Instabilities in the oscillatory flow of a complex fluid

The dynamics of a fluid in a vertical tube, subjected to an oscillatory pressure gradient, is studied experimentally for both a Newtonian and a viscoelastic shear-thinning fluid. Particle image velocimetry is used to determine the two-dimensional velocity fields in the vertical plane of the tube axis, in a range of driving amplitudes from $0.8\text{to}2.5\mathrm{mm}$ and of driving frequencies from $2.0\text{to}11.5\mathrm{Hz}$. The Newtonian fluid exhibits a laminar flow regime, independent of the axial position, in the whole range of drivings. For the complex fluid, instead, the parallel shear flow regime exhibited at low amplitudes [Torralba et al., Phys. Rev. E 72, 016308 (2005)] becomes unstable at higher drivings against the formation of symmetric vortices, equally spaced along the tube. At even higher drivings the vortex structure itself becomes unstable, and complex nonsymmetric structures develop. Given that inertial effects remain negligible even at the hardest drivings $(\mathrm{Re}<{10}^{-1})$, it is the complex rheology of the fluid that is responsible for the instabilities observed. The system studied represents an interesting example of the development of shear-induced instabilities in nonlinear complex fluids in purely parallel shear flow.

Figure 1

Schematic view of the experimental device, including the setup for particle image velocimetry (PIV) measurements.

Figure 2

Silicone oil: PIV results for $\nu =8.2\mathrm{Hz}$ and $z_0=1.2\mathrm{mm}$ $(\mathrm{Re}=3\times {10}^{-2})$. Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 3

100:60 CPyCl-NaSal solution: PIV results for $\nu =2.0\mathrm{Hz}$ and $z_0=0.8\mathrm{mm}$ ($\mathrm{Re}=4\times {10}^{-3}$, $\mathrm{Wi}=0.8$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 4

100:60 CPyCl-NaSal solution: PIV results for $\nu =2.0\mathrm{Hz}$ and $z_0=2.5\mathrm{mm}$ ($\mathrm{Re}={10}^{-2}$, $\mathrm{Wi}=2.4$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 5

100:60 CPyCl-NaSal solution: PIV results for $\nu =3.5\mathrm{Hz}$ and $z_0=0.8\mathrm{mm}$ ($\mathrm{Re}=8\times {10}^{-3}$, $\mathrm{Wi}=1.3$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 6

100:60 CPyCl-NaSal solution: PIV results for $\nu =3.5\mathrm{Hz}$ and $z_0=2.5\mathrm{mm}$ ($\mathrm{Re}=2\times {10}^{-2}$, $\mathrm{Wi}=4.2$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 7

100:60 CPyCl-NaSal solution: PIV results for $\nu =8.2\mathrm{Hz}$ and $z_0=0.8\mathrm{mm}$ ($\mathrm{Re}=2\times {10}^{-2}$, $\mathrm{Wi}=12.8$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 8

100:60 CPyCl-NaSal solution: PIV results for $\nu =11.5\mathrm{Hz}$ and $z_0=0.8\mathrm{mm}$ ($\mathrm{Re}=3\times {10}^{-2}$, $\mathrm{Wi}=18.6$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 9

100:60 CPyCl-NaSal solution: PIV results for $\nu =6.5\mathrm{Hz}$ and $z_0=1.2\mathrm{mm}$ ($\mathrm{Re}=2\times {10}^{-2}$, $\mathrm{Wi}=11.4$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 10

100:60 CPyCl-NaSal solution: PIV results for $\nu =8.2\mathrm{Hz}$ and $z_0=1.2\mathrm{mm}$ ($\mathrm{Re}=3\times {10}^{-2}$, $\mathrm{Wi}=19.3$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 11

Low-resolution azimuthal vorticity contours for the 100:60 CPyCl-NaSal solution driven at $\nu =8.2\mathrm{Hz}$ and $z_0=1.2\mathrm{mm}$ ($\mathrm{Re}=3\times {10}^{-2}$, $\mathrm{Wi}=19.3$), showing that several toroidal vortices form along the tube.

Figure 12

100:60 CPyCl-NaSal solution: PIV results for $\nu =6.5\mathrm{Hz}$ and $z_0=1.6\mathrm{mm}$ ($\mathrm{Re}=3\times {10}^{-2}$, $\mathrm{Wi}=15.1$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 13

100:60 CPyCl-NaSal solution: PIV results for $\nu =8.2\mathrm{Hz}$ and $z_0=1.6\mathrm{mm}$ ($\mathrm{Re}=4\times {10}^{-2}$, $\mathrm{Wi}=25.7$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 14

100:60 CPyCl-NaSal solution: PIV results for $\nu =6.5\mathrm{Hz}$ and $z_0=2.0\mathrm{mm}$ ($\mathrm{Re}=4\times {10}^{-2}$, $\mathrm{Wi}=18.9$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 15

Location of the quiescent flow points along the radial coordinate $x$ (solid lines), and minima of the permeability (dashed horizontal lines), as derived from a linear theory. The parameters $\rho$, $\eta$, and $t_m$ used to calculate the diagram are those given for CPyCl-NaSal 100:60 in the text, and the radius of the cylinder is $a=25\mathrm{mm}$. The different symbols (experimental data) give the radial location of the quiescent points of the flow, for $z_0=0.8\mathrm{mm}$ (◻); the maxima of the vorticity closer to the center of the tube, for $z_0=0.8\mathrm{mm}$ (●); the vortex center, for $z_0=1.2\mathrm{mm}$ (▵) and $z_0=1.6\mathrm{mm}$ (▴).

Figure 16

100:60 CPyCl-NaSal solution: PIV results for $\nu =8.2\mathrm{Hz}$ and $z_0=2.0\mathrm{mm}$ ($\mathrm{Re}=5\times {10}^{-2}$, $\mathrm{Wi}=32.1$). Top: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the gray level scale at the bottom of the figure.

Figure 17

100:60 CPyCl-NaSal solution: PIV results for $\nu =6.5\mathrm{Hz}$ and $z_0=2.5\mathrm{mm}$ ($\mathrm{Re}=4\times {10}^{-2}$, $\mathrm{Wi}=40.1$). Top: local divergence. Middle: velocity vector field. Bottom: azimuthal vorticity contours. The corresponding scales are given by the little arrow and the two gray level scales.

Figure 18

100:60 CPyCl-NaSal solution: $x$-averaged rms fluctuations of the $y$ component of the velocity as a function of time, ${\tilde{\sigma }}_{v_y}\left(t\right)$, for the viscoelastic fluid driven at $8.2\mathrm{Hz}$ and amplitudes $z_0=0.8$ (▴), $z_0=1.6$ (엯), and $2.0\mathrm{mm}$ (∎).

Figure 19

100:60 CPyCl-NaSal solution: time-averaged rms fluctuations of the $y$ component of the velocity, ${\tilde{\sigma }}_{v_y}\left(x\right)$, as a function of the radial coordinate $x$ for the viscoelastic fluid driven at an amplitude (a) 0.8, (b) 1.2, and (c) $1.6\mathrm{mm}$. The symbols correspond to the different driving frequencies (in Hz) 2.0 (∎), 3.5 (엯), 6.5 (⋆), 8.2 (▵), 10.5 (▾), and 11.5 (+). Notice the different magnitude of the vertical scales.

Figure 20

100:60 CPyCl-NaSal solution: space- and time-averaged rms fluctuations of the $y$ component of the velocity, as a function of the dimensionless control parameter $\chi \equiv \mathrm{Wi}∕t_m\nu$, at the different driving frequencies (in Hz) 2.0 (∎), 3.5 (엯), 6.5 (⋆), 8.2 (▵), 10.5 (▾), and 11.5 (+).

Figure 21

Birefringence map for $\nu =8.2\mathrm{Hz}$ and $z_0=1.2\mathrm{mm}$. Time phases: 0, $T∕4$, $T∕2$, $3T∕4$, and $T$.