Editing Navier–Stokes equations (section)

==Compressible flow==
Remark: here, the deviatoric stress tensor is denoted <math display="inline">\boldsymbol{\tau}</math> as it was in the [[#General continuum equations|general continuum equations]] and in the [[#Incompressible flow|incompressible flow section]].

The compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:<ref name="Batchelor_142_148"/>

<ul>
<li>the stress is '''[[Galilean invariance|Galilean invariant]]''': it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient <math display="inline">\nabla \mathbf{u}</math>, or more simply the [[strain-rate tensor|rate-of-strain tensor]]: <math display="inline">\boldsymbol{\varepsilon}\left(\nabla \mathbf{u}\right) \equiv \frac{1}{2}\nabla \mathbf{u} + \frac{1}{2} \left(\nabla \mathbf{u}\right)^T</math></li>
<li>the deviatoric stress is '''linear''' in this variable: <math display="inline">\boldsymbol{\sigma}(\boldsymbol \varepsilon) = -p \mathbf I + \mathbf{C} : \boldsymbol \varepsilon</math>, where <math display="inline">p</math> is independent on the strain rate tensor, <math display="inline">\mathbf{C}</math> is the fourth-order tensor representing the constant of proportionality, called the viscosity or [[elasticity tensor]], and : is the [[Dyadics#Double-dot product|double-dot product]].</li>
<li>the fluid is assumed to be [[isotropic]], as with gases and simple liquids, and consequently <math display="inline">\mathbf{C}</math> is an isotropic tensor; furthermore, since the deviatoric stress tensor is symmetric, by [[Helmholtz decomposition]] it can be expressed in terms of two scalar [[Lamé parameters]], the [[second viscosity]] <math display="inline">\lambda</math> and the [[dynamic viscosity]] <math display="inline">\mu</math>, as it is usual in [[linear elasticity]]:
{{Equation box 1
|indent=:
|title='''Linear stress [[constitutive equation]]''' ''(expression similar to the one for elastic solid)''
|equation=<math>\boldsymbol \sigma(\boldsymbol \varepsilon) =  - p \mathbf I + \lambda \operatorname{tr} (\boldsymbol \varepsilon) \mathbf I + 2 \mu \boldsymbol \varepsilon</math>
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where <math display="inline">\mathbf{I}</math> is the [[Identity matrix|identity tensor]], and <math display="inline">\operatorname{tr} (\boldsymbol \varepsilon)</math> is the [[trace (linear algebra)|trace]] of the rate-of-strain tensor. So this decomposition can be explicitly defined as:
<math display="block">\boldsymbol \sigma = -p \mathbf I + \lambda (\nabla\cdot\mathbf{u}) \mathbf I + \mu \left(\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T}\right).</math>
</li>
</ul>

Since the [[trace (linear algebra)|trace]] of the rate-of-strain tensor in three dimensions is the [[divergence]] (i.e. rate of expansion) of the flow:
<math display="block">\operatorname{tr} (\boldsymbol \varepsilon) = \nabla\cdot\mathbf{u}.</math>

Given this relation, and since the trace of the identity tensor in three dimensions is three:
<math display="block">\operatorname{tr} (\boldsymbol I) = 3.</math>

the trace of the stress tensor in three dimensions becomes:
<math display="block">\operatorname{tr} (\boldsymbol \sigma ) = -3p + (3 \lambda + 2 \mu )\nabla\cdot\mathbf{u}.</math>

So by alternatively decomposing the stress tensor into '''isotropic''' and '''deviatoric''' parts, as usual in fluid dynamics:<ref>{{cite book |last1=Chorin |first1=Alexandre E. |last2=Marsden |first2=Jerrold E. |date=1993 |title=A Mathematical Introduction to Fluid Mechanics |page=33}}</ref>
<math display="block">\boldsymbol \sigma = - \left[ p - \left(\lambda + \tfrac23 \mu\right) \left(\nabla\cdot\mathbf{u}\right) \right] \mathbf I + \mu \left(\nabla\mathbf{u} + \left( \nabla\mathbf{u} \right)^\mathrm{T} - \tfrac23 \left(\nabla\cdot\mathbf{u}\right)\mathbf I\right)</math>

Introducing the [[volume viscosity|bulk viscosity]] <math display="inline">\zeta</math>,
<math display="block"> \zeta \equiv \lambda + \tfrac23 \mu ,</math>

we arrive to the linear [[constitutive equation]] in the form usually employed in [[thermal hydraulics]]:<ref name=Batchelor_142_148/>

{{Equation box 1
|indent=:
|title='''Linear stress constitutive equation''' ''(expression used for fluids)''
|equation=<math>\boldsymbol \sigma = -[ p - \zeta (\nabla\cdot\mathbf{u})] \mathbf I + \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right]</math>
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which can also be arranged in the other usual form:<ref>Bird, Stewart, Lightfoot, Transport Phenomena, 1st ed., 1960, eq. (3.2-11a)</ref>
<math display="block">\boldsymbol \sigma = -p \mathbf I + \mu \left(\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T}\right) + \left(\zeta - \frac 2 3 \mu \right) (\nabla\cdot\mathbf{u}) \mathbf I.</math>

Note that in the compressible case the pressure is no more proportional to the [[hydrostatic stress|isotropic stress]] term, since there is the additional bulk viscosity term:
<math>p = - \frac 1 3 \operatorname{tr} (\boldsymbol \sigma)  + \zeta (\nabla\cdot\mathbf{u})</math>

and the [[deviatoric stress tensor]] <math>\boldsymbol \sigma'</math> is still coincident with the shear stress tensor <math>\boldsymbol \tau</math> (i.e. the deviatoric stress in a Newtonian fluid has no normal stress components), and it has a compressibility term in addition to the incompressible case, which is proportional to the shear viscosity:

<math>\boldsymbol \sigma' = \boldsymbol \tau = \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right]</math>

Both bulk viscosity <math display="inline">\zeta</math> and dynamic viscosity <math display="inline">\mu</math> need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these [[transport coefficient]] in the [[conservation variable]]s is called an [[equation of state]].<ref name="Batchelor 1967 p. 165"/>

The most general of the Navier–Stokes equations become
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' (''convective form'')
|equation=<math> \rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = - \nabla p  + \nabla \cdot \left\{ \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] \right\} + \nabla[\zeta (\nabla\cdot\mathbf{u})] + \rho\mathbf{a} .</math>
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in index notation, the equation can be written as<ref name="landau">Landau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6. Vol. 6. Elsevier, 2013.</ref>
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' (''index notation'')
|equation=<math> \rho \left(\frac{\partial u_i}{\partial t} + u_k \frac{\partial u_i}{\partial x_k}\right) = - \frac{\partial p}{\partial x_i} + \frac{\partial }{\partial x_k}\left[\mu \left(\frac{\partial u_i}{\partial x_k}+ \frac{\partial u_k}{\partial x_i}-\frac{2}{3}\delta_{ik} \frac{\partial u_l}{\partial x_l}\right)\right] + \frac{\partial }{\partial x_i}\left(\zeta \frac{\partial u_\ell}{\partial x_\ell}\right)+\rho a_i.</math>
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The corresponding equation in conservation form can be obtained by considering that, given the mass [[continuity equation]], the left side is equivalent to:

: <math> \rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \frac {\partial}{\partial t} (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u)</math>

to give finally:
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' (''conservative form'')
|equation=<math> \frac {\partial}{\partial t} (\rho \mathbf u) + \nabla \cdot \left(\rho \mathbf u \otimes \mathbf u + [p - \zeta (\nabla\cdot\mathbf{u})] \mathbf I - \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u}) \mathbf I\right] \right) = \rho\mathbf{a} .</math>
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Apart from its dependence of pressure and temperature, the second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. Example: in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the ''dispersion''. In some cases, the [[volume viscosity|second viscosity]] <math display="inline">\zeta</math> can be assumed to be constant in which case, the effect of the volume viscosity <math display="inline">\zeta</math> is that the mechanical pressure is not equivalent to the thermodynamic [[pressure]]:<ref>Landau & Lifshitz (1987) pp. 44–45, 196</ref> as demonstrated below.
<math display="block">\nabla\cdot(\nabla\cdot \mathbf u)\mathbf I=\nabla (\nabla \cdot \mathbf u),</math><math display="block"> \bar{p} \equiv p - \zeta \, \nabla \cdot \mathbf{u} ,</math>
However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves,<ref>White (2006) p. 67.</ref> where second viscosity coefficient becomes important) by explicitly assuming <math display="inline">\zeta = 0</math>. The assumption of setting <math display="inline">\zeta = 0</math> is called as the '''Stokes hypothesis'''.<ref>Stokes, G. G. (1845). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids.</ref> The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory;<ref>Vincenti, W. G., Kruger Jr., C. H. (1975). Introduction to physical gas dynamic. Introduction to physical gas dynamics/Huntington.</ref> for other gases and liquids, Stokes hypothesis is generally incorrect. With the Stokes hypothesis, the Navier–Stokes equations become
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' (''convective form, Stokes hypothesis'')
|equation=<math> \rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = - \nabla  p + \nabla \cdot \left\{ \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] \right\} + \rho\mathbf{a} .</math>
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If the dynamic {{mvar|μ}} and bulk <math>\zeta</math> viscosities are  assumed to be uniform in space, the equations in convective form can be simplified further. By computing the divergence of the stress tensor, since the divergence of tensor <math display="inline">\nabla \mathbf{u}</math> is <math display="inline">\nabla^2 \mathbf{u}</math> and the divergence of tensor <math display="inline">\left(\nabla \mathbf{u}\right)^{\mathrm T}</math> is <math display="inline">\nabla \left(\nabla \cdot \mathbf{u}\right) </math>, one finally arrives to the compressible Navier–Stokes momentum equation:<ref>Batchelor (1967) pp. 147 & 154.</ref>
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' with uniform shear and bulk viscosities (''convective form'')
|equation=<math> \frac {D \mathbf{u}}{D t} = - \frac 1 \rho \nabla p + \nu \, \nabla^2 \mathbf u + (\tfrac13 \nu + \xi) \, \nabla (\nabla\cdot\mathbf{u}) + \mathbf{a} .</math>
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where <math display="inline">\frac{\mathrm{D}}{\mathrm{D}t} </math> is the [[material derivative]]. <math>\nu=\frac \mu \rho</math> is the shear [[kinematic viscosity]] and <math>\xi=\frac \zeta \rho</math> is the bulk kinematic viscosity. The left-hand side changes in the conservation form of the Navier–Stokes momentum equation.
By bringing the operator on the flow velocity on the left side, one also has:

{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' with uniform shear and bulk viscosities (''convective form'')
|equation=<math> \left(\frac {\partial}{\partial t} + \mathbf u \cdot \nabla - \nu \, \nabla^2 - (\tfrac13 \nu + \xi) \, \nabla (\nabla\cdot) \right)\mathbf{u} = - \frac 1\rho \nabla p + \mathbf{a} .</math>
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The convective acceleration term can also be written as
<math display="block"> \mathbf u\cdot\nabla\mathbf u = (\nabla\times\mathbf u)\times\mathbf u + \tfrac12\nabla\mathbf u^2,</math>
where the vector <math display="inline"> (\nabla \times \mathbf{u}) \times \mathbf{u}</math> is known as the [[Lamb vector]].

For the special case of an [[incompressible flow]], the pressure constrains the flow so that the volume of [[fluid element]]s is constant: [[isochoric process|isochoric flow]] resulting in a [[Solenoidal vector field|solenoidal]] velocity field with <math display="inline"> \nabla \cdot \mathbf{u} = 0</math>.<ref>Batchelor (1967) p. 75.</ref>