Viscoelastic fluid-structure interactions between a flexible cylinder and wormlike micelle solution

It is well known that when a flexible or flexibly mounted structure is placed perpendicular to the flow of a Newtonian fluid, it can oscillate due to the shedding of separated vortices at high Reynolds numbers. Unlike Newtonian fluids, the flow of viscoelastic fluids can become unstable even at infinitesimal Reynolds numbers due to a purely elastic flow instability that can occur at large Weissenberg numbers. Recent work has shown that these elastic flow instabilities can drive the motion of flexible sheets. The fluctuating fluid forces exerted on the structure from the elastic flow instabilities can lead to a coupling between an oscillatory structural motion and the state of stress in the fluid flow. In this paper, we present the results of an investigation into the flow of a viscoelastic wormlike micelle solution past a flexible circular cylinder. The time variation of the flow field and the state of stress in the fluid are shown using a combination of particle image tracking and flow-induced birefringence images. The static and dynamic responses of the flexible cylinder are presented for a range of flow velocities. The nonlinear dynamics of the structural motion is studied to better understand an observed transition from a symmetric to an asymmetric structural deformation and oscillation behavior.

Figure 1

Various morphologies of wormlike micelles which impart viscoelasticity [12].

Figure 2

Schematic diagram of the experimental setup.

Figure 3

Steady and dynamic shear rheology data of the 50 mM CTAB, 25 mM NaSal wormlike micelle solution $T_{\mathrm{ref}}=25{}^{\circ }\text{C}$: (a) the storage modulus $G^{\prime }$ ($\blacksquare$) and the loss modulus $G^{″}$ ($\square$) as a function of frequency and (b) the viscosity as a function of shear rate.

Figure 4

(a) Representative plot of transient filament stretching rheometry experiments for the 50 mM CTAB, 25 mM NaSal wormlike micelle solution at $\text{Wi}=288$. The symbols represent the elastic contribution to the tensile stress difference as a function of Hencky strain. The experiment ends abruptly upon the rupture of the fluid filament. (b) Maximum extensional viscosity reached before filament rupture as a function of strain rate for the 50 mM $\mathrm{CTAB}$, 25 mM NaSal wormlike micelle solution.

Figure 5

(a) and (b) Displacements $x$ ($\blacksquare$) and $y$ ($•$) of the flexible cylinder for the range of flow velocities tested. Note that (a) represents data only for 1D oscillations of the cylinder and (b) represents just the 2D oscillations of the cylinder. The closed and open symbols represent the mean maximum and minimum positions during cylinder oscillations, respectively. The inset in (a) shows a close-up of cylinder displacement at the critical velocity for the onset of cylinder oscillations. The path followed by the flexible cylinder is shown during its (c) stable, (d) unstable 1D, and (e) unstable 2D oscillations at $U=4.3$, 10, and 11.5 mm/s, respectively.

Figure 6

(a), (c), and (e) The $x$ and $y$ displacements of the flexible cylinder plotted against time (dotted and solid lines respectively) and (b), (d), and (f) the $xy$ displacement orbits at (a) and (b) $U=4.3$ mm/s, (c) and (d) 10 mm/s, and (e) and (f) 11.5 mm/s. The sampling frequency is 8 Hz.

Figure 7

(a) Amplitude and (b) frequency for 1D ($\blacksquare$) and 2D ($•$) oscillations, plotted over the range of flow velocities tested.

Figure 8

Sequential PIV images of the flow around the flexible cylinder at $U=7.15$ mm/s for a symmetric flow field. The $+$ represents the position of the flexible cylinder at the start of a run while the $•$ represents the location of the cylinder within the flow field at the time of the oscillation. The flow is from left to right. Each image represents a time step of 0.06 s.

Figure 9

The PIV image sequence of the flow around the flexible cylinder at $U=11.5$ mm/s for an asymmetric fluid rupture in the flow field. The $+$ represents the position of the flexible cylinder at the start of a run. The flow is from left to right. Each image represents a time step of 0.06 s.

Figure 10

Contour map of transient velocity vectors along the flow field centerline starting at the flexible cylinder center (0 mm) up to 35 mm downstream mapped over the start to the end of the sudden breakdown of wormlike micelle solution in the wake of the flexible cylinder at $U=7.2$ mm/s.

Figure 11

The PIV image sequence of a 2D flexible cylinder oscillation at $U=11.5$ mm/s. The $+$ represents the position of the flexible cylinder at the start of a run. The flow is from left to right. Each image represents a time step of 0.06 s.

Figure 12

Contour map of transient velocity vectors across the flow channel cross section taken at the cylinder midsection mapped over a time interval during the dramatic flow transition shown in Fig. 11 at $U=11.5$ mm/s.

Figure 13

The FIB images for two types of flexible cylinder oscillations observed. (I) At $U=7.2$ mm/s, the flow field is symmetric and (II) at $U=11.5$ mm/s, the flow field is asymmetric when (a) the flexible cylinder has reached maximum deflection, (b) the wormlike micelle solution fractures, causing the cylinder to recoil, and (c) the cylinder has reached the minimum position. The $+$ represents the position of the flexible cylinder at the start of a run. The flow is from left to right.

Figure 14

(a) The PIV image of the flow around the flexible cylinder at $U=11.5$ mm/s during an asymmetric 2D oscillation. (b) The dashed region of interest is presented through an FIB image. The flow is from left to right.