Spreading of a Herschel-Bulkley Fluid Using Lubrication Approximation

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1 Journal of Undergraduate Research 7, 1-14 (214) Spreading of a Herschel-Bulkley Fluid Using Lubrication Approximation Nadiya Klep 1Department of Material Science and Engineering Clemson University, Clemson, South Carolina David D. Pelot Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, Illinois Alexander L. Yarin Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, Illinois and College of Engineering, Korea University, Seoul, South Korea In this work the flow of 1.5% Carbopol-94 solution was studied as it was transported through a wedge-like system at different angles, entrance heights, exit heights, and velocities. The high-speed video recordings of the flow were processed by tracking air bubbles entrapped in the solution. This data was used to find the velocity field of the solution beneath the wedge. Also, the force imposed by the solution on the inclined top plate of the wedge was measured. The theoretical description in the framework of the lubrication approximation for Newtonian fluids was used for comparison to the experimental results. It was found that Herschel-Bulkley fluids exhibit qualitatively similar behavior to the Newtonian fluid, such as the reverse flow near the top of the wedge. Overall, this work proves that the lubrication approximation can be effectively used to characterize the flow field of non-newtonian Herschel-Bulkley fluids at the wedge angles at least up to 2 degrees. Introduction The spreading of viscous fluids is something that is encountered in everyday life. In creams, lotions, foods, and construction materials, spreading behavior is one of the main flows considered during the preparation process. The motivation behind this study is to model the spreading of soft solid materials, which can be described in general as Herschel-Bulkley fluids. A Herschel-Bulkley fluid has a shear rate dependent viscosity and a yield stress, meaning that it can be spread onto a surface without running or dripping. Creating a product that can be applied smoothly and quickly is a competitive advantage for a company that can lead to a more profitable business for both the supplier and buyer. In order to model such a compound, a well-known transparent Herschel-Bulkley fluid was used. Carbopol- 94 R, a high molecular weight polyacrylic acid (PAA), is a commercially available thickener often used as an ideal non-newtonian fluid. 1 A study that was published in February shows that under basic 1-dimensional flow conditions the Herschel-Bulkley model describes the fluid flow of Carbopol sufficiently. 2 According to the manufacturer (Lubrizol R ), it has high relative viscosity, high yield stress, and is a clear gel when neutralized. 3 Carbopols shear thinning behavior can be described by the Herschel-Bulkley model τ < τ c γ = ; τ > τ c τ = τ c + K γ n (1) where τ and γ are the shear stress and shear rate, respectively, τ c is the yield stress, and K and n are the consistency index and the behavior index, respectively. 4 Experimental Fluid Preparation A 1.5% Carbopol-94 R (Lubrizol) aqueous solution was prepared by following Novean s specifications, and is specifically described next. First, 3 g of Carbopol powder was dissolved in 197 ml of water. Carbopol was slowly added to the water, while stirring on a hot plate at 5 C, with a magnetic stirrer set to the highest sustainable setting. The solution was allowed to stir overnight. Next, a 1 M NaOH solution was prepared by dissolving 2.8 g of NaOH pellets in 7 ml of water. Finally, the acidic Carbopol solution was neutralized by slowly adding 1 M NaOH and checking the ph with phstrips. Apparatus Setup The wedge-flow apparatus in Figure 1 consists of a mobile bottom plate with an adjustable velocity (U) and a stationary spreading plate, which can be adjusted to the desired angle, α, entrance height, h 1, and exit height, h 2. A force gauge (IMADA model DS2-11) is fitted normal to the edge of the top plate to measure the force on the fluid as it moves through the system. The set-up was powered by a LEESON A.C. motor (Model: CM38P17EZ12C). The exit height, h 2, was adjusted to a minimum value of

Journal of Undergraduate Research 7, 1-14 (214) x = x L (3) where x is the distance from the origin and L is the length created by the projection of the wedge onto the x-axis.

b) Entrance height, h 1, exit height, h 2, length, L, angle, α, and velocity of moving bottom plate, U, are labeled. Image of setup with Carbopol in the wedge.

2 Journal of Undergraduate Research 7, 1-14 (214) x = x L (3) where x is the distance from the origin and L is the length created by the projection of the wedge onto the x-axis. Lastly, the force gauge located at x = and y = h 1 collected data and the maximum value was recorded. Theoretical Discussion FIG. 1: a) Sketch of experimental setup. b) Entrance height, h 1, exit height, h 2, length, L, angle, α, and velocity of moving bottom plate, U, are labeled. Image of setup with Carbopol in the wedge. The hydrodynamic theory of lubrication is a wellstudied and understood phenomena when it comes to Newtonian fluids under limited conditions. It is applied to journal bearing and extruder screws, where the velocities and pressures are high, the angle between the moving and stationary parts is small, and the viscous forces predominate over inertia forces. Under these conditions the geometry of the system can be reduced to a wedge shape with a moving bottom plate of length, L, and an entrance and exit height, h 1 and h 2, respectively..8 mm and a maximum value of 1.4 mm and the entrance height was adjusted to create angles of 5, 1, and 2. Six experiments were completed combining different h 1 and h 2 values as well as two more experiments where the velocity was increased for one angle setting from.167 m s 1 to.24 m s 1. A Phantom Miro ex4 high-speed camera (model: MIROEX4-124MC) was used to record each trial. Then the video data was converted into JPEG images using the Phantom Video Player program. FIG. 2: Coordinate axis of the experimental setup with velocity field shown. Processing Techniques The thickness of the Carbopol gel film was measured with a wet-film thickness gauge after the Carbopol exited the wedge system in order to confirm the exit height of the wedge. The JPEG images were analyzed with the use of a MATLAB R212a program. The bubbles were tracked manually to allow for higher accuracy than an automatic tracking program would provide. Then, their velocities were plotted using OriginPro 8.6. Contour graphs were produced for each trial. A dimensionless parameter k was used to scale the y-axis of the graphs k = h 1 h 2 (2) where h 1 is the entrance height and h 2 is the exit height. Similarly the x-axis was scaled to become a dimensionless x scale Since the velocity in the y-direction, v, is much smaller than the velocity in the x-direction, u, the vertical motion under the wedge is ignored. Using Schlichtings approximations from Boundary Layer Theory 5 dp dx = µ 2 u y 2 (4) Where dp dx is the pressure gradient, µ is the viscosity, and 2 u y is the second derivative of the horizontal velocity 2 in the y direction. Then, applying boundary conditions of the system y =, u = U; y = h, u = (5) where U is the moving plate velocity and h is the equation of the wedge. The equation for velocity becomes 11 c 214 University of Illinois at Chicago

3 Journal of Undergraduate Research 7, 1-14 (214) ( u = U 1 y ) ( h2 dp y 1 y ) h 2µ dx h h (6) Due to conservation of mass, the volume flow in every part of the system must be constant Q = h(x) u dy = const. (7) Evaluating equation (7) and solving for the pressure gradient ( dp U dx = 12µ 2h 2 Q ) h 3 Using the boundary conditions of gives (8) x =, p = p ; x = L, p = p (9) p(x) = p + 6µU x dx x h 2 12µQ dx h 3 (1) where L is the length of the wedge in the x direction, p is the pressure outside the wedge, and Q = 1 L 2 U / L dx dx h 2 h 3 (11) The wedge is flat in this case, therefore, the equation of the wedge can be described as h = h 2 h 1 x + h 1 (12) L Inserting equations (12) and (11) into equation (1), then solving for the pressure distribution p(x) = p + ( 6µUL ) (h 1 h)(h h 2 ) h 2 1 h 2 2 h 2 (13) The force due to the normal stress, or pressure force, can be found as P = L p dx = 6µUL2 (k 1) 2 h 2 2 ln(k) 2(k 1) (k + 1) (14) The force due to the shear stress can be found as F = L ( ) du µ dy dx = µul y= (k 1)h 2 4ln(k) 6(k 1) (k + 1) (15) The coordinate for the center of pressure is given by 2k x c =.5L k 1 k 2 1 2kln(k) (k 2 l)ln(k) 2(k 1) 2 (16) Since the center of pressure and force gauge are at different distances from the pivot at x = L, the torque balance takes the form F exp (1 x exp ) = (P + F )(1 x c ) (17) where F exp is the force measured by the force gauge, and x exp is the position of the force gauge. In this experiment x exp =. Results and Discussion As bubbles moved within the fluid, their position was tracked in time and a velocity profile was created. The velocity profiles, represented as contour plots, are normalized using the velocity of the moving plate. Also, a thick black line is added to designate the position of the wedge. This was completed for each experimental trial. The experimental parameters and the calculated viscosity (17) are shown in Table 1. Note some locations inside the wedge were not able to be visualized. TABLE I: Summary of results indicating the name of the trial, (a)-(h), the angle of the wedge, α, the exit height, h 2, the dimensionless scaled entrance height, k, velocity of the bottom plate, U, the maximum force measured, F, and the calculated viscosity, µ, using (17) Trial Angle α h 2 k Velocity Force, F Calculated Fig.3 (deg) (mm) U, (m s 1 ) (N) Viscosity (Pa s) a b c d e f g h c 214 University of Illinois at Chicago

4 Journal of Undergraduate Research 7, 1-14 (214) As expected from the velocity profile in (6), the fluid near the moving plate up to the height of the exit moves at speeds near the maximum velocity, this is shown by colors yellow, orange, and red in Figure 3. When the fluid nears the wedge the movement of the bottom plate is not felt as strongly as the pressure gradient pushing the fluid backwards causing a negative velocity, shown by different shades of blue. The trend of increased and faster backflow can be seen as k increases in Figure 3. From Figure 3 (d): k = 19.6, (c): k = 33.5, (f): k = 37.6, to (e): k = 65.1 the concentration and the intensity (shade) of blue gets larger indicating a larger amount and faster reverse flow. The graphs are all scaled from -.2 (dark blue) where the negative indicates reverse flow at 2% of U to 1(red) which correlates to 1% or U in the forward direction. The force gauge data in Table 1 reveals three trends of the fluid that agree with the theory. Keeping the exit height, h 2, constant, the force decreases as k increases. Figure 3 (a), (c), and (e), k = 17.3, k = 33.5, k = 65.1, respectively all have h 2 =.8 mm (the low value) and the forces respectively decrease from 19.5 N, 14.5 N, to 8.4 N. Similarly in the case of h 2 equal to 1.4 mm (the high value), Figure 3 (b), (d), and (f), k = 1.3, k = 19.6, k = 37.6, respectively decrease in force from 13.4 N to 11.5 N to 7.5 N. The trend for keeping k constant but increasing h 2, as in trials (a) and (d), which had k set at 17.3 and 19.6, two relatively close values, the force decreases from 19.5 N to 11.5 N. The same trend was present in trials (c) and (f), where k was set to 33.5 and The force decreased from 14.5 N to 7.5 N as h 2 increased from.8 mm to 1.4 mm. The third trend observed was the increase in velocity while keeping both h 2 and k constant causing the force to increase. From trials (c) and (g), where h 2 was set at.8 mm and k was at 33.5, when the velocity was increased from.167 m s 1 (c) to.24 m s 1 (g) the force increased from 14.5 N to 16.5 N. Also from the contour (c) and (g) it can be seen that the reverse flow velocity increases as U increases due to the increase in the intensity of the blue shade. The same trend for both reverse flow and increased force were seen in trials (d) and (h) where h 2 was set at 1.4 mm and k was at 19.6, when the velocity was increased from.167 m s 1 (d) to.24 m s 1 (h) the force increased from 11.5 N to 14.5 N, and the contour map from (d) to (h) show increased intensity of blue. The forces due to the normal and shear stress are both proportional to velocity as in equations (14) and (15); therefore, any increase in velocity should be reflected in a directly proportional increase in force measured. Indeed, the velocity ratio is approximately equal to the force ratio between (d) and (h) as well as between (c) and (g). Inside the wedge there exists different velocity gradients, or shear rates. Since the viscosity of Carbopol is shear rate dependent (Figure 4), an estimation of the theoretical viscosity can be compared to the empirically calculated viscosity. An approximation of the maximum theoretical shear rate existing at the exit of the wedge is du/dy =.167 m s 1 /(8x1 4 m) = 29 s 1, corresponding to a viscosity of.9 Pa s. On the other hand, an approximate minimum theoretical shear rate existing at the entrance of the wedge is du/dy =.167 m s 1 /(5x1 2 m) = 3.3 s 1, corresponding to a viscosity of 36 Pa s. In Table 1, all calculated viscosities exist between the aforementioned theoretical viscosity values. FIG. 3: Contour plots showing velocity field for each experiment in Table 1. The legend shows the ratio of velocity to maximum velocity. FIG. 4: Viscosity of Carbopol 94 with respect to shear rate for several ramps (diamonds) using a vane type propeller in a TA instruments rheometer. The fitting line gives values of n =.1 and K = 14.5 Pa s n. 13 c 214 University of Illinois at Chicago

5 Journal of Undergraduate Research 7, 1-14 (214) Conclusion The lubrication approximation theory was successfully extended to include larger angles and the use of a non- Newtonian fluid. The trends that were observed in the experiments are supported by the theory. The velocity profiles showed forward velocity near the bottom moving plate as well as backflow near the back of the non-moving top late. Also, the three force trends follow the predicted force trends. The empirical viscosities were found to be in agreement with the theoretical found viscosity values. Acknowledgements The authors would like to thank the National Science Foundation for providing the funding and support for this research project through NSF Grant # We would also like to thank the program coordinators, Professor Christos Takoudis and Professor Gregory Jursich for running and organizing this Research Experience for Undergraduates program. 1 E. Weber, M. Moyers-Gonzalez, and T. Burghelea, Journal of Non-Newtonian Fluid Mechanics , 14 (212). 2 D. D. Pelot, R. Sahu, and A. Yarin, Journal of Rheology 57, 719 (213). 3 Tech. Rep., The Lubrizol Corporation (29). 4 P. Coussot, L. Tocquer, C. Lanos, and G. Ovarlez, Journal of Non-Newtonian Fluid Mechanics 158, 85 (29). 5 H. Schlichting, Boundary-Layer Theory (McGraw-Hill, Inc, 1987). 14 c 214 University of Illinois at Chicago

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