Steady Elongational Flow

For steady elongational flow (ic = constant) it is conventional to define a quantity tf, called the elongational viscosity (or Trouton viscosity ), as 

We assume that the forces acting on the boundaries of the material, other than those perpendicular to the z-direetion, are zero, a) that / + ixx = 0. Then we can compute rj from  

We want now to solve Eq. (15.1) for ic = constant. The solution may be developed as a power series in 67k as follows (in doing this Eq. (15.3) is used)  

Here we have included only that portion of the term in (62ic)3 which will be needed later. 

The stress difference can be calculated from Eq. (15.2), with the d/dt-term omitted  

Hence rj is given in terms of a normal stress difference. The occurrence of the stress difference explains why this quantity was given 

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However, in steady elongational flow, the maximum separating force in the connector is obtained when the dumbbell is aligned in the direction of flow and, again, for the case of two beads in contact is given by -... 

Deformation of a Sphere in Various Types of Flows A spherical liquid particle of radius 0.5 in is placed in a liquid medium of identical physical properties. Plot the shape of the particle (a) after 1 s and 2 s in simple shear flow with y 2s1 (b) after 1 s and 2 s in steady elongational flow with e = 1 s 1. (c) In each case, the ratio of the surface area of the deformed particle to the initial one can be calculated. What does this ratio represent ... 

Miiller-Plathe F (2002) Coarse-graining in polymer simulation From the atomistic to the mesoscopic scale and back. J Chem Phys Phys Chem 3 754—769 Muller R, Picot C, Zang YH, Froelich D (1990) Polymer chain conformation in the melt during steady elongational flow as measured by SANS. Temporary network model. Macromolecules 23(9) 2577—2582... 

The reliability of the experimental procedure (deformation in the melt, quenching, characterization of orientation at room temperature) could be verified in elongation (specimens quenched for different macroscopic extensions in steady elongational flow) as well as in simple shear (good correlation of the chain dimensions measured in all principal shear directions). 

They show that for elongational flow the SLLOD equations are identical to Newton s equations of motion with the inclusion of an additional external force that must exist in order to sustain a steady elongational flow. Their derivation shows that SLLOD is the correct set of equations to use when performing NEMD simulations of elongational flow. No doubt the issue will continue to be debated in the literature for some time to come. 

Stress Relaxation after Cessation of Steady Elongational Flow... 

We next consider stress relaxation after cessation of steady elongation flow. In this case there is flow at a steady elongational rate ic0 from time — oo to time 0. At time t = 0 the elongational flow ceases, and the fluid is held stationary. For t 0, k = 0 and the stresses relax as a function of time. The equation to be solved for the distribution function tp is... 

A suspension of rigid dumbbells is a rest for t < 0. For t St 0 it undergoes a steady elongational flow with steady elongational rate ic0. Therefore, we will solve the equation... 

We imagine a solution of rigid dumbbells in a state of steady elongational flow at a constant elongation rate k for time t < 0. Then at t = 0 the stress exerted on the fluid is suddenly removed and the fluid is allowed to recoil. We assume that the fluid will recoil with the same velocity profile as it had during the elongating process, but here k will be K+(t), a function of time. 

Stress growth at inception of steady elongational flow... 

In stress growth at inception of steady elongational flow, both the rigid dumbbells and elastic dumbbells give expressions for... 

The importance of extensional flow in the mixing process has been pointed out by Gotten (16) and thoroughly analyzed by Nakajima (8) in the case of carbon black-filled compounds. A steady elongational flow can be developed only if the extensional rate increases exponentially (versus time) (17). Nakajima demonstrated that this type of deformation induces an anisotropy of the material, enhanced in highly filled compounds or containing oriented fibers. Therefore, the steady state is nearly impossible and, with polymers, the elongational flow is not a pure deformation and necessarily involves a shear component. 

In subsequent sections we will need the expressions for the a tensor of Eq. (13.21) for steady shear flow and steady elongational flow... 

That IS, in steady elongational flow the velocity gradients can have an effect on the mass flux. 

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