In an [[isotropic]] Newtonian medium the viscosity tensor <math>\mu</math> is highly symmetric, and the physics is considerably simplified. The strain rate tensor <math>E</math> can always be separated into a term <math>E^{\mathrm{v}}</math> that describes a rate of uniform expansion, and a term <math>E^{\mathrm{s}}</math> that describes the rate of pure shearing deformation, without change in volume.  Likewise, the viscous stress tensor <math>\epsilon</math> can always be separated into an isotropic pressure component <math>\epsilon^{\mathrm{v}}</math> and a pure [[shearing (physics)|shearing]] stress component <math>\epsilon^{\mathrm{s}}</math>. If the medium is isotropic as well as Newtonian, each term of <math>\epsilon</math> must depend only on the corresponding term of <math>E</math>, by a scalar proportionality factor: <math>\epsilon^{\mathrm{v}} = \mu^{\mathrm{v}}E^{\mathrm{v}},</math> and <math>\epsilon^{\mathrm{s}} = \mu^{\mathrm{s}}E^{\mathrm{s}}</math>, where <math>\mu^{\mathrm{v}}</math> and <math>\mu^{\mathrm{s}}</math> are real numbers. Therefore, in such media the viscosity tensor <math>\mu</math> has only two independent scalar parameters: a [[bulk viscosity]] coefficient <math>\mu^{\mathrm{v}}</math>, that defines the resitance of the medium to gradual unform compression; and a [[dynamic viscosity]] coefficent <math>\mu^{\mathrm{s}}</math> that expresses its resistance to gradual shearing.

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the viscous stress tensor <math>(\varepsilon_{ij})</math> is related to the strain rate tensor <math>(E_{ij})</math> by the<math>(\mu_{ijk\ell})</math>:

:<math>\varepsilon_{ij}=\sum_{k\ell}\mu_{ijk\ell}E_{k\ell}</math>

<!-- FROM VISCOSITY STRESS TENSOR

 '''''r''''' is specified by the velocity field '''''v'''''('''''r'''''). The velocity at a small distance ''d'''r''''' from point '''''r''''' may be written as a [[Taylor series]]:

:<math>\mathbf{v}(\mathbf{r}+d\mathbf{r}) = \mathbf{v}(\mathbf{r})+\frac{d\mathbf{v}}{d\mathbf{r}}d\mathbf{r}+\ldots,</math>

where ''d'''v'''''&nbsp;/&nbsp;''d'''r''''' is the [[gradient]] of the velocity field:

:<math>\frac{d\mathbf{v}}{d\mathbf{r}} = \begin{bmatrix}
\displaystyle{\frac{\partial v_x}{\partial x}} & \displaystyle{\frac{\partial v_x}{\partial y}} & \displaystyle{\frac{\partial v_x}{\partial z}}\\
\displaystyle{\frac{\partial v_y}{\partial x}} & \displaystyle{\frac{\partial v_y}{\partial y}} & \displaystyle{\frac{\partial v_y}{\partial z}}\\
\displaystyle{\frac{\partial v_z}{\partial x}} & \displaystyle{\frac{\partial v_z}{\partial y}} & \displaystyle{\frac{\partial v_z}{\partial z}}
\end{bmatrix}.
</math>

This is also known as the [[Jacobian matrix|Jacobian]] of the velocity field.


If we represent ''x'', ''y'', and ''z'' by indices 1, 2, and 3 respectively, the ''i,j'' component of the Jacobian may be written as ''∂<sub>i</sub>''&nbsp;''v<sub>j</sub>'' where ''∂<sub>i</sub>'' is shorthand for ''∂''/''∂x<sub>i</sub>''. 
-->

<!--
 '''''r''''' is specified by the velocity field '''''v'''''('''''r'''''). The velocity at a small distance ''d'''r''''' from point '''''r''''' may be written as a [[Taylor series]]:

:<math>\mathbf{v}(\mathbf{r}+d\mathbf{r}) = \mathbf{v}(\mathbf{r})+\frac{d\mathbf{v}}{d\mathbf{r}}d\mathbf{r}+\ldots,</math>

where ''d'''v'''''&nbsp;/&nbsp;''d'''r''''' is the [[gradient]] of the velocity field:

:<math>\frac{d\mathbf{v}}{d\mathbf{r}} = \begin{bmatrix}
\displaystyle{\frac{\partial v_x}{\partial x}} & \displaystyle{\frac{\partial v_x}{\partial y}} & \displaystyle{\frac{\partial v_x}{\partial z}}\\
\displaystyle{\frac{\partial v_y}{\partial x}} & \displaystyle{\frac{\partial v_y}{\partial y}} & \displaystyle{\frac{\partial v_y}{\partial z}}\\
\displaystyle{\frac{\partial v_z}{\partial x}} & \displaystyle{\frac{\partial v_z}{\partial y}} & \displaystyle{\frac{\partial v_z}{\partial z}}
\end{bmatrix}.
</math>

This is also known as the [[Jacobian matrix|Jacobian]] of the velocity field.

If we represent ''x'', ''y'', and ''z'' by indices 1, 2, and 3 respectively, the ''i,j'' component of the Jacobian may be written as ''∂<sub>i</sub>''&nbsp;''v<sub>j</sub>'' where ''∂<sub>i</sub>'' is shorthand for ''∂''/''∂x<sub>i</sub>''. 

Any matrix may be written as the sum of an [[antisymmetric matrix]] and a [[symmetric matrix]], and this decomposition is independent of coordinate system, and so has physical significance. The velocity field may be approximated as:

:<math>v_j(\mathbf{r}+d\mathbf{r}) = v_j(\mathbf{r})+\frac{1}{2}\left(\partial_i v_j-\partial_j v_i\right)dr_i + \frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right)dr_i,</math>

where [[Einstein notation]] is now being used in which repeated indices in a product are implicitly summed. The second term from the right is the asymmetric part of the first derivative term, and it represents a rigid rotation of the fluid about '''''r''''' with angular velocity ''ω'' where:

:<math>\omega=\frac12 \mathbf{\nabla}\times \mathbf{v}=\frac{1}{2}\begin{bmatrix}
\partial_2 v_3-\partial_3 v_2\\
\partial_3 v_1-\partial_1 v_3\\
\partial_1 v_2-\partial_2 v_1
\end{bmatrix}.
</math>

 The remaining symmetric term is responsible for the viscous forces in the fluid, and is known as the strain rate tensor:

:<math>E_{ij}=\frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right)</math>


The strain rate tensor is the symmetric part of the velocity gradient:
:<math>E_{ij} = \frac12(\partial_j v_i + \partial_i v_j)</math>
where <math>v_i</math> is the component of the velocity parallel to [[coordinate axis|axis]] <math>i</math>; <math>\partial_j</math> denotes the derivative with respect to the space coordinate<math>x_j</math>; and all these derivatives are evaluated at the point <math>p</math> and time <math>t</math>.

The antisymmetric part of the velocity gradient, on the other hand, describes rotation of the material about the point as a rigid body, and therefore does not contribute to the viscous stress.

In a Newtonian fluid, the viscosity tensor is constant, independent of the velocity field and the stress.

-->

The strain rate tensor <math>E</math> can be broken down in a coordinate-independent ways as the sum of a scalar tensor that measures the rate of uniform expansion, and a symmetric tensor that, having zero [[trace (mathematics)|trace]], implies no change in the volume and therefore measures the rate of shear deformation:
<!--

:<math>
E_{ij} = \frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right) 
= 
\underbrace{\frac{1}{3}(\sum_k\partial_k v_k)\delta_{ij}}_{\text{rate-of-expansion tensor}}
+
\underbrace{\left(\frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right)-\frac{1}{3}(\sum_k\partial_k v_k)\delta_{ij}\right)}_{\text{rate-of-shear tensor}}
</math>

In a fluid that is [[isotropic]] as well as Newtonian, the viscous stress tensor <math>\epsilon</math> too separates into a linear combination of these two strain rate tensors:
:<math>\epsilon_{ij} = \zeta(\sum_k \partial_k v_k)\delta_{ij} +
\mu\left(\partial_i v_j+\partial_j v_i-\frac{2}{3}(\sum_k \partial_k v_k)\delta_{ij}\right),
</math>
-->

<!--FROM SECTION OF [[viscosity]] TO BE MERGED in [[viscous stress tensor]]:

{{details3|[[Hooke's law]] and [[strain tensor]]|an analogous development for linearly elastic materials}}

Viscous forces are the result of relative motion between elements of the fluid, and so are expressible as a function of the velocity field. In other words, the forces at '''''r''''' are a function of '''''v'''''('''''r''''') and all derivatives of '''''v'''''('''''r''''') at that point. In the case of linear viscosity, the viscous force will be a function of the Jacobian [[tensor]] alone. For almost all practical situations, the linear approximation is sufficient.

The forces in the fluid are due to the velocities of the individual molecules. The velocity of a molecule may be thought of as the sum of the fluid velocity and the thermal velocity. The viscous stress tensor described above gives the force due to the fluid velocity only. The force on an area element in the fluid due to the thermal velocities of the molecules is just the hydrostatic [[pressure]].
:<math>\sigma_{ij} = -p\delta_{ij}+\sigma_{\text{visc};ij}.\,</math>
-->


<math>\alpha</math> is the 
Attenuation is expressed in [[neper]] per meter in this equation.  This remarkable law does not contain unknown or unmeasurable parameters.

 

There has been substantial theoretical development in this field since Stokes’ pioneering work. It has brought one important correction to the Stokes law. It turns out that in addition to the [[dynamic viscosity]] the parameter of [[volume viscosity]] η<sup>v</sup> also affects the total attenuation according to the following relationship:

:<math> \alpha = \frac{2 (\eta+\eta^v)\omega^2}{3\rho V^3}</math> 

The parameter [[volume viscosity]] is surprisingly little known despite its fundamental role for [[fluid dynamics]] at high [[frequencies]]. This parameter appears in [[Navier-Stokes]] equation if it is written for [[compressible fluid]], as described in the most books on general hydrodynamics, and the acoustics,. 

Indeed, many rheological texts just assume the fluid to be incompressible and the volume viscosity therefore plays no role. 

The only values for the volume viscosity of simple Newtonian fluids known to us come from the old Litovitz and Davis review. They report a ''volume viscosity'' of water at 15 C<sup>o</sup> equals 3.09 [[centipoise]]

:
http://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Hookeslaw.ogg/Hookeslaw.ogg.360p.webm


    \epsilon_{1\,1}& \epsilon_{1\,2}& \epsilon_{1\,3}\\ 
    \epsilon_{2\,1}& \epsilon_{2\,2}& \epsilon_{2\,3}\\ 
    \epsilon_{3\,1}& \epsilon_{3\,2}& \epsilon_{3\,3}

    \sigma_{1\,1}& \sigma_{1\,2}& \sigma_{1\,3}\\ 
    \sigma_{2\,1}& \sigma_{2\,2}& \sigma_{2\,3}\\ 
    \sigma_{3\,1}& \sigma_{3\,2}& \sigma_{3\,3}
:
:<math>
  F = \begin{bmatrix} F_1\\ F_2 \\ F_3 \end{bmatrix}
  \quad\quad\quad\quad 
  X = \begin{bmatrix} X_1\\ X_2 \\ X_3 \end{bmatrix}
</math>
Then

:<math>\kappa = 
  \begin{bmatrix} 
    \kappa_{1\,1}& \kappa_{1\,2}& \kappa_{1\,3}\\ 
    \kappa_{2\,1}& \kappa_{2\,2}& \kappa_{2\,3}\\ 
    \kappa_{3\,1}& \kappa_{3\,2}& \kappa_{3\,3}
  \end{bmatrix},
</math>