Uniaxial extensional flow for a viscoelastic model that displays thixotropic yield stress behavior

Abstract

Homogeneous uniaxial extensional flow of a viscoelastic fluid, namely, the partially extending strand convection model combined with a Newtonian solvent, is investigated for large relaxation time. Initial value problems are addressed, for prescribed constant tensile stress. The limit of large relaxation time introduces a slow time scale of evolution, in addition to a fast time scale for flow. Numerical solutions of the original equations show distinct stages of evolution, which are mathematically analyzed with asymptotic analyses for multiple time scales. We discuss the stages of evolution from equilibrium, as well as unloading the applied stress from a yielded solution. The overall picture which emerges captures a number of features which are usually associated with thixotropic yield stress fluids, such as delayed yielding, and hysteresis for up and down stress ramping. Even at large applied tensile stress, there is persistence of an interval of parameters where the deformation rate increases quickly, only after a delayed response.

Appendix

Intersection of the initial fast curve with the slow curve

This section provides the calculation that leads to the statement at the end of the section titled “Fast dynamics,” for the initial fast curve to intersect the slow manifold. When the intersection occurs, the solution stays unyielded, and otherwise it yields. The initial fast curve intersects a slow curve when $\kappa =0$ or $\tau =\psi (s(t))(C_{11}(t)-C_{22}(t))$, where $C_{11}=C_{22}^{-2}$. This requires $\tau =\frac {1-C_{22}^{3}}{1+2C_{22}^3+\alpha C_{22}^{2} }$, which simplifies to

$$\begin{array}{@{}rcl@{}} &&{}r(C_{22}) = 0,\\ &&{}r(C_{22})=C_{22}^{3}(1+2\tau) +\tau\alpha C_{22}^2+\tau -1. \end{array} $$

(35)

Thus, given $\tau >0$ and $\alpha >-3$, we want to know if Eq. 35 is satisfied before $C_{22}$ hits 0.

We observe several properties about the curve $r(C_{22})$ vs. $C_{22}$: 1. $r(0)=\tau -1=\begin {cases} >0 &\text {if} \tau >1 \\ <0 & \text {if} \tau <1. \end {cases}$ 2. $r(1)=(3+\alpha )\tau >0$. 3. $r(-\infty )<0$.

Case $\tau <1$ has a root for $0<C_{22}<1$

Suppose $\tau <1$. Since $r(1)>0$ and $r(0)<0$, there is at least one real root for $0<C_{22}<1$. There is at least one real root in this interval. Since $r(-\infty )<0$ and $r(0)<0$, there are either no negative roots, or two negative roots; however, we are not concerned with negative roots. Figure 10a summarizes the behavior of $r(C_{22})$ for $\tau <1$ and $\alpha <0$. Figure 10b shows the two possible cases for $\alpha >0$.

Fig. 10

Schematic of $r(C_{22})$ vs. $C_{22}$, defined in Eq. 35. a $\tau <1$, $\alpha <0$. b $\tau <1$, $\alpha >0$. Two possibilities. In both, there is a transition from the fast curve to a slow curve at the nearest zero of $r(C_{22})$

Case $\tau >1$ has no roots for $\alpha >0$

If $\tau >1$, then $r(\infty )<0$ and $r(0)>0$ so there is at least one negative root. Since $r(1)>0$, there are either two real roots for $0<C_{22}<1$ or none. To proceed, we use further properties of $r(C_{22})$: 4. The extrema are at $r'=0$: $C_{22}=0,C_{22e}$, where

$$\begin{array}{@{}rcl@{}} C_{22e}= \frac{-2\tau\alpha}{3(1+2\tau)}. \end{array} $$

(36)

Since $\alpha >-3$, we have $\frac {-\alpha }{3}<1$, and together with $\frac {2\tau }{1+2\tau }<1$, we see that $C_{22e}<1$. If $\alpha >0$, then $C_{22e}<0$ so that the local extrema are at 0 and a negative value. In this case, the curve has a positive slope from $C_{22}=1$ down to 0 where $r(0)>0$, so that there are no zeros in the interval $0<C_{22}<1$. This is shown in the schematic of Fig. 11a. 5. Since $r''(0)=2\tau \alpha$, $C_{22}=0$ is a local minimum if $\alpha >0$ (see Fig. 11a). If, on the other hand, $\alpha <0$, then $C_{22}=0$ is a local maximum. 6. Since $r''( C_{22e})=-2\tau \alpha$, this is a local maximum if $\alpha >0$ (Fig. 11a) and a local minimum if $-3<\alpha <0$. In addition, we see that $r( C_{22e})= \tau -1 + \frac {4 \alpha ^{3} \tau ^{3}}{27 (1 + 2 \tau )^{2}}$. Thus, if $\tau >1$ and $\alpha >0$, then $r (C_{22e})>0$, and this means that there are no roots of the cubic equation between 0 and 1; therefore, the flow yields. This is the case of Fig. 11a.

Fig. 11

Schematic of $r(C_{22})$ vs. $C_{22}$, defined in Eq. 35. a $\tau >1$, $\alpha >0$. The fast curve does not meet the slow curve, but proceeds to yielding. b $\tau >1$, $\alpha <0$. Two possibilities. There is a root if $r(C_{22e})<0$ (solid line), where $C_{22e}$ is the local extremum, so the solution lands on a slow manifold. There is no root (dashed line) if $r(C_{22e})>0$ so the solution yields

Fig. 12

The curve $r(C_{22e})=0$ in the $\tau$ vs. $\alpha$ plane, where $C_{22e}$ is the local extremum of $r(C_{22})=0$. To the left of the curve, the fast curve intersects the slow curve. To the right of the curve, the fast curve does not intersect the slow curve. The vertical dashed line represents $\tau =1$

However, if $\alpha <0$, then the last term in $r(C_{22e})$ is ${\frac {- \alpha }{3}^{3}} \frac {-2\tau }{(1 + 2 \tau )^{2}} > 0$. Therefore, for the case $\tau >1$, the value of r at the local minimum could be positive or negative. If $r(C_{22e})<0$, then there is a root of $r(C_{22})=0$ for $0<C_{22}<1$ and the fast curve meets a slow curve. Figure 12 shows the case $\alpha <0$ and $\tau >1$. The $\alpha$ vs. $\tau$ plane is cut by the curve $r(C_{22e})=0$. The region where $r(C_{22e})<0$ is to the left of the curve, and this is the region where the initial fast curve meets a slow curve. The initial fast dynamics chooses the root of $r(C_{22})=0$ closest to 1. There are two possibilities in Fig. 10b: the dashed line corresponds to the right of the curve in Fig. 12 and the solid line to the left of the curve.