Particle image velocimetry in viscoelastic fluids and ...ltces.dem.ist.utl.pt/lxlaser/lxlaser2012/upload/24_paper_igblgc.pdf · interaction effects, ... viscoelastic and shear thinning

16th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 09-12 July, 2012

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Particle image velocimetry in viscoelastic fluids and particle interaction effects

Alireza Mirsepassi1*, Derek Dunn Rankin2

1: Department of Mechanical and Aerospace Engineering, University of California Irvine, USA 2: Department of Mechanical and Aerospace Engineering, University of California Irvine, USA

* Correspondent author: [email protected]

Abstract Two main assumptions in laser Doppler velocimetry (LDV) and particle image velocimetry (PIV) are that tracer particles are following the flow faithfully and that they are not interacting hydrodynamically or affecting the flow field. It has been shown, however, that particles in polymeric fluids even in low volumetric concentrations (ϕ <1% v/v) can interact profoundly because of the viscoelastic nature of the background fluid, leading even to particle chain or string formation. The objective of this study is to determine how these interactions can affect the flow conclusions drawn from PIV measurements. We chose a highly viscoelastic and shear thinning fluid and confirmed that particle string formation does occur rapidly in a simple shear flow of this fluid. Then we measured via PIV the fully developed, velocity profile in the mid-plane of a square channel over a wide range of particle loadings 0.005%- 0.2% v/v. The measurements confirm that despite the chaining process, there is no discernible effect of particle interaction on PIV velocity measurements.

1. Introduction

Experimental velocity measurements are essential in understanding the fluid dynamics of complex fluids. Precise experimental velocity measurements can help with understanding the correlation between mechanical stress and the velocity gradient tensor, rheology of complex fluids, and with finding reliable constitutive equations and numerical schemes (Bird, et al., 1987;Crochet, 1988). In processing of polymeric materials, velocity measurements can determine the regions of high shear rates or recirculation zones which can affect the final quality of the product (Hatzikiriakos & Migler, 2005). As has been well-established, particle image velocimetry (PIV) has the ability to measure velocity in a plane using a sheet of laser light by computing the auto-correlations or cross-correlations of scattered light from tracer particles (Adrian, 1991;Willert & Gharib, 1991). Pakdel and Mckinley utilized digital particle image velocimetry (DPIV) for investigating velocities in viscoelastic solutions (Pakdel & Mckinley, 1997). Other investigators used the same technique for measuring the velocity of other complex fluids (Arigo & Mckinley, 1998;Baumert, et al., 1997). Based on indirect evidence from these and similar measurements, it appears that polymeric materials can be seeded with particles up to a volume fraction of ϕ = 0.1% v/v before the effect of particle interaction becomes significant enough to affect the PIV results. There are no studies, however, that directly evaluate the effect of particle loading in a polymeric solution even though particle interactions in viscoelastic and shear thinning polymer solutions are sufficiently profound that particles form chain and strings, even at low volumetric fractions in simple shear flows (Scirocco, et al., 2004; Mirsepassi, et al., 2012). In this work we first confirm particle string formation in a shear thinning viscoelastic fluid (where particle interaction is strong), and then, changing the volumetric concentration about two order of magnitudes (0.2% - 0.005 % v/v), fluid velocity in the mid-plane of a square channel is measured by means of PIV. Despite the clear particle interaction effects, ensemble velocity profiles do not show deviations with changing particle loading in regions away from the wall. Near the wall, the low velocity increases the uncertainty in the measurement so that any effects of particle interaction are insignificant by comparison anyway.


- 2 - Figure 2. Shear viscosity measurements of fluids

2. Experimental Approach PIV measurements in polymer solutions require a transparent viscoelastic fluid; specifically any absorption at the laser wavelength should be insignificant. For amplifying the hydrodynamic interaction between particles, an elastic and shear thinning fluid should be used that has shown strong particle string and chain formation (Feng & Joseph, 1996). Considering these factors, 1.7% w/w of poly-(ethylene oxide) in DI water with specific gravity of SG = 1.0 was chosen as the transparent viscoelastic fluid. Particle chaining using 5µm diameter particles at a low volume fraction of 0.2% v/v is confirmed in a simple shear flow geometry (cone-plate) for shear rates of 𝛶 = 5 and 10(𝑠!!) as shown in Figure (1). Additional details of the experimental setup and conditions can be found in (Mirsepassi, et al., 2012) Figure 1. Confocal microscopy imaging with 5µm particles in 1.7% PEO(10s-1) at (a) the flow startup, (b) after 30 minutes of shearing- 0.2% v/v 3. Fluid Medium and Tracer Particles Performance of our PIV system is verified using glycerin (Fisher Scientific Inc.) as a high viscosity Newtonian fluid which does not show particle string formation. As mentioned before, high molecular weight poly-(ethylene oxide) (PEO-WSR301, Mw= 4 × 106 g/mol, Dow Inc.) in DI water is the selected polymeric fluid medium. Stock PEO solutions were prepared by dispersing the dry polymer in deionized water, and allowing it to equilibrate for 24 hours. A homogeneous polymer distribution (1.7% w/w) was then obtained by thoroughly mixing the solutions using a single stirrer turbine at 30 rpm for 96 hours; higher rotational speeds were seen to promote polymer degradation. This solution has strong viscoelastic and shear thinning behavior, the combination of which can cause particle string formation. Viscosity of both fluids is measured with a controlled stress rheometer (Physica MCR301, Anton Paar GmBH, Germany) using a cone-and-plate geometry (2° sandblasted cone, 25 mm diameter). All experiments, including the rheological measurements were performed at 25°C. The shear rate dependent viscosity of the PEO solution is shown in Figure (2). Prominent shear thinning behavior of 1.7% PEO is well described by the power law equation η!=kγ!!!, where n =1 is for Newtonian fluids and n < 1 shows shear thinning behavior. The exponent n and power law index k are respectively 0.48, and 6.49(Pa.sn-1). Glycerin has a constant viscosity of 1.1 (Pa.s). Tracer

𝒗!!⃗

𝜴→

a b


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particles are silver coated hollow glass spheres with average diameter of 13µm and density of 1.6 g/cc and also silver coated hollow ceramic spheres with average diameter of 83µm and density of 1.1 g/cc. Silver coating increases the signal to noise ratio in PIV measurements which is critical for reliable measurements. 4. Experimental Setup Optical distortions can significantly introduce noise to PIV measurements. Hence, in order to avoid curved surfaces, a square channel was selected for the velocity measurements. A schematic of the experiment is shown in Figure (3). A transparent polycarbonate square channel with width of 9.38 mm and length of 65 cm is connected to a fluid reservoir. The reservoir is connected to a high pressure nitrogen supply with a venting mechanism for creating low pressures. The supply pressure delivering the fluid was controlled by a high precision pressure gage (Wallace-Tiernan Inc.) with 0.02 psi graduations. Far downstream of the entrance (37cm) velocity in the mid-plane of the square channel is measured by passing a laser sheet with 1mm thickness across the channel. A continuous low power diode laser beam (HLM1845) with 655nm wavelength, after passing through a plano-convex lens and a cylindrical lens, forms the planer laser sheet. The laser sheet hits a 45° mirror which is placed on a micron precision linear stage for calibration and locating the mid-plane. A high-speed CMOS Phantom camera (version 4.3) with a macro lens (7000 series, Navitar Inc.) records an image from the laser sheet. Tracer concentration in PIV measurements is usually in the range of (0.001% < ϕ < 0.1% v/v). The low particle concentration limit comes from insufficient particle numbers in the interrogation window for reliable cross-correlation. The high concentration limit is related to not affecting the flow field dynamics and not creating speckle for PIV measurements. Experiments are performed to cover a wide range of particle volumetric concentration from O(10-3)% to O(10-1)%. Working in a very low particle concentration, necessitated using small particles (13µm) to ensure relatively high particle number density and enough particles in the interrogation volume. Working at high volumetric concentrations pushes the seed requirements toward using larger particles (83µm), in order to avoid speckle.

Figure 3. Schematic of experimental setup for PIV measurments


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It is possible to be more quantitative with particle loading by defining Ns and Ni as source density and image density respectively, and C is particle number density (Equations 4.1, 4.2). Laser sheet thickness is ∆𝑧, 𝑑𝑒 is the particle image diameter, and 𝐴!"# is the interrogation area in the object plane. Particle image diameter (𝑑𝑒) is the convolution of diffraction-limited image (𝑑𝑠) with the geometric image assuming both have Gaussian distribution and M is optical magnification (Equations 4.2, 4.3). The criterion for not creating speckle is Ns ≪1 and, for performing a reliable PIV measurement, we need to have enough particles in the interrogation volume, Ni > 7, assuming no in-plane or out-of-plane loss of particles (Keane & Adrian, 1992). 𝑁𝑠 = 𝐶.∆𝑧.𝜋𝑑𝑒

!

4𝑀! (4.1) 𝑁𝑖 = 𝐶.∆𝑧.𝐴!"# (4.2) 𝑑𝑒 = (𝑑𝑝.𝑀)! + 𝑑𝑠! (4.3) 𝑑𝑠 = 2.44 1+𝑀 𝐹# 𝜆 (4.4) Velocity measurements are performed with 3 different tracer concentrations as shown in Table (1). Optical magnification of the system is M=0.7 (32 pixels/mm) with an 𝐹#of 8. The field of view is 9.38×7.80 𝑚𝑚!(H×W) and the interrogation volume in all cases is 0.75×0.75×1 𝑚𝑚!(H×W×D). The camera records at 300fps with an exposure time of 100 µs. After collecting the images, by using the open source software (PIVlab), images are imported into MATLAB. After preprocessing and choosing the region of interest, images are analyzed by a two-pass interrogation of 48 pixels to 24 pixels and 50% overlap at each time. Table1. Tracer concentrations, image density (Ni), and source density (Ns)

Case φ(%) v/v dp (µm) Ni Ns 1 0.005 13 9 0.01 2 0.02 13 35 0.05 3 0.2 83 9 0.09

5. Results and Discussion Performance of the PIV setup is first evaluated by comparing the experimental results to the analytical solution of a steady, fully-developed Newtonian flow in a rectangular channel. The analytical solution is based on solving the Navier-Stokes equations with a constant pressure gradient along the channel and no slip boundary conditions. Analytical results are normalized to match maximum experimental velocity on the centerline (y = 0, z = 0). As reviewed by Berker (1963), the analytical solution of a steady state laminar Newtonian fluid in a rectangular channel is (5.1):

−𝑎 ≤ 𝑦 ≤ 𝑎 , − 𝑏 ≤ 𝑧 ≤ 𝑏

𝑈 𝑦, 𝑧 =𝑈! (!!! , !!!)

−1(!!!) !

𝑖! 1− 1cosh ( 𝑖𝜋𝑏 2𝑎)

!!!!,!,!,…

×…

(−1)(!!!) ! 1−cosh( 𝑖𝜋𝑧 2𝑎)cosh( 𝑖𝜋𝑏 2𝑎) ×

cos(𝑖𝜋𝑦 2𝑎)𝑖!

!

!!!,!,!,…

In the current work, we are looking at the ensemble-averaged velocity profile at z = 0, which is the mid-


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plane. Velocity in the middle of the channel is calculated for glycerin, and compared with the theoretical velocity profile. In all the cases, Reynolds number, based on the hydraulic diameter (𝐷!) of the channel and mean velocity(𝑈!), is less than 100 which ensures a laminar flow. Mean velocity is obtained by measuring the flow rate by mass on a Sartorius scale (CP22025). Fully developed flow will be achieved based on the development length formula (Beavers, et al., 1969) 𝑋 𝐷! = 0.03 𝑅𝑒 𝐷! ,𝑈! , where X is the development length. Therefore velocity measurements 37cm downstream of the entrance represent a fully developed velocity field. Velocity calculations show good agreement with experiment in particle volumetric concentrations of 0.005% (13µm) and 0.2% (83µm) away from the wall. Profiles are non-dimensionalized with the centerline velocity of 36mm/s and half-width of the channel Figure (4). Error bars show the standard deviation of the velocity measurements at each point normalized by the centerline velocity. The average ratios of standard deviation to mean velocity in both cases is about 5% in regions away from each wall (1mm). Standard deviation in terms of pixel displacement with 13 µm particles is about 0.15pixels corresponding to 1.4 mm/s and 0.17pixels with 83 µm particles, corresponding to 1.6mm/s. In regions close to the walls, because of high shear rate, particle slip, and the laser sheet reflection off the surfaces, the error is much higher, and PIV measurements were not reliable. The root mean square error as a measure of accuracy in regions away from the wall is about 1 mm/s. After finding the precision and accuracy limits of the PIV system, we continue the velocity measurements with the non-Newtonian fluid. Unsurprisingly, the shear thinning nature of the fluid

Figure 4. Velocity profile in glycrine in two different tracer loadings

causes the velocity profile in 1.7% PEO to show a dramatic deviation from that of Newtonian fluid, as seen in Figures(5,6) in three different flow rates of 14 gr/min, 90gr/min, and 177gr/min. Each experiments were performed with 3 different particle volumetric concentrations. Furthermore, experiments covered a wide range of Deborah number (4 < 𝐷𝑒(!) < 50). Deborah number is defined as the relaxation time of the polymeric fluid, which is obtained from rheological measurements 𝜆 ≈ 7𝑠,

0

0.2

0.4

0.6

0.8

1

1.2

-‐1 -‐0.5 0 0.5 1

𝑼ʹ′=𝑼 𝒚 /𝑼

(𝒚=𝟎

)

𝒚ʹ′=𝟐𝒚/𝒘_(𝒄𝒉𝒂𝒏𝒏𝒆𝒍 𝒘𝒊𝒅𝒕𝒉)

13µm-‐ 0.005% v/v

Experiment Theory

0

0.2

0.4

0.6

0.8

1

1.2

-‐1 -‐0.5 0 0.5 1

𝑼ʹ′=𝑼 𝒚 /𝑼

(𝒚=𝟎

)

𝒚ʹ′=𝟐𝒚/𝒘_(𝒄𝒉𝒂𝒏𝒏𝒆𝒍 𝒘𝒊𝒅𝒕𝒉)

83µm-‐ 0.2% v/v

Experiment Theory


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Figure 5. Left: Images of 1.7% PEO flowing along the channel with a) 0.2% v/v , b) 0.005% v/v , and c) 0.02% v/v tracer loadings- Right: Non-dimensional velocity profile with 1.7% PEO with 3 different tracer loadings with De = 25, flow rate = 90 gr/min Figure 6. Non-dimensional velocity profile with 1.7% PEO at De = 4, flow rate = 14 gr/min (left) and De = 50, flow rate = 177 gr/min (right). times by the average shear rate (𝐷𝑒=𝛶.λ). This non-dimensional parameter indicates the presence of viscoelastic forces in the medium. Average shear rate is obtained by average velocity based on flow rate divided by the half-width of the channel. Velocity profiles are obtained after averaging over 900 image pairs. In all cases, velocity profiles have steep gradients in regions close to the wall, with a fairly flat region across the bulk of the channel. This behavior is quite reasonable because next to walls, the high shear rates cause the viscosity to decrease, creating a rapid velocity increase. In the low shear region near the center, the viscosity increases, thereby reducing the velocity gradient. Bird et al. obtained the exact solution of a pressure driven power law fluid in a circular cross section channel and presented the same behavior (Bird, et al., 1987). There is no exact solution for flow of viscoelastic fluids in a rectangular channel, numerical simulations are needed, so the Newtonian solution is used for comparison to show the effect of viscoelasticity on the flow. In the case of De = 25 the standard deviation of tracer displacement away from walls remains roughly constant (0.17 pixels) with 83µm and 0.10 pixels in the cases with 13µm. Similar qualitative trends are observed in two other cases. The decreasing precision with tracer loading of 0.2% v/v might be associated with multiple-scattering phenomena because of high particle concentrations and a large ratio of particle image diameter to interrogation spot size (Willert, 1996). By comparing the velocity profiles away from walls, however, there is excellent agreement between the mean velocities at each point with different particle concentrations. This result indicates that although particle interaction exists in viscoelastic fluids those interactions do not measurably affect the velocity profile over a wide range of particle concentration.

x

y


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Particle interaction in viscoelastic fluids can be strong, even forcing the particles to form strings and chains in the flow, but these interactions have a very long characteristic time in the order of O(102)s (Feng & Joseph, 1996) and a commensurately small characteristic velocity. The characteristic time is much longer than the experimental time scales in typical PIV measurements, which is on the order of seconds. Considering the initial distance between particles which is on the order of O(103)µm, chaining velocity can be estimated as on the order of 10 µm/s. This value is much smaller than the precision of our PIV measurements and is much smaller than the precision of most such measurements. In summary, this study shows that increasing particle volumetric concentration does not measurably affect the ensemble velocity profile in regions away from the wall, and the order of magnitude of the chaining velocity is much smaller than the precision of typical PIV measurements. Hence, the particle interaction effects are not a significant concern except in flows of extremely low velocities but high velocity gradients. Even in these cases uncertainties from in-plane and out-of-plane velocity gradients are likely to be substantially higher than those resulting from the chaining velocity. Consequently particle interaction is unlikely to ever play a large detrimental role in PIV measurements in typical tracer loadings of less than 0.1% v/v. Bibliography

1) Adrian, R., 1991. Particle-Imaging Techniques for Experimental Fluid Mechanics. Annu. Rev. Fluid Mech, Volume 23, pp. 261-304.

2) Arigo, M. & Mckinley, G., 1998. An experimental investigation of negative wakes behind spheres settling in a shear-thinning viscoelastic fluid. Rheol Acta, Volume 37, p. 307–327.

3) Baumert, B., Liepmann , D. & Muller, S., 1997. Digital particle image velocimetry in flows with nearly closed pathlines: The viscoelastic Taylor-Couette instability. Non-Newton Fluid Mech, Volume 69, pp. 221-237.

4) Beavers, G., Sparrow, E. & Magnuson, R., 1969. Experinents on Hydrodynamically Developing Flow in Rectangular Ducts of Arbitrary Aspect Ratio. Int. J. Heat Mass Transfer., Volume 13, pp. 689-702.

5) Berker, R., 1963. Integration des equations du mouvement d'un fluide visqueux incompressible.In Handbuch der Physik. Berlin: Springer-Verlag.

6) Bird, R., Armstrong, R. & Hassager, O., 1987. Dynamics of Polymeric Liquids. 2 ed. s.l.:Wiley Interscience.

7) Crochet, M., 1988. Numerical Simulation of Highly Viscoelastic Flows. Sydney, Proc. Znt. Cong. on Rheology, Vol. 1, p. 19.

8) Feng, J. & Joseph, D., 1996. The motion of solid particles suspended in viscoelastic liquids under torsional shear. J . Fluid Mech., Volume 374, pp. 199-227.

9) Hatzikiriakos, S. & Migler, K., 2005. Polymer processing instabilities: control and understanding.

10) Keane, R. D. & Adrian, R. J., 1992. Theory of cross-correlation analysis of PIV images. Applied Scientific Research , Volume 49, pp. 191-215.

11) Mirsepassi, A., Rajaram, B., Mohraz, A. & Dunn Rankin, D., 2012. Particle chaining and chain dynamics in viscoelastic liquids. J. Non-Newtonian fluid mech. http://dx.doi.org/10.1016/j.jnnfm.2012.04.005

12) Pakdel, P. & Mckinley, G., 1997. Digital particle imaging velocimetry of viscoelastic fluids. AIChe, 43(2).

13) Scirocco, R., Vermant, J. & Mewis, J., 2004. Effect of the viscoelasticity of the suspending fluid on structure formation in suspensions. J. Non-Newtonian Fluid Mech, Volume 117, p. 183–192.

14) Willert , C. & Gharib, M., 1991. Digital particle image velocimetry. Experiments in Fluids, Volume 10, pp. 181-193.

15) Willert, C., 1996. The Fully Digital Evaluation of Photographic PIV Recordings. Appl. Sci. Res., Volume 56, pp. 79-102.

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Particle image velocimetry in viscoelastic fluids and ...ltces.dem.ist.utl.pt/lxlaser/lxlaser2012/upload/24_paper_igblgc.pdf · interaction effects, ... viscoelastic and shear thinning