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Material Properties

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

η=−τyxγ˙\eta = - \frac{\tau_{yx}}{\dot{\gamma}} the viscosity function

Ψ1=−(τxx−τyy)γ˙2\Psi_1 = -\frac{(\tau_{xx} - \tau_{yy})}{\dot{\gamma}^2} the first-normal stress coefficient

Ψ2=−(τyy−τzz)γ˙2\Psi_2 = -\frac{(\tau_{yy} - \tau_{zz})}{\dot{\gamma}^2} the second-normal stress coefficient

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

η=−τyxγ˙\eta = - \frac{\tau_{yx}}{\dot{\gamma}} the viscosity function

Ψ1=−(τxx−τyy)γ˙\Psi_1 = -\frac{(\tau_{xx} - \tau_{yy})}{\dot{\gamma}} the first-normal stress coefficient

Ψ2=−(τyy−τzz)γ˙\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 Ψ1\Psi_1 and Ψ2\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:

<r2>=<(rN−r1)⋅(rN−r1)><r^2>=<(\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.