Fingering instability in non-Newtonian fluids during ...

Fingering instability in non-Newtonian fluids during squeeze flow in a Hele-Shaw cell M Dutta Choudhury & S Tarafdar

Indian Journal of Physics ISSN 0973-1458 Volume 89 Number 5 Indian J Phys (2015) 89:471-477 DOI 10.1007/s12648-014-0606-3

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Author's personal copy Indian J Phys (May 2015) 89(5):471–477 DOI 10.1007/s12648-014-0606-3

ORIGINAL PAPER

Fingering instability in non-Newtonian fluids during squeeze flow in a Hele-Shaw cell M Dutta Choudhury and S Tarafdar* Condensed Matter Physics Research Centre, Jadavpur University, Kolkata 700032, India Received: 07 March 2014 / Accepted: 26 September 2014 / Published online: 30 October 2014

Abstract: Instability at the interface separating different fluids, may develop under different conditions, leading to increased roughness of the boundary. A difference in viscosity of the fluids is usually responsible for viscous fingering, this occurs when the pressure on the low viscosity side is higher. We report here a reverse effect when a non-Newtonian fluid is squeezed between two plane surfaces by applying a force. We observe that a wave-like irregularity develops on the interface, though the viscosity of the air surrounding the fluid is negligible compared to the apparent viscosity of the thick potato starch gel under study. Development of the wavelength of the undulations as a function of the fluid composition and other factors is studied. We suggest a qualitative explanation for this effect, which is observed only in non-Newtonian fluids. Keywords:

Squeeze flow; Fingering; Hele-Shaw cell; Interfacial instability

PACS Nos.: 47.15.gp; 47.50.-d; 47.54.-r; 47.55.-t

1. Introduction Viscous fingering is a well known surface instability, which arises when a less viscous fluid is forced under pressure into a fluid with higher viscosity [1, 2]. The phenomenon has been widely studied in the Hele-Shaw cell, which is a very simple set up, consisting of two plane plates separated by a narrow gap. In the normal Hele-Shaw cell the gap is fixed [3], while in the lifting Hele-Shaw cell (LHSC) the separation between the plates can be varied at a constant velocity [4, 5], or with a constant force [6]. A drop of the high viscosity fluid is placed between the two plates and the low viscosity fluid, usually air, surrounds it. In the compression mode the upper plate is lowered until it squeezes the fluid drop down to the desired thickness. Next comes the separation mode, when the upper plate is slowly lifted. Viscous fingering patterns are observed during lifting, provided certain criteria are satisfied [6–8]. During compression, on the other hand, the fluid blob spreads out in a near-perfect circle, the periphery remaining smooth and stable. Most of the experiments cited are done with

Newtonian fluids. We report here a new observation seen when the fluid is a non-Newtonian gel. In this case, the interface develops an instability on compression, forming finger-like protrusions, which grow and may even exhibit tip-splitting. Potato starch gel of different concentrations is the viscous fluid in the present experiments. A similar experiment has been reported earlier [9], where it has been shown that the area of contact between the plate and fluid does not increase monotonically, but oscillates, showing a visco-elastic behavior and the oscillations are clearly seen in the video clip [10]. Such instabilities are of interest in practical problems related to wetting and spreading such as, electrowetting [11], Rayleigh–Taylor instability [12] and spin-coating [13]. ‘Normal’ viscous fingering under varied conditions is still a topic of active research and new theoretical as well as experimental studies have been published. For example, He et al. [14] have studied the instability between the interface of two immiscible reactive fluids. Mishra et al. [15] have studied double diffusive effect in miscible fingering. Rocha and Miranda [16] have studied the effect of a curvature dependent surface tension. Lifting Hele-Shaw flow with a yield stress fluid [17] and criteria for determining the number of fingers [18] in a theory including

*Corresponding author, E-mail: [email protected]

Ó 2014 IACS

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capillary action and wetting effects have been reported along with a study with an elastic interface [19]. The ‘reverse’ fingering, i.e. instability in a fluid-fluid interface with high pressure on the higher viscosity side in several different situations has been reported before [19– 22]. Troian et al. [20] have reported spreading of a surfactant on a thin layer of water, which they attribute to the Marangoni effect. Chang and Liang [21] have reported reverse fingering in a surfactant at very high interface velocities, in the presence of a wetting layer. Presence of a wetting surfactant layer has been shown to induce reverse fingering by Krechetnikov and Homsy [23]. They have also shown that Marangoni stresses and surfactant concentration gradients are responsible for the phenomenon. Tang et al. [22] have found such an effect in a particulate suspension, which became inhomogeneous under pressure. The material under study, potato starch and similar starches have very complex rheology, which leads to interesting effects. When a paste is made in cold water, a shear thickening occurs leading to the entertaining behavior shown in several videos e.g. [24] and are reported earlier [25]. On the other hand, if an aqueous solution is heated, resulting in gelatinization of the starch, one has a shear thinning fluid, which we have studied in this paper.

M Dutta Choudhury and S Tarafdar

Fig. 1 Stress plotted against shear rate for x = 2.5. Inset shows the average radius of the approximate circular blob as a function of time, the initial and final shear rates calculated from regions marked by arrows, are matched with corresponding points on the stress versus strain rate curve to show that the squeezing experiments are conducted in the shear thinning regime

recordings were analyzed using the software Image-ProPlus, one video clip clearly showing two cycles of oscillations in area of contact can be seen in Ref. 10.

3. Results and discussion 2. Experimental details The non-Newtonian fluid under study was an aqueous gel of potato starch (Lobachemie, Mumbai). In our experiment, x g of potato starch was dissolved in 100 ml of distilled water and was heated up for 10 min and boiled for 2 min, stirring continuously to form a clear gel. The solution was allowed to cool for 1 h and 0.6 g of VESCO food colour was added to enhance the contrast. The food color contained 3.65 wt% of dye. All spreading and rheological measurements, where done with the same sample containing the same amount of dye. Experiments were performed for x = 1.5, 2.5 and 3.5 g, which were repeated several times to ensure reproducibility. Rheology measurements for the samples were done in a Rheometer (Gemini Rheometer, Bohlin) using cone-plate geometry. A micropipette was used to place a droplet of the nonNewtonian fluid on a smooth circular glass plate of diameter 15.5 cm and thickness 1.5 cm. The gel looked homogeneous and did not show any phase separation during extraction by the micropipette. The mass of the droplet was varied from 0.04 to 0.06 g. A similar circular glass plate was placed on top of the drop. The weight of the upper plate was 0.56 kg. After 2 s a weight W was placed on the upper plate and a CCD camera (WATEC-202D, Japan) below the lower plate recorded the spreading of the area of contact. In different experiments, W was varied from 2 to 5 kg. The video

There is a noticeable deviation from linearity in the stress– shear strain curve. A strong non-Newtonian behavior is seen in the plots of apparent viscosity against strain rate. The log-log plot of shear stress against strain-rate for x = 0.5, 1.5 and 2.5 has a linear nature for low strain rates, showing shear thinning obeying a power-law [9]. r g_ m

ð1Þ

_ the strain rate and m the where r is the shear stress, g, power-law exponent. The slopes are m = 0.5 for x = 0.5 and 1.5. For x = 2.5 m = 0.6 for low strain rates, but for higher strain rates there is a deviation in linearity of the log–log plot and an approximate slope of 1.05 is obtained, indicating weak shear-thickening. The samples also exhibit weak thixotropy. We have shown in Fig. 1 the stress plotted against strain rate for the x = 2.5 sample. Initial shear thinning is followed by a slight shear thickening for high strain rates. In the inset of Fig. 1 we show increase in the average radius for the present squeeze flow experiment with time. The correlation between the two sets of data is discussed later. It may be noted that in Dutta Choudhury et al. [9] the apparent viscosity is plotted against strain rate. In this plot the initial fall in viscosity corresponding to shear-thinning is clearly evident, but the very small change in slope in Fig. 1 for high strain rate (which corresponds to weak shear thickening)

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Fig. 2 Successive stages of developing surface instability in a squeezed drop of potato starch gel for x = 2.5 Fig. 3 The change observed in successive trials on the same plate for x = 2.5 is shown. (a), (b) The 2nd (T-2) and 4th (T-4) trials respectively. (c) A closeup of trial 7 (T-7), exhibiting strong fingering and tip splitting marked by arrows

becomes insignificant. So shear-thinning is the predominant behavior of this sample and we show later that the range of strain rates we are concerned with in the present squeezeflow experiment is well within the shear-thinning regime. When the upper plate is placed on the fluid droplet, it spreads out in a circle. On placing the weight on the plate, the circle expands. Initially the expansion is monotonic, but after a point the increase becomes oscillatory and undulations appear on the periphery. The oscillatory spreading of the area has been analyzed in detail in Dutta Choudhury et al. [9]. Here we focus on the interfacial instability. For x values of 1.5, 2.5 and 3.5 a smooth circle is initially formed with a radius close to 0.8 cm in each case. After placing, the weight W = 5 kg the periphery first increases smoothly, but later starts showing an instability as reported by Dutta Choudhury et al. [9, 10]. One can see roughly periodic undulations on the periphery which become more pronounced with time. The corrugated appearance of the boundary

becomes stronger as x increases. Figure 2 shows three successive stages of the evolving instability for x ¼ 2:5. The experiments have been repeated several times on the same plate after washing and careful drying. However, it appears that later observations for the same fluid show more prominent undulations. Figure 3(a)–3(c) show respectively the 2nd, 4th and 7th trials for the same sample on the same plate. Figure 3 compares the contour of the 4th trial with the 2nd trial for x = 2.5. A close-up of a part of the periphery for the 7th trial is also shown, where sharper growth and tip splitting of the fingers, very similar to the appearance of miscible viscous fingering [26] is observed. There is a reproducibility observed for the nth trial in different experiments done under identical conditions starting with a fresh substrate. The instability becomes significant and measurable simultaneously with the appearance of oscillations [9] in the time development of the area of the blob.



Fig. 6 The number of peaks versus average radius for trial number 2 shown for three separate experiments with x = 2.5 potato starch during squeeze flow. The open circle, triangle and star symbols corresponding to the three sets, performed under identical conditions, lie very close and overlap in many cases demonstrating reproducibility of the results Fig. 4 The number of peaks versus time for different concentrations of potato starch during squeeze flow

Video clips showing the development of the fluid blob are separated into still frames. When the fingers are discernible, the number of fingers (Nf ) at an instant of time (s) and the average radius of the blob (Rav ) are noted. This gives the wavelength for the instability as k ¼ 2pRav =Nf

Fig. 5 The number of peaks versus average radius for different concentrations of potato starch during squeeze flow. The result is higher for later trials

3.1. Analysis of finger development The following procedure has been adopted to measure the details of the contour of the liquid blob precisely. The edge of the blob is digitised and the size of each pixel is determined by comparing with a known ‘scale’ photographed at exactly the same magnification. This leads to 1 pixel 3 lm, which corresponds to the precision of our measured values. The wavelength of viscous fingers is easily measured by simply counting the number of peaks in a pattern [27, 28].

ð2Þ

We show in Figs. 4 and 5 respectively Nf as a function of time and as function of Rav for different concentrations. To demonstrate reproducibility for a given trial number n, experiments are performed several times under identical conditions, using a fresh substrate for different sets. Results for Nf of three such sets for x = 2.5 are shown in Fig. 6 for trial number n ¼ 2. The data points lie very close to each other, often overlapping. It must be noted that the radius does not increase monotonically with time due to the oscillations in area with time. To demonstrate the time development, we have also plotted in Fig. 7 the graph with a line joining points in chronological order for one of the data sets shown in Fig. 6. The resulting plot looks like a random walk, but here the back and forth motion shows clearly the non-monotonic variation of the wavelength as the radius increases and decreases repeatedly with time. When the oscillations set in, for greater accuracy we have defined an inner and outer radii for the fluid blob, by fitting an outer and an inner circle to the blob, as illustrated in Fig. 8(a) and 8(b). The wavelength calculated considering the inner and outer radii are slightly different, if calculated from kðinÞ ¼ 2pRin =Nf ; kðoutÞ ¼ 2pRout =Nf

ð3Þ

The inner and outer radii as a function of time are shown in Fig. 9. The variation of the amplitude of the disturbance with

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have employed this method too for the pattern shown in Fig. 2 for 10 s. The periodogram in Fig. 11, shows that the principal peak gives the same frequency (and hence wavelength of 0.3 cm) as the simpler ‘peak counting’ method. Due to the inherent noise in the data a number of frequencies appear in the periodogram, but the wavelength from the most prominent peak matches the ‘peak counting’ result. 3.2. Calculating shear rate

Fig. 7 Chronological development of the number of peaks as function of the average radius of the blob is seen by following the line joining the points. The evidence of oscillations in the area of contact and hence average radius is clear

time can be estimated from the vertical difference between the outer and inner radii. The maximum variation is about 4 %, the variation in the area calculated with and without the undulations also comes out near 4 %. The amplitude is seen to increase and later saturate after a certain time. The amplitudes of oscillation are around 0.10–0.14 for x = 1.5 and 2.5. Variation of the corresponding wavelengths with time for x = 2.5 are shown in Fig. 10. Since oscillations in area have set in, the radius Rav oscillates with time. The nonmonotonic behavior of k with time is not revealed in Fig. 10 . In spite of the fluctuations, there is an overall decrease in k with the radius. All results shown are for the second trial. In the later trials, where the fingers split it is not possible to count the fingers accurately, so we do not attempt wavelength calculation in these cases. A more objective method of identifying the wavelength would be to perform a Fourier transform or Lomb periodogram of the outer contour of the fluid blob [29]. We Fig. 8 (a) represention of squeeze-flow fingering, showing our definitions of inner and outer radii, (b) together with the photograph of a real sample with the inner circle fitted to it

The approximate shear rate for the squeeze flow can be calculated as follows. We plot the average radius of the blob as a function of time in the inset of Fig. 1, which gives a smooth curve. The oscillations are not visible as the relative oscillation in total area is quite small. We have calculated the shear rate as Vinterface =d, where Vinterface is the velocity of the interface and d the film thickness averaged over the range over which the velocity is calculated. The radius is seen to increase sharply at first and much more slowly at a later time. We have calculated the velocity for the slow and fast regimes as marked in Fig. 1. Unlike earlier reports of reverse fingering, we find in particular, that this effect is observed for non-Newtonian fluids, where the apparent viscosity changes appreciably with shear rate. A careful scrutiny of the shear stress versus strain rate in Fig. 1 shows that after the initial rise with a sub-linear exponent, the stress variation becomes almost linear, but towards the highest strain rates there is a small upward deviation showing a weak shear-thickening. So shear-thinning as well as shear-thickening are seen on varying the strain rate over a wide range. One may ask therefore in which regime is the squeeze flow experiment done? To answer this, we consider the following argument— The average film thickness is larger in the fast regime and smaller in the slow regime, but the variation in d is much less compared to variation in Vinterface , so the strain rate in the initial fast regime is larger than the later slow regime. We have indicated the corresponding strain rates by similar arrows in Fig. 1.



Fig. 11 Lomb periodogram of the undulating outline in a second squeeze flow trial with x ¼ 2:5 potato starch. The wave-number corresponding to the highest peak (indicated by the arrow) matches the value k * 0.3 cm found by ‘peak counting’ Fig. 9 The inner and outer radii shown as function of time for x ¼ 2:5. potato starch during squeeze flow. Development of the vertical difference between the two traces represents the amplitude of the instability as a function of time

We conclude that the squeeze experiment is conducted in the shear-thinning regime. In earlier studies of squeeze flow and viscous fingering [6, 30] ‘reverse fingering’ has not been observed. The instability occurs. Moreover only where an oscillatory spreading is seen, which is interpreted as a non-Newtonian visco-elastic effect [9]. Under such a condition, the reverse fingering may tentatively be explained as follows. When the viscous gel flows outward under compression, a smooth interface is seen, but when oscillation starts and the area contracts, the pressure is higher on the low viscosity side (air, in this case). So the instability appears and undulations form. In the following expansion phase, the time span is not enough to smoothen out the rough interface, before the next contraction starts. So the fingers continue to grow.

4. Conclusions We have experimentally demonstrated a reverse fingering, with a higher viscosity fluid penetrating into a lower viscosity fluid, (air in this case) in the form of undulating fingers. Variation of the finger wavelength with fluid concentration and time are studied. The effect is prominent when the high viscous fluid shows shear-thinning in the regime of the squeeze flow experiment and appears to be related to the oscillatory spreading in such fluids observed earlier [9]. We observe further that the instability is stronger as experiments are repeated on the same substrate, showing that a microscopically thin layer of adsorbed fluid molecules remains on the surface to enhance the effect. This is similar to the results obtained by Chang and Liang [21]. A tentative physical explanation for the phenomenon is offered. Acknowledgments M. Dutta Choudhury is grateful to Alumni Association, Jadavpur University for Research Grants endowed by S Das and to CSIR for award of a senior research fellowship. A Giri is sincerely acknowledged for technical help and advice. Authors thank T Dutta, S Kitsunezaki and A Nakahara for helpful suggestions and discussion. We also thank CGCRI, Kolkata for helping Rheology measurement.

References

Fig. 10 The wavelengths versus time for x ¼ 2:5 of potato starch during squeeze flow

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