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# User:Taylorgasher: Difference between revisions

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My user page | My user page | ||

== test == | |||

:<math> \mathbf y_t + \mathbf A(\mathbf y) \mathbf y_x = 0.</math> | |||

==General continuum equations== | |||

{{Main|Derivation of the Navier–Stokes equations}} | |||

{{see also|Cauchy momentum equation#Conservation form}} | |||

The Navier–Stokes momentum equation can be derived as a particular form of the [[Cauchy momentum equation]], whose general convective form is | |||

:<math> \frac{D \mathbf{u}}{D t} = \frac 1 \rho \nabla \cdot \boldsymbol{\sigma} + \mathbf{g}</math> | |||

By setting the [[Cauchy stress tensor]] <math>\boldsymbol\sigma</math> to be the sum of a viscosity term <math>\boldsymbol\tau</math> (the [[Cauchy stress tensor#Stress deviator tensor|deviatoric stress]]) and a pressure term <math>-p\mathbf{I}</math> (volumetric stress) we arrive at | |||

{{Equation box 1 | |||

|indent=: | |||

|title='''Cauchy momentum equation''' ''(convective form)'' | |||

|equation=<math> | |||

\rho\frac{D\mathbf{u}}{Dt} = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{g} | |||

</math> | |||

|cellpadding | |||

|border | |||

|border colour = #50C878 | |||

|background colour = #ECFCF4 | |||

}} | |||

where | |||

* <math>\frac{D}{Dt}</math> is the [[convective derivative]], | |||

* {{math|''ρ''}} is the density, | |||

* {{math|'''u'''}} is the flow velocity, | |||

* <math>\nabla\cdot</math> is the [[divergence]], | |||

* {{math|''p''}} is the [[pressure]], | |||

* {{math|''t''}} is [[time]], | |||

* <math>\boldsymbol\tau</math> is the derivatoric stress tensor, which has order two, | |||

* {{math|'''g'''}} represents [[body force|body acceleration]]s acting on the continuum, for example [[gravity]], [[Fictitious force|inertial accelerations]], [[Coulomb's law|electrostatic accelerations]], and so on, | |||

:<math> \mathbf | In this form, it is apparent that in the assumption of an inviscid fluid -no deviatoric stress- Cauchy equations reduce to the [[Euler equations (fluid dynamics)|Euler equations]]. | ||

Assuming [[conservation of mass]] we can use the [[Continuity_equation#Differential_form|continuity equation]], <math>\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho\,\mathbf{u}) = 0</math> to arrive to the conservation form of the equations of motion. This is often written:<ref>Batchelor (1967) pp. 137 & 142.</ref> | |||

{{Equation box 1 | |||

|indent=: | |||

|title='''Cauchy momentum equation''' ''(conservation form)'' | |||

|equation=<math> | |||

\frac {\partial}{\partial t} (\rho\,\mathbf{u}) | |||

+ \nabla \cdot (\rho\,\mathbf{u} \otimes \mathbf{u}) | |||

= - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{g} | |||

</math> | |||

|cellpadding | |||

|border | |||

|border colour = #50C878 | |||

|background colour = #ECFCF4 | |||

}} | |||

where {{math|⊗}} is the [[outer product]]: | |||

::<math>\mathbf u \otimes \mathbf v = \mathbf u \mathbf v^{\mathrm T}.</math> | |||

The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity). | |||

All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a [[constitutive relation]]. By expressing the deviatoric (shear) stress tensor in terms of [[viscosity]] and the fluid [[Shear velocity|velocity]] gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below. | |||

===Convective acceleration=== | |||

{{see also|Cauchy momentum equation#Convective acceleration}} | |||

[[Image:ConvectiveAcceleration vectorized.svg|thumb|An example of convection. Though the flow may be steady (time-independent), the fluid decelerates as it moves down the diverging duct (assuming incompressible or subsonic compressible flow), hence there is an acceleration happening over position.]] | |||

A significant feature of the Cauchy equation and consequently all other continuum equations (including Euler and Navier–Stokes) is the presence of convective acceleration: the effect of acceleration of a flow with respect to space. While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle. |

## Revision as of 13:33, 13 August 2018

My user page

## test

## General continuum equations

Template:Main Template:See also

The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation, whose general convective form is

By setting the Cauchy stress tensor to be the sum of a viscosity term (the deviatoric stress) and a pressure term (volumetric stress) we arrive at

where

- is the convective derivative,
- Template:Math is the density,
- Template:Math is the flow velocity,
- is the divergence,
- Template:Math is the pressure,
- Template:Math is time,
- is the derivatoric stress tensor, which has order two,
- Template:Math represents body accelerations acting on the continuum, for example gravity, inertial accelerations, electrostatic accelerations, and so on,

In this form, it is apparent that in the assumption of an inviscid fluid -no deviatoric stress- Cauchy equations reduce to the Euler equations.

Assuming conservation of mass we can use the continuity equation, to arrive to the conservation form of the equations of motion. This is often written:^{[1]}

where Template:Math is the outer product:

The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity).

All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation. By expressing the deviatoric (shear) stress tensor in terms of viscosity and the fluid velocity gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below.

### Convective acceleration

A significant feature of the Cauchy equation and consequently all other continuum equations (including Euler and Navier–Stokes) is the presence of convective acceleration: the effect of acceleration of a flow with respect to space. While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.

- ↑ Batchelor (1967) pp. 137 & 142.