Without using the Stokes assumption, the stress tensor \(T_{ij} = -p \delta_{ij} + 2\mu S_{ij} + \lambda S_{mm} \delta_{ij}\) is:
\[
T_{ij} = -p\delta_{ij} + \tau_{ij} = -p\delta_{ij} + 2\mu \left(S_{ij} - \frac{1}{3}S_{mm}\delta_{ij} \right) + \mu_v S_{mm}\delta_{ij}
\]
This linear relation between \(\mathbf{T}\) and \(\mathbf{S}\) is consistent with Newton’s definition of the viscosity coefficient \( \mu \) in a simple parallel flow \( u(y) \),
for which the equation above gives a viscous shear stress of \( \tau = \mu \left( \frac{du}{dy} \right) \). A fluid obeying this equation is called a Newtonian fluid where \( \mu \) and \( \mu_v \) may only depend on the local thermodynamic state.
The linear Newtonian friction law might only be expected to hold for small strain rates since it is essentially a first-order expansion of the stress in terms of \( S_{ij} \) around \( T_{ij} = 0 \). However, the linear relationship is surprisingly accurate for many common fluids such as air, water, gasoline, and oils.
Yet, other liquids display non-Newtonian behavior at moderate rates of strain.
shear stress in a non-Newtonian flow may be a nonlinear function of the local strain rate.
The simplest model of this behavior for a unidirectional shear flow \( \mathbf{u} = (u_1(x_2), 0, 0) \) is a power law
\[
\tau_{12} = \eta \frac{\partial u_1}{\partial x_2} = \left(m \dot{\gamma}^{n-1} \right) \frac{\partial u_1}{\partial x_2} = m \dot{\gamma}^n
\]
where \( \eta = m\dot{\gamma}^{n-1} \) is the non-Newtonian viscosity, \( m \) is the power law coefficient,
\( \dot{\gamma} = \partial u_1 / \partial x_2 \) is the shear rate, and \( n \) is the power law exponent.
For a Newtonian fluid, \( n = 1 \) and \( m \) is the fluid's viscosity
When \( n < 1 \): shear thinning or pseudoplastic (viscosity drops with increasing strain rate)
When \( n > 1 \): shear thickening or dilatant
The current stress on a non-Newtonian fluid particle may depend on the local strain rate and on its history.These memory effects give the fluid some elastic properties that may allow it to mimic solid behavior over short times.
Such fluids are called viscoelastic substances.
For linear viscoelastic materials, the general linear stress-strain rate law \(\tau_{ij} = K_{ijmn} S_{mn}\) is replaced by
\[
\tau_{ij}(t) = \int_{-\infty}^{t} K_{ijmn}(t - t') S_{mn}(t') \, dt'
\]
where \( K_{ijmn} \) is the tensorial relaxation modulus and the integral accounts for the strain-rate history
Finally, flowing non-Newtonian fluids may develop normal stresses that do not occur in Newtonian fluids.
For example, in simple shear flow \( \mathbf{u} = (u_1(x_2), 0, 0) \), one may observe:
A nonzero first normal-stress difference, \( T_{11} - T_{22} \)
A nonzero second normal-stress difference, \( T_{22} - T_{33} \)
In polymeric fluids, the first normal stress difference is typically negative, indicating tensile stress along streamlines.
The second is typically positive and smaller in magnitude. These phenomena explain extrudate swell and rod climbing in polymeric fluids
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