Dynamics of particle migration in confined viscoelastic Poiseuille flows

Particles migrate in the transverse direction of the flow due to the existence of normal stress anisotropy in weakly viscoelastic liquids. We test the ability of theoretical predictions to predict the transverse velocity migration of particles in a confined Poiseuille flow according to the viscoelastic constitutive parameters of dilute polymer solutions. First, we carefully characterize the viscoelastic properties of two families of dilute polymer solutions at various concentrations using shear rheometry and capillary breakup experiments. Second, we develop a specific three-dimensional particle tracking velocimetry method to measure with a high accuracy the dynamics of particles focusing in flow for Weissenberg numbers Wi ranging from ${10}^{-2}$ to ${10}^{-1}$ and particle confinement $\beta$ of 0.1 and 0.2. The results show unambiguously that the migration velocity scales as $\text{Wi}{\beta }^2$, as expected theoretically for weakly elastic flows of an Oldroyd-B liquid. We conclude that classic constitutive viscoelastic laws are relevant to predict particle migration in dilute polymer solutions whereas detailed analysis of our results reveals that theoretical models overestimate by a few tenths the efficiency of particle focusing.

Figure 1

Migration is observed in a thin rectangular microchannel with an inverted microscope. Observation is made at the $xy$ plane at different $x$ locations.

Figure 2

$z$ detection of particles. (a) Fluorescence imaging of a particle at different distances $d_f$ to the focal plane. (b) Calibration map obtained by orthoradial averaging of the images acquired at various $d_f$. (c) Example of image obtained under flow. Numbers are the $z$ position ($z=d_f-H/2$) determined with the calibration map. (d) Velocity profile determined by coupling PTV and $z$ detection. The solid line corresponds to the theoretical Poiseuille velocity profile. (e) PDF of the $z$ position of the particles in Newtonian liquid at 1 cm of the inlet.

Figure 3

Characterization of viscoelastic properties. (a) Viscosity as a function of the shear rate (for each shear rate, one point is obtained for an increasing ramp, and one point for a decreasing ramp). Solid lines correspond to the best Carreau fits, used to determine the zero-shear viscosity of the solutions shown in Table 1. (b) Image sequence of capillary breakup (HPAM 10%); white scale bar represents 1 mm. (c) Evolution of the streched filament radius as a function of time, for all the solutions used. The solid lines correspond to the best exponential fits of the data. Equation (2) is used to determine the relation time $\tau$. (d) Linear oscillatory rheometry compared to the Oldroyd-B predictions (dotted lines) using the values of ${\eta }_0$ and $\tau$ determined by rotational rheometry and capillary breakup respectively.

Figure 4

Examples of evolution of the PDFs of the $z$ position along the flow direction $x$ (see legends). The higher the Weissenberg number Wi and the confinement ratio $\beta$ are, the more efficient the focusing is. The solid lines are the best fits to the data (see text and Appendix pp1).

Figure 5

(a) Standard deviation of the $z$ position of the particles ${\sigma }_p^{*}$, normalized by the channel height $H$, as a function of the normalized distance $x^{*}=x/H$ from the inlet, for various Wi numbers. (b) Master curve obtained by plotting ${\sigma }_p^{*}$ as a function of $\text{Wi}{\beta }^2{x}^{*}$, taking the best fit of the prefactor $K$ (see text) and the theoretical value coming from [4, 7]. All the data are gathered in the figure; red and blue symbols correspond to PVP and HPAM solutions, respectively.

Figure 6

Values of the effective compliance $K\tau /\eta$, where the prefactor $K$ is the best fit parameter obtained by fitting all the PDFs for a given experiment (same solution, particle, and pressure drop). Blue symbols correspond to the HPAM solutions and red symbols to the PVP ones. Color darkness is related to the pressure drop, the darker the higher. Stars are for the small particles ($\beta \simeq 0.1$) and squares for the biggest ($\beta \simeq 0.2$). The two dotted lines correspond to the two available theoretical predictions [4, 7] while the solid line is the best linear fit to the experimental data, of slope 10.91.

Figure 7

(a) Theoretical particle trajectories. (b) Theoretical PDF evolution along the channel. (c) Theoretical prediction, after convolution with a Gaussian function whose standard deviation ${\sigma }_m^{*}$ is representative of the uncertainty of the $z$ measurement. PDFs are represented for $\mathrm{Wi}{\beta }^2{x}^{*}\in \left[0,0.122,0.244,0.366,0.488,0.61\right]$.