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1-6stressstrain.tex,
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\section{Relations between stress and rate-of-strain  tensors}
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When the fluid is at rest on a  macroscopic scale, no tangential stress
acts on a surface.  There is only the normal stress, i.e., the pressure
$-p \delta_{ij} $ which is thermodynamic in origin, and is maintained by
molecular collisions.   Denoting the additional stress by $\tau_{ij}$
which is due to the relative motion on the continuum scale, 
\be 
\sigma_{ij} =   -p \delta_{ij} +   \tau_{ij}  
  \label{Eq:6.1} \ee 
The second  part is called the viscous
stress $\tau_{ij}$  and must depend on gradients of velocity,
\[ \f{\p q_i}{\p x_j} \, , \f{\p^2 q_i}{\p x_j\p x_k} \cdots . \]

\subsection{Newtonian fluid}
For many fluids in nature such as air and water,  the relation between $\tau_{ij}$ and  $\p q_i/\p x_i$ are linear under most circumstances. Such fluids are called Newtonian. For second-rank tensors, the most general linear relation is, 
\be \tau_{ij} = C_{ij\ell m} \f{\p q_{\ell}}{\p x_m} . \label{Eq:6.2} \ee
where $C_{ij\ell m}$ is a coefficient tensor of rank 4. In principle
there are $3^4=81$ coefficients. It can be shown (Spain, {\it Cartesian
Tensors}) that in an isotropic fluid the fourth-rank tensor is of the
following form:
\be C_{ij\ell m} = \lambda\delta_{ij}\delta_{\ell m}+
\mu(\delta_{i\ell}\delta_{jm} + \delta_{im}\delta_{j\ell} ) \ee
Eighty  one coefficients in $C_{ij\ell m}$ reduce to two:
$\lambda$ and
$\mu$, and 
\be \tau_{ij} = \mu \lp \f{\p q_i}{\p x_j} + \f{\p q_j}{\p x_i} \rp +
\lambda \f{\p q_{\ell}}{\p x_{\ell}} \delta_{ij} . \label{Eq:5.8} \ee
where $\mu$, $\lambda  $ are viscosity coefficients depending empirically
on temperature.  

Note that the velocity gradient is made up of two parts
\[
\f{\p q_i}{\p x_j} =  \f{1}{2} \lp \f{\p q_i}{\p x_j} + \f{\p q_j}{\p
x_i} \rp + \f{1}{2} \lp \f{\p q_i}{\p x_j} - \f{\p q_j}{\p x_i}
\rp   \]
where 
\be  e_{ij} =  \f{1}{2} \lp \f{\p q_i}{\p x_j} + \f{\p q_j}{\p
x_i} \rp\ee
is the rate of strain tensor, and 
\be  \Omega_{ij}=  \f{1}{2} \lp \f{\p q_i}{\p x_j} - \f{\p q_j}{\p x_i}
\rp \ee is the vorticity tensor.  
Note also that (\ref{Eq:5.8}) depends only on the rate of strain but not 
on vorticity. This is reasonable since a fluid in rigid-body rotation should not experience any viscous stress. In a  rigid-body
rotation with angular velocity
$\omega$,  
the fluid velocity is 
\[ \vec{q} = \vec{\omega} \times \vec{r} \qquad q_i = 
\left| \begin{array}{ccc}
\vec{i} & \vec{j} & \vec{k} \\ \omega_1 & \omega_2 & \omega_3 \\ x_1 &
x_2 & x_3 \end{array} \right| \] The vorticity components are not zero;
for example, 
\begin{eqnarray*} 2\Omega_{12} = \f{\p q_1}{\p x_2} - \f{\p q_2}{\p x_1}
& = & \f{\p}{\p x_2} \lp \omega_2 x_3 - \omega_3 x_2 \rp - \f{\p}{\p x_1}
\lp \omega_3 x_1 - \omega_1 x_3 \rp = -2\omega_3   .
\end{eqnarray*} Hence
$\tau_{ij}$ cannot depends on  $\Omega_{ij}$ and only on $e_{ij}$.



The trace of $\tau_{ij}$ is
\[ \tau_{ii} = (2\mu + 3\lambda ) \f{\p q_i}{\p x_i} = (3\lambda + 2\mu
) \nabla \cdot \vec{q} . \] where $k = 3\lambda + 2\mu =$ is called the
bulk viscosity.

For incompressible fluids $\nabla \cdot \vec{q} = 0$; the viscous stress
tensor is 
\be \tau_{ij} = \mu \lp \f{\p q_i}{\p x_j} + \f{\p q_i}{\p x_i} \rp
\label{Eq:5.9} \ee The total stress tensor is therefore
\be \sigma_{ij} = -p \, \delta_{ij} + \mu \lp \f{\p q_i}{\p x_j} + \f{\p
q_j}{\p x_i} \rp \label{Eq:6.10} \ee
 
The governing equations    for an incompressible Newtonian fluid may now be summarized:
\be \f{D \rho}{D t } = 0, ~~~\mbox{(incompressibility)}\label{Eq:6.12}\ee
\be \f{\p q_i}{\p x_i} = 0, ~~~\mbox{(continuity)} \label{Eq:6.13}\ee
\be   
\rho \f{Dq_i}{Dt} = -\f{\p p}{\p x_i} + \mu \f{\p^2 q_i}{\p x_j \p x_j}   +
\rho f_i, ~~~\mbox{(momentum conservation)} \label{Eq:6.11} \ee
 after using continuity. The last equation (\ref{Eq:6.11})\, and sometimes
the  set of equations (\ref{Eq:6.11}), (\ref{Eq:6.12}) and
(\ref{Eq:6.13}),  is called the Navier-Stokes equation(s). Now we have just five scalar 
equations for five unknowns $\rho, p, $ and $q_i$. Boundary and initial conditions must be further specified. For example on the surface of a stationary rigid body, no slippage is allowed, so that 
\be q_i=0, ~~~\mbox{on a rigid stationary surface}\label{eq:noslip}\ee

\subsection{Non-Newtonian fluids}
Many fluids such as toothpaste, gel, honey, heavy oil (DNAPL), etc., flows like a fluid if  the shear stress is  above a critical value,  and behaves like a solid if below.  Of  geological interest is the mud which is a mixture of water with highly cohesive clay particles. From  volcanic eruption, lava  can mix with rain,   melting snow, or  lake water to form     mud, which flows down the hill slope, carries along stones, trees and other debris, to cause  severe damages. In some mountainous areas,   heavy rainfall infiltrates the top soil  and causes mud to slide and flow  into rivers to form hyper concentrated fluid-mud.    
These fluids  are called non-Newtonian since  the relation between $\sigma_{ij}$ and $e_{ij}$ is nonlinear.

We shall only discuss  a special model of non-Newtonian fluid, i.e., the  Bingham plastic model. For simple shearing flow $u=u(y)$, the constitutive relation for a Bingham plastic is 
\begin{eqnarray} \f{\p u}{\p y} &=& 0 ,\quad \tau\leq \tau_c; \\ \nonumber
  \f{\p u}{\p y} &=& \f{1}{\mu}(\tau - \tau_c), \quad \tau>\tau_c\end{eqnarray}
where $\tau_c$ is called the yield stress and $\mu$ the  Bingham viscosity, both of which  depend on the clay concentration $C$. 


In  three dimensions,   the Bingham model can be generalized by   introducing   the second invariants of the stress and rate-of-strain tensors. 
The second invariant of the viscous stress tensor is  
\begin{eqnarray} \lefteqn{II_T \equiv\f{1}{2}
\lb \tau_{ij}\tau_{ij} -\lp\tau_{kk}\rp^2 \rb}\\ \nonumber
&& =\tau^2_{12}+\tau^2_{23}+\tau^2_{31}-\lp\tau_{11}\tau_{22} + 
\tau_{11}\tau_{33} + \tau_{22}\tau_{33}\rp \end{eqnarray}
Similarly the second invariant of the rate of strain tensor is 
\begin{eqnarray} \lefteqn{II_E \equiv\f{1}{2}
\lb e_{ij}e_{ij} -\lp e_{kk}\rp^2 \rb}\\ \nonumber
&& =e^2_{12}+e^2_{23}+e^2_{31}-\lp e_{11}e_{22} + 
e_{11}e_{33} + e_{22}e_{33}\rp\\ \nonumber
&& =    \f{1}{4}\lb \lp \f{\p u}{\p y} + \f{\p v}{\p x} \rp^2+ \lp \f{\p v}{\p z} + \f{\p w}{\p y} \rp^2+ \lp \f{\p w}{\p x} + \f{\p u}{\p z} \rp^2
\rb\\ 
\nonumber
&& -\lp \f{\p u}{\p x}\f{\p v}{\p y}+ \f{\p u}{\p x} \f{\p w}{\p z}+ \f{\p v}{\p y}\f{\p w}{\p z}\rp \\\nonumber
 \end{eqnarray}
The Bingham plastic law is then 
\begin{eqnarray} e_{ij} &=& 0, \quad \mbox{if}\quad \sqrt{II_T} <\tau_c,\\ \nonumber
\tau_{ij} &=& 2\mu e_{ij} + \tau_c\f{e_{ij}}{\sqrt{II_E}}, \quad \mbox{if}\quad \sqrt{II_T} \geq \tau_c.\end{eqnarray}
This is due to Hohenemser and Prager (1936). 


In simple shear $u=u(y)$, the only non-zero components of $\tau_{ij}$ and $e_{ij}$ are $\tau_{xy}$ and $e_{xy}$. The Bingham law reduces to 
\begin{eqnarray} e_{xy}&=& 0, \quad \mbox{if}\quad |\tau_{xy}|<\tau_c,\\ \nonumber
\tau_{xy}&=& 2\mu e_{xy} + \tau_c, \quad \mbox{if}\quad |\tau_{xy}|\geq \tau_c. \end{eqnarray}
In other words, 
\begin{eqnarray} \f{\p u}{\p y}&=& 0, \quad \mbox{if}\quad   |\tau_{xy}|<\tau_c,\\ \nonumber
\tau_{xy}&=&  \mu \f{\p u}{\p y} + \tau_c \sgn \lp\f{\p u}{\p y} \rp, \quad \mbox{if}\quad |\tau_{xy}|\geq \tau_c. \end{eqnarray}

We shall examine some examples later. 



 

 

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