Editing Newtonian fluid (section)

==Definition==
An element of a flowing liquid or gas will endure forces from the surrounding fluid, including [[stress (mechanics)|viscous stress forces]] that cause it to gradually deform over time. These forces can be mathematically [[Taylor series|first order approximated]] by a [[viscous stress tensor]], usually denoted by <math>\tau</math>.

The deformation of a fluid element, relative to some previous state, can be first order approximated by a [[strain tensor]] that changes with time. The time derivative of that tensor is the [[strain rate tensor]], that expresses how the element's deformation is changing with time; and is also the [[gradient]] of the velocity [[vector field]] <math>v</math> at that point, often denoted <math>\nabla v</math>.

The tensors <math>\tau</math> and <math>\nabla v</math> can be expressed by 3×3 [[matrix (mathematics)|matrices]], relative to any chosen [[coordinate system]]. The fluid is said to be Newtonian if these matrices are related by the [[matrix equation|equation]]
<math>\boldsymbol{\tau} = \boldsymbol{\mu} (\nabla v)</math>
where <math>\mu</math> is a fixed 3×3×3×3 fourth order tensor that does not depend on the velocity or stress state of the fluid.

===Incompressible isotropic case===
For an [[Incompressible flow|incompressible]] and isotropic Newtonian fluid in '''[[laminar flow]] only in the direction x''' (i.e. where viscosity is isotropic in the fluid), the shear stress is related to the strain rate by the simple [[constitutive equation]]
<math display="block">\tau = \mu \frac{du}{dy}</math>
where
*<math>\tau</math> is the [[shear stress]] ("[[drag (physics)|skin drag]]") in the fluid,
*<math>\mu</math> is a scalar constant of proportionality, the [[Viscosity#Dynamic viscosity|dynamic viscosity]] of the fluid
*<math>\frac{du}{dy}</math> is the [[derivative]] in the direction y, normal to x, of the [[flow velocity]] component u that is oriented along the direction x.

In case of a general 2D incompressibile flow in the plane x, y, the Newton constitutive equation become:

<math display="block">\tau_{xy} = \mu \left( \frac{\partial u}{\partial y} +\frac{\partial v}{\partial x} \right)</math>
where:
*<math>\tau_{xy}</math> is the [[shear stress]] ("[[drag (physics)|skin drag]]") in the fluid,
*<math>\frac{\partial u}{\partial y}</math> is the [[partial derivative]] in the direction y of the [[flow velocity]] component u that is oriented along the direction x.
*<math>\frac{\partial v}{\partial x}</math> is the partial derivative in the direction x of the flow velocity component v that is oriented along the direction y.

We can now generalize to the case of an [[incompressible fluid|incompressible flow]] with a general direction in the 3D space, the above constitutive equation becomes
<math display="block">\tau_{ij} = \mu \left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right)</math>
where
*<math>x_j</math> is the <math>j</math>th spatial coordinate
*<math>v_i</math> is the fluid's velocity in the direction of axis <math>i</math>
*<math>\tau_{ij}</math> is the <math>j</math>-th component of the stress acting on the faces of the fluid element perpendicular to axis <math>i</math>. It is the ij-th component of the shear stress tensor

or written in more compact tensor notation
<math display="block">\boldsymbol{\tau} = \mu\left(\nabla\mathbf{u}+\nabla\mathbf{u}^{T}\right)</math>
where <math>\nabla \mathbf{u}</math> is the flow velocity gradient.

An alternative way of stating this constitutive equation is:
{{Equation box 1
|indent=:
|title='''Stokes' stress [[constitutive equation]]''' ''(expression used for incompressible elastic solids)''
|equation=:<math>\boldsymbol \tau = 2 \mu \boldsymbol \varepsilon</math>
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|border colour = #0073CF
|background colour=#DCDCDC
}}
where
<math display="block">\boldsymbol{\varepsilon} = \tfrac{1}{2} \left( \mathbf{\nabla u} + \mathbf{\nabla u}^\mathrm{T} \right)</math>
is the rate-of-[[strain tensor]]. So this decomposition can be made explicit as:<ref name=Batchelor_142_148>Batchelor (1967) pp. 137 & 142.</ref>
{{Equation box 1
|indent=:
|title='''Stokes's stress constitutive equation''' ''(expression used for incompressible viscous fluids)''
|equation=:<math>\boldsymbol \tau = \mu \left[\nabla\mathbf{u} +  (\nabla\mathbf{u}) ^\mathrm{T}\right]</math>
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|border colour = #0073CF
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}}

This constitutive equation is also called the '''Newton law of viscosity'''.

The total [[Cauchy stress tensor|stress tensor]] <math>\boldsymbol{\sigma}</math> can always be decomposed as the sum of the [[hydrostatic stress|isotropic stress]] tensor and the [[deviatoric stress tensor]] (<math>\boldsymbol \sigma '</math>):

<math>\boldsymbol \sigma = \frac 1 3 \operatorname{tr}(\boldsymbol \sigma) \mathbf I + \boldsymbol \sigma'</math>

In the incompressible case, the isotropic stress is simply proportional to the thermodynamic [[pressure]] <math>p</math>:
<math display="block">p = - \frac 1 3 \operatorname{tr}(\boldsymbol \sigma) = - \frac 1 3 \sum_k \sigma_{kk}</math>

and the deviatoric stress is coincident with the shear stress tensor <math>\boldsymbol \tau</math>:
<math display="block">\boldsymbol \sigma' = \boldsymbol \tau = \mu\left(\nabla\mathbf{u}+\nabla\mathbf{u}^{T}\right)</math>

The stress [[constitutive equation]] then becomes
<math display="block"> \sigma_{ij} = - p \delta_{ij} + \mu \left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right)</math>
or written in more compact tensor notation
<math display="block">\boldsymbol{\sigma} = - p \mathbf{I} + \mu\left(\nabla\mathbf{u}+\nabla\mathbf{u}^{T}\right)</math>
where <math>\mathbf{I}</math> is the identity tensor.

===General compressible case===
The Newton's constitutive law for a compressible flow 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 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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|border colour = #FF0000
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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>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}

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>

Note that the incompressible case correspond to the assumption that 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>
So one returns to the expressions for pressure and deviatoric stress seen in the preceding paragraph. 

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">Batchelor (1967) p. 165.</ref>

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. (2007). 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.

Finally, note that Stokes hypothesis is less restrictive that the one of incompressible flow. In fact, in the incompressible flow both the bulk viscosity term, and the shear viscosity term in the divergence of the flow velocity term disappears, while in the Stokes hypothesis the first term also disappears but the second one still remains.

===For anisotropic fluids===
More generally, in a non-isotropic Newtonian fluid, the coefficient <math>\mu</math> that relates internal friction stresses to the [[spatial derivative]]s of the velocity field is replaced by a nine-element [[viscous stress tensor]] <math>\mu_{ij}</math>.

There is general formula for friction force in a liquid: The vector [[Differential (mathematics)|differential]] of friction force is equal the viscosity tensor increased on [[vector product]] differential of the area vector of adjoining a liquid layers and [[Rotor (mathematics)|rotor]] of velocity:
<math display="block"> d \mathbf{F} = \mu _ {ij} \, d\mathbf{S} \times\nabla\times \, \mathbf {u} </math>
where <math> \mu _ {ij} </math> is the viscosity [[tensor]]. The diagonal components of viscosity tensor is molecular viscosity of a liquid, and not diagonal components – [[turbulence eddy viscosity]].<ref name=Volobuev>{{cite book | first=A. N. |last=Volobuev| title=Basis of Nonsymmetrical Hydromechanics| publisher=[[Nova Science Publishers, Inc.]] |location=New York | year=2012 | isbn=978-1-61942-696-2}}</ref>