Term information
In physics the Navier Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term. ... The derivation of the Navier Stokes equations begins with an application of Newton's second law: conservation of momentum (often alongside mass and energy conservation) being written for an arbitrary portion of the fluid. In an inertial frame of reference, the general form of the equations of fluid motion is:[2] \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f}, where \mathbf{v} is the flow velocity, is the fluid density, p is the pressure, \mathbb{T} is the (deviatoric) stress tensor, and \mathbf{f} represents body forces (per unit volume) acting on the fluid and \nabla is the del operator. This is a statement of the conservation of momentum in a fluid and it is an application of Newton's second law to a continuum; in fact this equation is applicable to any non-relativistic continuum and is known as the Cauchy momentum equation. source: http://en.wikipedia.org/wiki/Navier-Stokes_equations
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\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f}
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi> ρ </mi> </mrow> <mfenced> <mrow><mrow> <mfrac> <mrow><mo>∂</mo></mrow> <mrow><mo>∂</mo><mi>t</mi></mrow> </mfrac> <mrow><mi>v</mi><mo> + </mo><mi>v</mi><mo> ⋅ </mo><mfenced> <mrow><mrow><mtext>DEL</mtext></mrow> <mfenced> <mrow><mi>v</mi></mrow> </mfenced></mrow> </mfenced></mrow> </mrow></mrow> </mfenced><mo> = </mo><mo> − </mo<mfenced> <mrow><mn>1</mn></mrow> </mfenced><mo> ⋅ </mo><mrow><mtext>DEL</mtext></mrow> <mfenced> <mrow><mi>p</mi></mrow> </mfenced><mo> + </mo><mtext>DEL</mtext><mo> ⋅ </mo><mi>T</mi><mo> + </mo><mi>f</mi></math>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mi>ρ</mi> <mfenced close=")" open="("> <mrow> <mfrac> <mrow> <mo>∂</mo> <mstyle mathvariant="bold"> <mi>v</mi> </mstyle> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>+</mo> <mstyle mathvariant="bold"> <mi>v</mi> </mstyle> <mo>⋅</mo> <mo>∇</mo> <mstyle mathvariant="bold"> <mi>v</mi> </mstyle> </mrow> </mfenced> <mo>=</mo> <mo>-</mo> <mo>∇</mo> <mi>p</mi> <mo>+</mo> <mo>∇</mo> <mo>⋅</mo> <mi mathvariant="double-struck">T</mi> <mo>+</mo> <mstyle mathvariant="bold"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="SnuggleTeX">\[ \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f} \]</annotation> </semantics> </math>
Term relations
- equation of fluid dynamics
- has_variable some (
variable and
is_about some stress quality) - has_variable some (
variable and
is_about some (
velocity and
quality_of some motion)) - has_about some (
graph and
has_quality some non-linear) - has_variable some (
variable and
is_about some (
mass density and
quality_of some material_entity)) - has_variable some (
variable and
is_about some pressure) - has_variable some (
variable and
is_about some (
force and
quality_of some anatomical part)) - describes some diffusion