Extensional rheology of a dilute particle-laden viscoelastic solution

Recent discoveries have allowed a mathematical and computational foundation for understanding the rheology of particles suspended in viscoelastic fluids. We employ these new tools to understand the extensional rheology of such suspensions. We accomplish this by first calculating the renormalized particle contribution to the extensional viscosity in such a suspension in the dilute particle limit over a wide range of extensional Weissenberg number and Hencky strain. The models we use for the suspending fluids are the simplest dumbbell models—the Oldroyd-B, FENE-P, and Giesekus models—such that our results are general for polymer solutions which exhibit strong strain hardening at values of the Weissenberg number above those which engender the coil-stretch transition, $\mathrm{Wi}\ge 0.5$. We demonstrate that the effect of particles on the “extra elongational viscosity” relative to the fluid is nonmonotonic in strain (increasing for small strain and then decreasing for large strain). Thus at a fixed strain, the particle “extra viscosity” relative to the fluid may increase or decrease with Wi. We demonstrate that this interesting behavior is due to the interplay between the two contributions of the “particle-induced fluid stress” (PIFS) and the “stresslet” to the extra viscosity. The contribution of the particle-induced fluid stress to the suspension viscosity increases at small strain but plateaus and then decreases at higher values of the strain. Thus, at small strain the local velocity gradients near a particle increase the polymer stretch, while for greater strain, polymers which have undergone the coil-stretch transition collapse in the neighborhood of a given particle. On the other hand, the stresslet contribution to the viscosity relative to the fluid decreases monotonically as the polymer stretch surrounding a given particle “shields” the particle and thus reduces the local surface tractions. Beyond Hencky strain of approximately 2 the decreasing value of the stresslet coupled with the plateauing of the PIFS, causes the overall reduction in the particle-induced extra viscosity relative to that of the fluid.

Figure 1

Comparison of small strain theory with transient simulations as a function of strain for (a) stresslet and (b) particle-induced fluid stress. The parameter $\beta =0.68$ for the Oldroyd-B model used in simulations.

Figure 2

Ratio of particle viscosity to suspending fluid viscosity with strain for a range of Weissenberg numbers for the FENE-P, Oldroyd-B, and Giesekus models. The parameters of the FENE-P model are $\beta =0.68,L=100$, of the Oldroyd-B model are $\beta =0.68$, and of the Giesekus model are $\alpha ={10}^{-3},\beta =0.68$. The inset also shows the ratio of particle viscosity to fluid viscosity for the Oldroyd-B model and Wi = 1. The two curves correspond to two different $\beta$ values. The $\beta$ value for the dotted curve is equal to 0.68 and the $\beta$ value for the dashed curve is equal to 0.4. We see that $\beta$ does not have a significant effect on the ratio.

Figure 3

Ratio of PIFS and stresslet to suspending fluid viscosity with strain for different Wi for the FENE-P model. The parameters of the FENE-P model used in simulations are $\beta =0.68,L=100$.

Figure 4

Eulerian contours of $C_{kk}-C_{kk}^{f0}$ in the $x_1\text{−}{x}_2$ plane for FENE-P model parameters $L=100,\beta =0.68,\text{Wi}=0.8$ and strain $\epsilon =2$ in panel (a) and $\epsilon =4$ in panel (b).

Figure 5

Evolution of polymer conformation trace along the same streamline at different starting times for $\mathrm{Wi}=0.8$. Inset shows streamline plotted on top of discriminant $D$ for uniaxial extension flow past a sphere with $D<0$ indicating extension-dominated and $D>0$ indicating rotation-dominated regions.