Dumbbells Hookean

If a dilute polymer solution is subjected to a imidirectional or steady flow with a velocity gradient large enough to stretch out the polymer molecule, nonlinear viscoelastic effects are observed. The simple Hookean dumbbell model, described in Section 3.4.4, can predict... 

EXTENSIONAL FLOW. In steady extensional flows, such as uniaxial extension, the single-relaxation-time Hookean dumbbell model and the multiple-relaxation-time Rouse and Zimm models predict that the steady-state extensional viscosity becomes infinite at a finite strain rate, s. With the dumbbell model, this occurs when the frictional drag force that stretches the dumbbell exceeds the contraction-producing force of the spring—that is, when the extension rate equals the critical value Sc. ... 

As indicated above, such an agreement is perhaps expected. On the other hand, it is remarkable that a rather complex phenomenological theory postulated for an LC continuum can be reconciled with an even more complex molecular theory built on the concept of intermolecular potential. Perhaps the only other such happy instance is the agreement between the continuum Oldroyd-B model for viscoelastic liquids and the molecular model based on a dilute suspension of linear Hookean dumbbells in a Newtonian solvent. ... 

H) Hookean dumbbells The beads are joined by a Hookean spring which has zero length if there are no forces acting on the beads the tension in the connector F(c) = HR, where H is a Hookean spring constant. 

Before leaving the stress-tensor expressions we should note that for the special case of Hookean dumbbells, it is possible to eliminate the quantity between Eqs. (4.19) and (4.20) and obtain an equation for r in terms of kinematic quantities — that is, a properly formulated constitutive equation (or rheological equation of state) is obtained10 ... 

It was pointed out in Eq. (4.23) that it is possible to obtain a constitutive equation for the elastic (Hookean) dumbbell from the kinetic theory equations. For rigid dumbbells such a constitutive equation has not been obtained. Eq. (20.4) above represents the only attempt so for to do this. 

In the preceding sections many results have been presented for the bead-rod (rigid dumbbell) suspensions these results were obtained by solving the equation for the distribution function and then calculating the components of the stress tensor. It was pointed out in Eq. (4.23) for bead-spring (Hookean dumbbell) suspensions, that there is a constitutive equation which can be used to calculate the stresses directly without any need for finding the distribution function. Hence obtaining the Hookean dumbbell suspension results presents no difficulty. 

Time constant for Hookean dumbbell model Time constants for Rouse chain model Solvent contnbution to thermal conductivity Tensor virial multiplied by 2 Momentum space distribution function Integration variable in Taylor series Stress tensor (momentum flux tensor) External force contribution to stress tensor Kinetic contribution to stress tensor Intramolecular contribution to stress tensor Intermolecular contribution to stress tensor Fluid density... 

Dilute Solutions of Hookean Dumbbells at Constant Temperature... 

We now give the solution to Eq. (13.17) for the simplest possible model, namely Hookean dumbbells, for which there is but one element in the D-matnx and it is identically equal to zero. For this equation we can postulate a Gaus-sian-form solution ... 

Once the singlet distribution function has been found, we are in a position to evaluate the various contributions to the fluxes that depend on (see Table 1). In this section we discuss the contnbutions to the stress tensor, and in the next two sections the contnbutions to the mass and heat flux vectors. In these sections, for illustrative purposes, we restrict ourselves to the Rouse bead-spring chain and the Hookean dumbbell models, for which we can use the singlet distribution functions , given in Eqs. (13.5) and (13.8). 

This is the same as DPL, Eq. (15.3-17). Thus the stress tensor is given in terms of the Finger strain tensor via Eq. (13.10). The polymer contribution to the stress tensor for the Hookean dumbbell model is obtained by replacing in Eq. (14.12) by the a tensor defined m Eq. (13.7)... 

For isothermal systems it i (2) is zero. For nonisothermal solutions of Hookean dumbbells Eq. (13.8) can be used to show that ... 

For isothennal Rouse chains this contribution is zero. For Hookean dumbbells in nonisothermal solutions it follows from Eq. (13.18) that n (2) is zero. 

The third-order terms for the contributions to the stress tensor are, for Hookean dumbbells ... 

The Constitutive Equation for an Isothermal Solution of Hookean Dumbbells in a System with Concentration Gradients... 

We begin by writing Eq. (13.4) for Hookean dumbbells, multiplying this equation by — HQQ, and integrating over Q Then, when Eqs. (14.10) and (14.16), appropriately simplified for N = 2 and isothermal systems, are used, we find that - through terms of third order ... 

---