Measurements of the bulk and interfacial velocity profiles in oscillating Newtonian and Maxwellian fluids

We present the dynamic velocity profiles of a Newtonian fluid (glycerol) and a viscoelastic Maxwell fluid (CPyCl-NaSal in water) driven by an oscillating pressure gradient in a vertical cylindrical pipe. The frequency range explored has been chosen to include the first three resonance peaks of the dynamic permeability of the viscoelastic-fluid–pipe system. Three different optical measurement techniques have been employed. Laser Doppler anemometry has been used to measure the magnitude of the velocity at the center of the liquid column. Particle image velocimetry and optical deflectometry are used to determine the velocity profiles at the bulk of the liquid column and at the liquid-air interface respectively. The velocity measurements in the bulk are in good agreement with the theoretical predictions of a linear theory. The results, however, show dramatic differences in the dynamic behavior of Newtonian and viscoelastic fluids, and demonstrate the importance of resonance phenomena in viscoelastic fluid flows, biofluids in particular, in confined geometries.

Figure 1

(Color online) Schematic view of the experimental device, including the setups for PIV and deflectometry measurements.

Figure 2

(Color online) Diagram showing the location of the quiescent flow points along the radial coordinate of the cylinder, $r$, as a function of driving frequency, $\nu$, for a Maxwell fluid. The dashed horizontal lines separate the frequency axis in intervals of constant number of quiescent flow points. The parameters $\rho ,t_m$, and $\eta$ used to compute the diagram are the ones listed in the text for the 60:100 CPyCl-NaSal solution, together with the cylinder radius $a=25\mathrm{mm}$.

Figure 3

(Color online) A typical picture recorded in optical deflectometry measurements. The image of a regular array of black dots (grid spacing $d=5.0\mathrm{mm}$) is distorted by the interface deformation.

Figure 4

(Color online) Sketch of the path followed by a light ray in the system.

Figure 5

Theoretical deformation profiles of the Maxwell fluid driven at 2 Hz (dashed line) and at 6.5 Hz (solid line).

Figure 6

Dimensionless response function at the center of the cylinder, as a function of dimensionless driving frequency, for two concentrations of the CPyCl-NaSal solution. The vertical dashed lines give the location of the resonance frequencies predicted by a linear theory.

Figure 7

Glycerol: PIV results at 2 Hz (dots), measured at 6 cm from the upper free interface, and the corresponding theoretical prediction (solid line) at $t=0.125\mathrm{s}$.

Figure 8

60:100 CPyCl-NaSal solution: PIV results at 2 Hz (dots), measured at 6 cm from the upper free interface, and the corresponding theoretical prediction (solid line) at $t=0.375\mathrm{s}$.

Figure 9

Glycerol: PIV results at 6.5 Hz (dots), measured at 6 cm from the upper free interface, and the corresponding theoretical prediction (solid line) at $t=0.115\mathrm{s}$.

Figure 10

60:100 CPyCl-NaSal solution: PIV results at 6.5 Hz (dots), measured at 6 cm from the upper free interface, and the corresponding theoretical prediction (solid line) at $t=0.038\mathrm{s}$.

Figure 11

Glycerol: PIV results at 10 Hz (dots), measured at 6 cm from the upper free interface, and the corresponding theoretical prediction (solid line) at $t=0.075\mathrm{s}$.

Figure 12

60:100 CPyCl-NaSal solution: PIV results at 10 Hz (dots), measured at 6 cm from the upper free interface, and the corresponding theoretical prediction (solid line) at $t=0.025\mathrm{s}$.

Figure 13

60:100 CPyCl-NaSal solution: PIV results at 2 Hz, measured at 6 cm from the upper free interface and at different time phases within a driving period.

Figure 14

60:100 CPyCl-NaSal solution: PIV results at 6.5 Hz (dots), obtained at 10 cm from the upper free interface, and the corresponding theoretical prediction (solid line) at $t=0.057\mathrm{s}$.

Figure 15

60:100 CPyCl-NaSal solution: PIV results at 10 Hz (dots), obtained at 10 cm from the upper free interface, and the corresponding theoretical prediction (solid line) at $t=0.022\mathrm{s}$.

Figure 16

Glycerol: deflectometry results at 2 Hz, measured at two different time phases within an oscillation.

Figure 17

60:100 CPyCl-NaSal solution: deflectometry results at 2 Hz, measured at two different time-phases within an oscillation.

Figure 18

Glycerol: deflectometry results at 6.5 Hz, measured at two different time phases within an oscillation.

Figure 19

60:100 CPyCl-NaSal solution: deflectometry results at 6.5 Hz, measured at two different time–phases within an oscillation.