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Research Papers

Experimental and Numerical Models of Three-Dimensional Gravity-Driven Flow of Shear-Thinning Polymer Solutions Used in Vaginal Delivery of Microbicides

[+] Author and Article Information
Vitaly O. Kheyfets

Department of Mechanical Engineering,
University of Kansas,
Lawrence, KS 66045

Sarah L. Kieweg

Department of Mechanical Engineering,
University of Kansas,
Lawrence, KS 66045;
Department of Obstetrics and Gynecology,
University of Kansas Medical Center,
Kansas City, KS 66160
e-mail: kieweg@ku.edu

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received November 23, 2012; final manuscript received March 19, 2013; accepted manuscript posted April 4, 2013; published online May 9, 2013. Assoc. Editor: Ram Devireddy.

J Biomech Eng 135(6), 061009 (May 09, 2013) (14 pages) Paper No: BIO-12-1580; doi: 10.1115/1.4024140 History: Received November 23, 2012; Revised March 19, 2013; Accepted April 04, 2013

HIV/AIDS is a growing global pandemic. A microbicide is a formulation of a pharmaceutical agent suspended in a delivery vehicle, and can be used by women to protect themselves against HIV infection during intercourse. We have developed a three-dimensional (3D) computational model of a shear-thinning power-law fluid spreading under the influence of gravity to represent the distribution of a microbicide gel over the vaginal epithelium. This model, accompanied by a new experimental methodology, is a step in developing a tool for optimizing a delivery vehicle's structure/function relationship for clinical application. We compare our model with experiments in order to identify critical considerations for simulating 3D free-surface flows of shear-thinning fluids. Here we found that neglecting lateral spreading, when modeling gravity-induced flow, resulted in up to 47% overestimation of the experimental axial spreading after 90 s. In contrast, the inclusion of lateral spreading in 3D computational models resulted in rms errors in axial spreading under 7%. In addition, the choice of the initial condition for shape in the numerical simulation influences the model's ability to describe early time spreading behavior. Finally, we present a parametric study and sensitivity analysis of the power-law parameters' influence on axial spreading, and to examine the impact of changing rheological properties as a result of dilution or formulation conditions. Both the shear-thinning index (n) and consistency (m) impacted the spreading length and deceleration of the moving front. The sensitivity analysis showed that gels with midrange m and n values (for the ranges in this study) would be most sensitive (over 8% changes in spreading length) to 10% changes (e.g., from dilution) in both rheological properties. This work is applicable to many industrial and geophysical thin-film flow applications of non-Newtonian fluids; in addition to biological applications in microbicide drug delivery.

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References

Figures

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Fig. 1

Coordinate system and definitions of spreading characteristics in the x (axial) direction and the y (lateral) direction. All experiments and numerical simulations in this study used an α = 30 deg inclination angle, except for the steep inclination angle α = 60 deg required for the similarity solution in Fig. 3.

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Fig. 2

Domain (Ω) for the numerical method. The domain is divided into N points along the axial direction and M points along the lateral direction. Points along the axial and lateral direction are divided by Δx and Δy, respectively. To improve computational performance, the domain size (and thus N and M) expanded over time to accommodate the spreading fluid, such that the fluid never neared the boundaries, and calculations never used boundary values for h. The spatial index for points in the computational domain is k, and the M1/2 indicial notation (used in Eqs. (11) and (13)) indicates the temporary designation of the flux at the node interfaces between the k, k+M, and k-M grid nodes.

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Fig. 4

Example of experimental images and reconstruction of surface topography. (a) Comparison of photo image (immediately after the gel was dispensed on the spreading surface) with resulting digital surface topography of three repeated experiments (runs 1–3) of the 2.7% HEC gel. Left: Top and side view of the photograph image of experiment initial condition. Right: Top contour and side profile of the resulting digital surface topography. (b) Isometric views of the evolution of the digital surface topographies of a spreading experiment at t = 0, 60 and 120 s (RUN1 of 2.7% HEC gel). (c) Diagram showing how the experimental top and side view were used to define the footprint, height, and cross-sectional shape of the surface topography at any time point (see Appendix). The values for dt, db, and hpeak (in mm) were measured from the footprint (top view) and side-profile images along each value of x, and used to define the approximate lateral shape not seen by the camera [see Eq. (A1) in the Appendix]. The resulting measurements of h(x,y) at t = 0 s defined the initial condition for the experimental shape (real IC) and were used as input in the numerical model. The maximum length, width, and height obtained from t = 0 s footprint and side-profile images were also used to describe the approximate IC.

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Fig. 5

Example experimental result with corresponding numerical simulations. Sample spreading characteristics (Lfront, Wmax, hmax: maximum length, width, and height at each time point) and the axial velocity of the moving front for a sample run of 2.4% HEC (run 1) with real IC used in the numerical simulations. This figure is an example showing that both the 2D and 3D numerical models overestimate axial spreading when compared with experiment, but the 3D model is an improvement over the 2D model (see Table 3 for quantified errors). (a) Black (left axis): Maximum axial length (Lfront) over time for 2D and 3D models, and experiment. (a) Red (right axis): Velocity of the moving front over time for 2D and 3D models, and experiment. (b) Black (left axis): Maximum lateral width (Wmax) over time for 3D model and experiment. (b) Red (right axis): Maximum height (hmax) over time for 2D and 3D models, and experiment.

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Fig. 6

Experimental and numerical results for all nine experiments. Comparison of axial spreading of all nine experimental runs (three runs [R1, R2, R3] of 2.4%, 2.7%, and 3.0% HEC concentration), and the matched numerical simulations that used the corresponding real experimental initial condition (real IC). Experiment (dot), 3D numerical model (solid), 2D numerical model (dashed).

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Fig. 7

Comparison of real versus approximate IC in the 3D numerical model. An example of 3D numerical results using two different initial conditions (IC) and compared with experiment for a 2.7% HEC gel. Top left: Surface topography of real IC. Bottom left: Surface topography of approximate IC, obtained using Eq. (24) with bulk geometry dimensions of real IC. Top right: Axial spreading (Lfront) versus time for computations using real and approximate initial conditions, compared with experiment (same as Figs. 4—2.7% HEC run 1). Bottom right: Axial spreading versus time for computations using approximate and real initial conditions at early stage of spreading. This figure also illustrates how using the real IC removes the waiting time solution.

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Fig. 8

Parametric study of axial spreading. (a) Contour lines of axial spreading (Lfront [=] cm) as a function of power-law rheological parameters. Contour lines were drawn using axial spreading values at t = 90 s from nine 3D numerical simulations (combinations of n = 0.3, 0.6, and 0.9 and m = 200, 400, and 600) with a matlab interpolation algorithm. (b) Dimensionless plot of parametric study of axial spreading. The dimensionless Lfront/H is plotted against the dimensionless time t'=t/T=t/(m/ρgH)1/n for three values of the dimensionless power-law parameter n. The lines represent all time points for each of the simulations in the parametric study. The nine dots indicate the t = 90 s end time points used in the dimensional results shown in (a) and Fig. 9, and the base of the sensitivity analysis in Fig. 10 Each dot on each of the lines represent the t = 90 s data point for m = 600, 400, and 200 P sn−1, moving left to right across the constant n line. This figure shows that within the ranges of m and n considered, both parameters have an impact on axial spreading. Also, maximum spreading in the axial direction occurred at higher values of n (near Newtonian) and lower values of m (i.e., larger values of t').

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Fig. 9

The effect of the shear-thinning index n and the consistency m (P sn−1) on the change-of-rate indicator ζ. A larger ζ represents a deviation from a constant spreading velocity. This figure shows that increasing m and decreasing n resulted in a more constant axial spreading velocity (i.e., ζ →0). The figure data points were calculated using axial spreading values at t = 90 s for nine 3D numerical simulations (combinations of n = 0.3, 0.6, and 0.9 and m = 200, 400, and 600), and lines connecting points were drawn to aid viewing data at the same n values.

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Fig. 10

Percent (%) sensitivity of axial spreading (Lfront (t = 90 s)] to changes in m and n. (a) Percent (%) sensitivity to 10% increase in n. This figure shows highest sensitivity for these simulations at m = 600 P sn−1 and n = 0.6. (b) Percent (%) sensitivity to 10% decrease in m. This figure shows highest sensitivity for these simulations at m = 200 P sn−1 and n = 0.3. (c) Percent (%) sensitivity to 10% decrease in m and increase in n. This figure shows highest sensitivity for these simulations at m = 400 P sn−1 and n = 0.6. Sensitivity calculations used 27 additional simulations and were based on percent changes from the original nine 3D numerical simulations in the parametric study (combinations of n = 0.3, 0.6, and 0.9 and m = 200, 400, and 600). Lines connecting points were drawn to aid viewing data at the same n values.

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Fig. 3

Numerical model verified through comparison with similarity solution for power-law fluid. Comparison of side-view height profile, at 90 s of spreading, obtained from the 3D numerical model and the 2D similarity solution. There is agreement between the numerical and similarity solution, verifying the numerical method. Input parameters for both solutions: α = 60 deg, m = 1000 P sn−1, n = 0.75. Note: The profile of the 3D numerical model is cut at the centerline along the lateral axis.

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