Editing Newtonian fluid (section)

===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>
|cellpadding
|border
|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>
|cellpadding
|border
|border colour = #0073CF
|background colour=#DCDCDC
}}

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.