left({\partial v_z \over \partial t}+ v_x {\partial v_z \over \partial x}+ v_y {\partial v_z \over \partial y}+ v_z {\partial v_z \over \partial z}\right)= \mu \left[{\partial v_z^2 \over \partial x^2}+{\partial v_z^2 \over \partial y^2}+{\partial v_z^2 \over \partial z^2}\right]-{\partial P \over \partial z} +\rho g_z$

Continuity Equation (assuming incompressibility):

${\partial v_x \over \partial x}+{\partial v_y \over \partial y}+{\partial v_z \over \partial z}=0$

Simplified version of the N-S equations. Adapted from Incompressible Flow, second editon by Ronald Panton

The equations are derived by considering the mass, momentum, and energy balances for an infinitesimal control volume. The Navier-Stokes equations need to be augmented by an equation of state for compressible flows. The variables to be solved for are the velocity components, the fluid density, static pressure, and temperature. The flow is assumed to be differentiable and continuous, allowing these balances to be expressed as partial differential equations. The equations can be converted to Wilkinson equations for the secondary variables vorticity and stream function. Solution depends on the fluid properties (such as viscosity, specific heats, and thermal conductivity), and on the boundary conditions of the domain of study. For a derivation of the Navier-Stokes equations, see some of the external links listed below.

Note that the Navier-Stokes equations can only describe fluid flow approximately and that, at very small scales or under extreme conditions, real fluids made out of mixtures of discrete molecules and other material, such as suspended particles and dissolved gases, will produce different results from the continuous and homogeneous fluids modelled by the Navier-Stokes equations. Depending on the Knudsen number of the problem, statistical mechanics may be a more appropriate approach. However, the Navier-Stokes equations are useful for a wide range of practical problems, providing their limitations are borne in mind.

Although the full, unsteady Navier-Stokes equations correctly describe nearly all flows of practical interest they are too complex for practical solution in many cases and a special "reduced" form of the full equations is often used instead — these are the Reynolds-averaged Navier-Stokes (RANS) equations. The solution of the full steady Navier-Stokes equations is sufficiently accurate alone for cases where the fluid flow is laminar. For turbulent flows the Reynolds-averaged form of the equations are most commonly used. The RANS form of the equations introduce new terms that reflect the additional modelling of the small turbulent motions.

Solution of flow equations by numerical methods is called computational fluid dynamics.

It is a famous open question whether smooth initial conditions always lead to smooth solutions for all times; a $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute for the answer to this question.

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Categories: Partial differential equations | Fluid dynamics

Last updated: 10-31-2004 21:51:50

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Navier-Stokes equations

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