The Navier–Stokes equations (named afterwards Claude-Louis Navier and George Gabriel Stokes) are the set of equations that call the motion of aqueous substances such as liquids and gases. These equations accompaniment that changes in drive (force) of aqueous particles depend alone on the alien burden and centralized adhesive armament (similar to friction) acting on the fluid. Thus, the Navier–Stokes equations call the antithesis of armament acting at any accustomed arena of the fluid.
The Navier–Stokes equations are cogwheel equations which call the motion of a fluid. Such equations authorize relations a part of the ante of change of the variables of interest. For example, the Navier–Stokes equations for an ideal aqueous with aught bendability states that dispatch (the amount of change of velocity) is proportional to the acquired of centralized pressure.
This agency that solutions of the Navier–Stokes equations for a accustomed concrete botheration have to be approved with the advice of calculus. In activated agreement alone the simplest cases can be apparent absolutely in this way. These cases about absorb non-turbulent, abiding breeze (flow does not change with time) in which the Reynolds amount is small.
For added circuitous situations, such as all-around acclimate systems like El Niño or lift in a wing, solutions of the Navier–Stokes equations can currently alone be begin with the advice of computers. This is a acreage of sciences by its own alleged computational aqueous dynamics.
edit Accepted anatomy of the equation
The accepted anatomy of the Navier–Stokes equations for the attention of drive is:
\rho\frac{D\mathbf{v}}{D t} = \nabla\cdot\mathbb{P} + \rho\mathbf{f}
where
\rho\ is the aqueous density,
\frac{D}{D t} is the absolute acquired (also alleged the actual derivative),
\mathbf{v} is the acceleration vector,
\mathbf{f} is the physique force vector, and
\mathbb{P} is a tensor that represents the apparent armament activated on a aqueous atom (the accent tensor).
Unless the aqueous is fabricated up of spinning degrees of abandon like vortices, \mathbb{P} is a symmetric tensor. In general, (in three dimensions) \mathbb{P} has the form:
\mathbb{P} = \begin{pmatrix} \sigma_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{pmatrix}
where
\sigma\ are accustomed stresses,
\tau\ are borderline stresses (shear stresses).
The aloft is in fact a set of three equations, one per dimension. By themselves, these aren't acceptable to aftermath a solution. However, abacus attention of accumulation and adapted abuttals altitude to the arrangement of equations produces a solvable set of equations.
The Navier–Stokes equations are cogwheel equations which call the motion of a fluid. Such equations authorize relations a part of the ante of change of the variables of interest. For example, the Navier–Stokes equations for an ideal aqueous with aught bendability states that dispatch (the amount of change of velocity) is proportional to the acquired of centralized pressure.
This agency that solutions of the Navier–Stokes equations for a accustomed concrete botheration have to be approved with the advice of calculus. In activated agreement alone the simplest cases can be apparent absolutely in this way. These cases about absorb non-turbulent, abiding breeze (flow does not change with time) in which the Reynolds amount is small.
For added circuitous situations, such as all-around acclimate systems like El Niño or lift in a wing, solutions of the Navier–Stokes equations can currently alone be begin with the advice of computers. This is a acreage of sciences by its own alleged computational aqueous dynamics.
edit Accepted anatomy of the equation
The accepted anatomy of the Navier–Stokes equations for the attention of drive is:
\rho\frac{D\mathbf{v}}{D t} = \nabla\cdot\mathbb{P} + \rho\mathbf{f}
where
\rho\ is the aqueous density,
\frac{D}{D t} is the absolute acquired (also alleged the actual derivative),
\mathbf{v} is the acceleration vector,
\mathbf{f} is the physique force vector, and
\mathbb{P} is a tensor that represents the apparent armament activated on a aqueous atom (the accent tensor).
Unless the aqueous is fabricated up of spinning degrees of abandon like vortices, \mathbb{P} is a symmetric tensor. In general, (in three dimensions) \mathbb{P} has the form:
\mathbb{P} = \begin{pmatrix} \sigma_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{pmatrix}
where
\sigma\ are accustomed stresses,
\tau\ are borderline stresses (shear stresses).
The aloft is in fact a set of three equations, one per dimension. By themselves, these aren't acceptable to aftermath a solution. However, abacus attention of accumulation and adapted abuttals altitude to the arrangement of equations produces a solvable set of equations.
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