Polymer Solutions

Abstract

In the microstructure approach to the quest for a relevant constitutive equation for the complex-structure fluid, a relevant model for the microstructure is postulated, and the consequences of the micromechanics are then explored at the macrostructural level, with appropriate averages being taken to smear out the details of the microstructure.

Problems

Problem 7.1

Use the solution (7.24) in (7.27) to show that

$$ \mathbf {D}=kT\varvec{\zeta }^{-1}. $$

This is the Stokes–Einstein relation, relating the diffusivity to the mobility of a Brownian particle.

Problem 7.2

Starting from the Langevin equation in configuration space, in the limit $\mathbf {m}\rightarrow \mathbf {0},$

$$\begin{aligned} {\dot{\mathbf x}}=-\varvec{\zeta }^{-1}\cdot \mathbf {Kx}+\varvec{\zeta }^{-1}\mathbf {F }^{(b)}\left( t\right) , \end{aligned}$$

(7.87)

show that

$$\begin{aligned} \varDelta \mathbf {x}\left( t\right) =-\varvec{\zeta }^{-1}\cdot \mathbf {Kx} \varDelta t+\int _{t}^{t+\varDelta t}{\varvec{\zeta }^{-1}\mathbf {F}^{(b)}\left( { t^{\prime }}\right) dt^{\prime }}. \end{aligned}$$

(7.88)

From this, show that

$$\begin{aligned} \left\langle {\varDelta \mathbf {x}}\right\rangle =-\varvec{\zeta }^{-1}\cdot \mathbf {Kx},\;\;\;\left\langle {\varDelta \mathbf {x}\left( t\right) \varDelta \mathbf {x}\left( t\right) }\right\rangle =2kT\varvec{\zeta }^{-1}\varDelta t \end{aligned}$$

(7.89)

and conclude that the Fokker–Planck equation is

$$ \frac{{\partial \phi }}{{\partial t}}=\frac{\partial }{{\partial \mathbf {x}}} \cdot \left[ {kT\varvec{\zeta }^{-1}\frac{{\partial \phi }}{{\partial \mathbf {x}}}+\varvec{\zeta }^{-1}\cdot \mathbf {Kx}\phi }\right] . $$

Problem 7.3

Investigate the migration problem in a plane Poiseuille flow.

Problem 7.4

Show that the solution to (7.56) is

$$\begin{aligned} \left\langle {\mathbf {R}\left( t \right) } \right\rangle = e^{ - t/2\lambda } e^{\mathbf {L}t} \mathbf {R}_{0} . \end{aligned}$$

(7.90)

Thus conclude that the flow is strong if

$$ \text {eigen }\left( \mathbf {L} \right) \ge 1/2\lambda , $$

where eigen ($\mathbf {L}$) is the maximum eigenvalue of $\mathbf {L}$.

Problem 7.5

Using the result (7.70), show that the following solves the Maxwell equation (7.62):

$$ \mathbf {S}^{(p)}\left( t\right) =\frac{G}{\lambda }\int _{-\infty }^{t}{ e^{(s-t)/\lambda }\mathbf {C}_{t}\left( s\right) ^{-1}ds}=G\mathbf {I}+\varvec{ \tau }^{(p)}. $$