Development of Film Interference Flow Imaging Method (FIFI) Studying Polymer Stretching Effects on Thin Liquid Layer∗

The addition of very small amounts of flexible polymer to a flow reduces drag in turbulent flow by almost 60% at the maximum. The phenomena has been tried to explain by velocity profile near the wall. Thus, shear strain and shear viscosity gathered much attentions. However, viscoelasticity of polymers that is extensional viscosity should also important to affect fluids. It is said that the extensional viscosity is increased by causing polymer entangled, however, the extensional viscosity should be also increased when the polymer is stretched. In this study, flexible polymer that is easily stretched under strain and naturally stretched rigid polymer are added to the fluids to observe how it effects on turbulent dynamics. For the visualization and single-image analysis of turbulence, our original method Film Interference Flow Imaging method is used. Inner structure of sample solution is analyzed by Scanning Microscopic Light Scattering to investigate how polymers expand in fluids. Here, it is assumed that sudden stretch of the flexible polymer in flow cause sudden increase of extensional viscosity, and gradual orientation of the rigid polymer cause gradual increase, which is related to the turbulent drag reduction. Such a fluids control can work not only to improve energy efficiency in industries but also to treat micro fluids in medical area. [DOI: 10.1380/ejssnt.2012.335]


I. INTRODUCTION
The addition of very small amounts of flexible polymer to a flow reduces drag in turbulent flow by almost 60% at the maximum.The phenomena is known well as Toms' effect or drag reduction (DR), and used in many industries to improve the energy efficient.Lumley emphasized that visco-elastic effects due to polymers can occur only at the higher hydrodynamic frequencies in turbulent flow.This notion of elastic behavior at high frequencies is historically for molten and entangled polymer chain [1].He also mentioned the strongly enhanced viscosity in the regions of flow, while the viscosity in the laminar sublayer near the wall should remain low.This suggestion is related to shear thinning near the wall, that Lumley explained the reduction of turbulent losses.That is why, shear viscosity or velocity profile near the wall have been got much attention in turbulent researches [2].On the other hand, there were experiments with polymer injection at the center of a pipe, in the case also the drag reduction was founded where wall effects are not involved [3,4].By this experiment, de Gennes claimed that not only the viscosity enhancements near the wall but also the elastic modu-lus of polymers were important [5].In the explanation of viscoelasticity, the relaxation time of polymer is important to see the Deborah number corresponding to a "coil-stretch transition".The evidence of flexible polymer stretch in extensional flows is shown experimentally by Chu et al. [6] as a first time.For rigid polymer, orientation is observed by birefringence technique in rather concentrated solution [7].Indeed, such an extension of polymers is expected to increase the extensional viscosity of flows.Figure 1 shows the comparison between an intrinsic viscosity and an extensional viscosity of polymer solution.In the case of extensional flow, the polymer stretch in the flow, thus even the dilute solution can show the higher viscosity.In recent years, extensional viscosity described by the way of fluids dynamics also gathered the attention in DR researches [8].
Here, we consider that since the polymer stretching is important to increase extensional viscosity, naturally stretched rigid polymers also contribute to the increase.Thus, in this study, flexible and rigid polymers were added to turbulence to see the effects.In order to make turbulence, flowing soap films were used.
Historically, flowing soap films is good test bed for twodimensional fluids [9,10].As shown in Fig. 2(d), the water layer is sandwiched by surfactants.The thickness of water layer is much larger than vanishingly small surfactant molecules.Besides, the surface area is infinity compared to the thickness, thus, soap film is considered as 2D water layer flows.Soap films reflect illumination light at the front and the back of the surface, which make interference patterns of the film.Since interferences of the illumination lights are affected by the thickness of the water layer, the interference patterns has information of dynamics of water layer as 2D flows as shown in Figs.

2(b)-(d). Flows were visualized and analyzed by interference patterns originally proposed by our previous work as
Film Interference Flow Imaging method (FIFI) [11].FIFI is the system composed of 2D flow visualization apparatus and single-image processing software.
Internal structure of sample solution is also observed by Scanning Microscopic Light Scattering (SMILS) originally proposed by Furukawa et al. that is specialized for structural analysis in micro scale without experimental noises [12].By the experiments of SMILS, static hydrodynamic radius of polymers is observed.

II. EXPERIMENTAL ARRANGEMENT AND MATERIALS A. For the observation by Film Interference Flow Imaging (FIFI)
Soap solutions contain Sodium Dodecylbenzenesulfonate (SDBS) as a surfactant at concentration of 2 wt%.The polymer used is polyethyleneoxide (PEO, molecular weight of 3.5 × 10 6 ) as a flexible polymer and Hydroxypropyl cellulose (HPC, molecular weight is greater than 1.0×10 6 ) as a rigid polymer at concentrations of 3.0×10 −3 wt%.All materials were obtained from Wako Pure Chemical Industries.
The experiments were carried out in an apparatus of FIFI method shown in Fig. 2(a).The channel frame was made of nylon wires of diameter 1 mm.For a grid of equally spaced eight cylinders, the diameter of the cylinders is 0.3 cm and 0.6 cm spacing between the teeth.For measurements presented here, the film thickness h(t) was about 3.85 µm with the mean velocity V (t) was 130 cm/s when the flow flux Q(t) was 0.5 ml/s, since there is the relationship like h(t) = Q(t)/V (t)W , where W is width of the soap film channel.The interference images of soap films were recorded with a digital video camera (Panasonic TM700) at the data acquisition area which was 20 cm behind the grid.The shutter speed of the video camera was 1/3000 s.A time interval in a series of images is adjusted to 1/60 s.Each of the frames acquired by the camera was converted into RGB form files with a spatial resolution of 640 × 360 pixels.A part of the image that corresponds to the data acquisition area was clipped.The size of the image is 256 × 256 pixels, which corresponds to 2.56 × 2.56 cm 2 .Five images were acquired for each soap solution as experimental data.

B. For the experiments by Scanning Microscopic
Light Scattering (SMILS) For SMILS experiments, PEO and HPC were used at the concentrations of 3.0×10 −3 wt%.All sample solutions contains the same concentrations of PEO or HPC, and SDBS at the concentrations of 0, 0.1, 0.3, 1.0 and 2.0 wt%.The critical micelle concentration of SDBS is about 0.1 wt%.
The SMILS is the dynamic light scattering system specialized for inhomogeneous solutions or gels, as described in Ref [12].The prepared solutions in a test tube were observed by SMILS with 532 nm laser beam.Hydrodynamic radius of internal structure of solution was calculated.With this experiment, it is observed that how polymers expand in solution.

III. IMAGE PROCESSING BY SOFTWARE IN FIFI
The interference images were analyzed by 2D-FFT and Curvature analysis as software in FIFI described in our previous work [13].
The 2D-FFT software calculates the power spectrum of the interference images.Since the RGB pixel intensity of the images is related to the thickness of water layer, the power spectrum of the image shows the dynamics of 2D fluids.Here, the power spectrum ⟨I 2 (k x , k y )⟩ is calculated by the pixel intensity G with the hamming window, where the k x and k y are the spatial frequency that are perpendicular and parallel to the flow direction in the image.For measurements presented here, the data acquisition area is 256 × 256 pixels, which correspond to 2.56 × 2.56 cm 2 , thus, the k x and k y are ranging from 1/2.56 to 1/0.02, that is, 0.391 to 5.00 cm −1 as the frequency.⟨I 2 (k x , k y )⟩ of the pixel intensity G characterizes the strength of the thickness fluctuation of water layer on spatial frequency, k x and k y .Here the power spectrum ⟨I 2 (k x , 0)⟩ is used for calculating the scaling exponent.
The Curvature analysis software calculates the curvatures of the interference pattern of the turbulence.The interference patterns have the information of the thickness of the water layer, besides the thickness is related to the instantaneous streamlines [14].Streamlines is the http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology family of curves that are instantaneously tangent to the velocity vector of the flow.The curvature is defined by κ =| θ 2 − θ 1 | /ds, where θ 1 and θ 2 are the angles between the axis and the tangent of the line at two contacts and ds is the distance between the two contacts meaning the variation of the tangential direction on the line.That is why, the curvature of the interference pattern is related to the variations of the velocity vector, that is, velocity fluctuations.The characteristic patterns of the interference image are obtained, the contour of the interference pattern is traced to calculate the curvature κ of the contour as shown in Fig. 5(b).The precise explanation is written in our previous paper [13].

IV. RESULTS AND DISCUSSION
Figure 3 shows the examples of interference patterns of 2D turbulence.When PEO is added to the solution, the eddies in turbulence were deformed, while the effects of HPC is hardly seen.These images were analyzed by FIFI software.

A. Analysis by FIFI
The power spectrum ⟨I 2 (k x , 0)⟩ of the interference pattern for SDBS 2 wt% solution in the spatial frequency k x is shown in Fig. 4, which is related to the fluctuation of water layer on spatial frequency k x .The k x has the information of the fluctuation directed perpendicular to the mean flow.Indeed, the ⟨I 2 (k x , 0)⟩ shows the scaling behavior, the power component is −1.57, which almost fits with the −5/3 that is predicted by theory for 2D turbulence.With the addition of PEO, the interference patterns become thin and long in the mean flow direction, while it is slightly changed with the HPC additives (See Figs.3(b) and (c)).The power components are changed from −5/3 to close to −1 by addition of PEO especially concentrations as shown in inset figure of Fig. 4.This variation can be explained by theory of fluid dynamics which suggests that PEO prohibit the inverse transfers of energy in 2D turbulence [15].This situation was visualized as Fig. 3, which means the eddies did not have enough energy to become bigger.
Figure 5 shows the curvature histogram of the each in- terference images.The curvature histogram is fitted by a distribution function theoretically suggested for velocity fluctuation, where κ is the curvature, A 1 and A 2 are prefactor, γ is the power component.Indeed Eq. ( 1) fits well to the curvature histogram, γ is decreased by addition of PEO and HPC as shown in the inset of Fig. 5(c).Therefore, when the histogram shows the distribution of the velocity fluctuation, the decrease of velocity fluctuation is determined by the index γ.All the fitting parameters are listed in Table I.

B. Analysis by SMILS
As described before, the effect of PEO on 2D turbulence is larger than that of HPC.We assume that this is due to the flexibility of polymers.In order to quantify how polymer works in turbulent, the static inner structure is

C. Polymer effects on 2D turbulence described by FIFI and SMILS
It is assumed that the effects of polymer are due to the extensional viscosity.Fig. 8 shows the close up figure of the flow channel around the grid.A simple calculation of extensional strain is described by Eq. (2).
where S 0 is the cross-section area before deformation, S(t) is the cross-section area after deformation, t is the time required for deformation and ε is the extensional rate.In this study, S 0 can be calculated as 3.85×10 −7 m 2 as shown in Fig. 8. S(t) is 2/3 of S 0 when the deformation area is determined as the middle of the grid, since the diameter of the grids is 0.3 cm and the spacing between the grids is 0.6 cm as shown in Fig. 8. Thus S(t) is 2.57 × 10 −7 m 2 .
Here the mean velocity of the turbulence is 130 cm/s, thus t is 1.15 × 10 −3 s calculated by 0.15 cm divided by 130 cm/s, then the ε is 351 s −1 , which can be enough large to deform the polymers.In the case of PEO as a flexible polymer, it is assumed that the polymer is stretched at the grid, and the deformation can be large from 10 −9 mscaled to 10 −7 m-scaled.On the other hand, in the case of HPC as a rigid polymer, the molecules have already expanded in SDBS solution.Thus, the expansion scale is not so changed even under the extensional strain.That is why, it is assumed that the polymer is not stretched but oriented under the extensional strain, and also the deformation can be confined in 10 −7 m-scaled as shown in Fig. 8.This different scale deformation and same scale orientation can be a reason why the effects of each polymer on turbulence are different.We now assume that the extensional viscosity suddenly increase with flexible polymer at the critical strain rate such as strain hardening, and gradually increase with rigid polymer.The former phenomena will effects on DR efficiently.

V. CONCLUSIONS
PEO and HPC effect on turbulence differently as observed by FIFI.The R h of the PEO is 10 −9 m-scaled in SDBS solution, and the polymer can be stretched under the extensional strain at the grid.HPC is already expanded 10 −7 m-scaled in SDBS solution, thus it is assumed that HPC is rather oriented under the extensional strain.For sudden increase of extensional viscosity, sudden stretch of flexible polymers may be important, which will work for effects on fluid.Polymer effects on fluids, that is, fluids control can work not only to improve energy efficiency in industries but also to treat micro fluids in medical area.

FIG. 1 :
FIG.1:The schematic of (a) intrinsic viscosity and (b) extensional viscosity.The ν is the mean velocity, the ε is the extension rate.

FIG. 2 :
FIG. 2: Experimental set up in FIFI.(a) is the whole image, (b) is the close up figure of the flowing soap film channel, that is 2D flow, (c) is the example of interference pattern.(d) shows the cross section of 2D flow.

FIG. 5 :
FIG. 5: (a) is the example of characteristic pattern of the interference pattern whose contour is picked out as (b) to calculate the curvature.(c) is Curvature histogram for different polymer concentrations.Inset figure shows the variation of the index γ at each solution.

TABLE I :
Fitting parameters of Eq. (1) to the curvature histogram for each PEO concentrations.