A. Shear flow

In this final section we calculate the viscosity of a dilute polymer solution.   Shear flows, for which the velocity components are given by

$$v_{\alpha }\left( \vec{r},t\right) =\sum_{\beta }\kappa _{\alpha \beta }(t)r_{\beta }$$ (6.48)

are commonly used for studying viscoelastic  properties. If the shear rates  $\kappa _{\alpha \beta }\left( t\right)$ are small enough, the stress tensor  depends linearly on $\bar{\kappa}\left( t\right)$ and can be written as

$$S_{\alpha \beta }\left( t\right) =\int_{-\infty }^{t}d\tau G(t-\tau )\kappa _{\alpha \beta }\left( \tau \right)$$ (6.49)

where G(t) is called the shear relaxation modulus .

An important special case is a stepwise shear  flow, which is switched on at t=0:

   

$$\Theta (t) \dot{\gamma}r_{y}$$ v_x(t) = (6.50)

v_y(t) = 0 (6.51)

v_z(t) = 0 (6.52)

where $\dot{\gamma}$ is called the shear rate and $\Theta$ is the Heaviside function ( $\Theta (t)=0$ for t<0, $\Theta (t)=1$ for $t\ge 0$). From Eqs. ( 5.13) and (6.49) we see that

$$\eta =\lim_{t\rightarrow \infty }\frac{S_{xy}(t)}{\dot{\gamma... ...t-\tau \right) =\int_{0}^{\infty }d\tau G\left( \tau \right) .$$ (6.53)

The limit $t\rightarrow \infty$ must be taken because during the early stages elastic stresses are built up.

The calculation now consists of two steps. First we shall bring Eq. (5.64) into a form applicable to the present case. In the second step we shall formulate the Langevin approach of section 6.3 in the case when the system is under shear, and calculate the stress tensor.