# Material Properties

We define three material properties which we use for the flow fields describe above. For flows with $a \le 0$, these material properties are given by:

$\eta = - \frac{\tau_{yx}}{\dot{\gamma}}$ the viscosity function

$\Psi_1 = -\frac{(\tau_{xx} - \tau_{yy})}{\dot{\gamma}^2}$ the first-normal stress coefficient

$\Psi_2 = -\frac{(\tau_{yy} - \tau_{zz})}{\dot{\gamma}^2}$ the second-normal stress coefficient

For flows with $a > 0$, these material properties are given by:

$\eta = - \frac{\tau_{yx}}{\dot{\gamma}}$ the viscosity function

$\Psi_1 = -\frac{(\tau_{xx} - \tau_{yy})}{\dot{\gamma}}$ the first-normal stress coefficient

$\Psi_2 = -\frac{(\tau_{yy} - \tau_{zz})}{\dot{\gamma}}$ the second-normal stress coefficient

The difference is only in the power on the flow rate for $\Psi_1$ and $\Psi_2$ which has been adjusted so that for planar elongation we recover the traditional definitions of the elongational material functions. The average end-to-end distance of the chain can also be calculated using the following equation:

$=<(\textbf{r}_N-\textbf{r}_1)\cdot(\textbf{r}_N-\textbf{r}_1)>$

This give the averaged square distance from the first to the last bead in the chain.