Bistability and metabistability scenario in the dynamics of an ellipsoidal particle in a sheared viscoelastic fluid

The motion of an ellipsoidal particle in a viscoelastic liquid subjected to an unconfined shear flow is addressed by numerical simulations. A complex dynamics is found with different transients and long-time regimes depending on the Deborah number $\text{De}$ ($\text{De}$ is the product of the viscoelastic liquid intrinsic time times the applied shear rate). Spiraling orbits toward a log-rolling motion around the vorticity are observed for low Deborah numbers, whereas the particle aligns with its major axis near to the flow direction at high Deborah numbers. The transition from vorticity to flow alignment is characterized by a periodic regime with small amplitude oscillations around orientations progressively shifting from vorticity to flow direction by increasing $\text{De}$. A range of Deborah numbers is detected such that the periodic solution coexists with the flow alignment regime (bistability). A further range of $\text{De}$ is found where flow alignment is attained differently for particles starting far from or next to the shear plane: in the latter case, very long transients are found; hence an effective bistability (metabistability) is predicted to occur in a large time lapse before reaching the fully aligned state. Finally, the computed Deborah number values for flow alignment favorably compare with available experimental data.

Figure 1

Schematic representation of the problem investigated in this work. On the right, the generated flow field without the particle (simple shear flow) is reported.

Figure 2

Orbits described by the orientation vector for an ellipsoid with aspect ratio $\mathrm{AR}=4$ and $\mathrm{De}=1$ (a and b) and $\mathrm{De}=3$ (c and d). In the leftmost panels, the initial orientation vector (red circles) is $(p_{x,0},p_{y,0},p_{z,0})=(0,0.31,0.95)$ whereas in the rightmost panels it is $(p_{x,0},p_{y,0},p_{z,0})=(0,0.95,0.31)$.

Figure 3

Orbits described by the orientation vector for an ellipsoid with aspect ratio $\mathrm{AR}=4$ and $\mathrm{De}=2.2$ (a–c), $\mathrm{De}=2.5$ (d–f), and $\mathrm{De}=2.7$ (g–i). In the leftmost panels, the initial orientation vector (red circles) is $(p_{x,0},p_{y,0},p_{z,0})=(0,0.31,0.95)$ whereas in the central panels it is $(p_{x,0},p_{y,0},p_{z,0})=(0,0.95,0.31)$. In the rightmost panels, the time evolution of the $z$ component of the orientation vector corresponding to the orbits in the left and central columns is reported. Insets: The details of the oscillations over a time window of 100 dimensionless units.

Figure 4

The $z$ component of the orientation vector corresponding to equilibrium regimes as a function of the Deborah number. The solid blue curves denote stable steady-state regimes, the closed blue circles correspond to stable periodic regimes, the horizontal dashed red line refers to unstable steady-state regimes, and the open red circles denote unstable periodic regimes. Four regions corresponding to different behaviors are identified and are denoted by different colors. The vertical dashed lines refer to the orbits reported in Fig. 3.

Figure 5

Critical Deborah number for flow alignment as a function of the aspect ratio. The lower (upper) triangles are the minimum (maximum) simulated values such that flow alignment is a stable (unstable) solution. The dashed line passes through the average value of the lower and upper triangles and serves as guide for the eye. The solid red curve is the theoretical prediction in Ref. [22]. The closed blue circles are experimental data taken from Ref. [21].

Figure 6

Average angular velocity as a function of the Deborah number for different aspect ratios. An ellipsoid initially released with its long axis on the flow-gradient plane is considered.