The relationship between the stress and deformation in a continuum is called a constitutive equation, and a linear constitutive equation between stress \( T_{ij} \) and \( \partial u_i / \partial x_j \) is examined here.
A fluid that follows the simplest possible linear constitutive equation is known as a Newtonian fluid
In a fluid at rest, there are only normal components of stress on a surface, and the stress does not depend on the orientation of the surface; the stress is isotropic.
The only second-order isotropic tensor is the Kronecker delta \( \delta_{ij} \), the Kronecker delta is defined as
\(\boxed{
\delta_{ij} =
\begin{cases}
1 & \text{if } i = j \\
0 & \text{if } i \ne j
\end{cases} }
\)
The stress in a static fluid must be of the form
\(\boxed{
T_{ij} = -p \delta_{ij}}
\)
\( p \) is the thermodynamic pressure related to \( \rho \) and \( T \) by an equation of state such as that for a perfect gas \( p = \rho R T \). The negative sign occurs because the normal components of \( \mathbf{T} \) are regarded as positive if they indicate tension rather than compression
A moving fluid develops additional stress components \( \tau_{ij} \) because of viscosity, and these stress components appear as both diagonal and off-diagonal components within \( \mathbf{T} \).
A simple extension of \(\boxed{
T_{ij} = -p \delta_{ij}}
\) that captures this phenomenon and reduces to it when fluid motion ceases is
\[
T_{ij} = -p \delta_{ij} + \tau_{ij}
\]
This decomposition of the stress into fluid-static (\( p \)) and fluid-dynamic (\( \tau_{ij} \)) contributions is approximate,
because \( p \) is only well defined for equilibrium conditions. However, molecular densities, speeds, and collision rates are typically high enough,
so that fluid particles reach local thermodynamic equilibrium conditions in nearly all fluid flows so that \( p \) in \(T_{ij} = -p \delta_{ij} + \tau_{ij}\) is still the thermodynamic pressure
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