category: literaturenote citekey: chunexperimentalinvestigationflowrate2025 title: Experimental investigation of the flow rate--pressure drop relation of a viscoelastic Boger fluid in a deformable channel authors: "Chun, SungGyu; Christov, Ivan C.; Feng, Jie" year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2506.10926v1" zotero_key: IPA6ZP69 zotero_storage: TWTHAG5X collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Chun, SungGyu"

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Experimental investigation of the flow rate–pressure drop relation of a viscoelastic Boger fluid in a deformable channel

SungGyu Chun,¹ Ivan C. Christov,² and Jie Feng1, ∗

¹Mechanical Science and Engineering, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801, USA ²School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA (Dated: June 13, 2025)

We study the steady flow-induced deformation between an incompressible non-Newtonian fluid and a threedimensional (3D) deformable channel. Specifically, we provide a comprehensive experimental–theoretical framework for such flows of constant-viscosity viscoelastic (i.e., Boger) fluids, which allows us to quantify the influence of the fluid's viscoelasticity on the flow rate–pressure drop ( − Δ) relation. For a flow-ratecontrolled regime and weakly viscoelastic flow, we find excellent agreement between the theoretical prediction and experimental results, specifically providing an experimental demonstration of how both wall compliance and fluid viscoelasticity decrease the pressure drop. To definitively demonstrate the coupled effect of wall compliance and fluid viscoelasticity (change of cross-sectional area and curving of streamlines), we perform experiments in an equivalent rigid channel, showing that both the Boger fluid and a viscosity-matched Newtonian fluid have the same pressure drop. Thus, the experimentally verified theory for the −Δ relation of weakly viscoelastic flows of Boger fluids in compliant channels can now be used for microfluidic system design involving complex fluids.

I. INTRODUCTION

The field of soft hydraulics—characterized by the intricate interplay between flow-induced forces on the inherently compliant boundary of microscopic conduits—has gained considerable attention within the scientific community in recent years. This growing attention stems from the critical role of soft hydraulics across a wide range of applications, such as the design of organ-on-a-chip platforms [1], wearable electronics [2, 3], and soft robotics [4, 5], to name a few. In these applications, understanding the relationship between the flow rate and pressure drop Δ in various compliant geometries is crucial for device performance and functionality. Although the flow rate–pressure drop ( − Δ) relations for the laminar flow of Newtonian fluids in common geometries are well established in textbooks, the same is not true for non-Newtonian, specifically viscoelastic, fluids flowing through non-uniform rigid [6 –9] and deformable [10–12] conduits, despite their relevance in many practical applications such as microfluidics [13 –15] and tribology [16–18]. Importantly, both the rheology of the complex fluids and the flow-induced non-uniform conduit deformation lead to nonlinear −Δ relations [12, 19– 23], which is important to characterize for microfluidic applications [15, 24, 25]. Furthermore, microfluidic techniques have been increasingly proposed and employed to measure complex fluids' rheology and specifically their viscoelasticity [24, 26, 27]. Yet, an experimental measurement of how the interplay between viscoelasticity and wall compliance sets the − Δ relation in steady low-Reynolds-number flow within a deformable configuration remains elusive.

Previous investigations of the − Δ relations for the pressure-driven flow of viscoelastic fluids in rigid channels have predominantly relied on numerical simulations [28, 29] and experimental measurements [30, 31], providing key insights into the complex interplay between fluid elasticity and flow resistance. For deformable geometries, two-dimensional (2D) numerical simulations [32, 33] using the Oldroyd-B [34, 35], finite-extensibility nonlinear elastic (FENE) type, and Owens constitutive equations have shown that the viscoelasticity of the fluid strongly influences the forces acting on a short deformable wall segment in a channel. The viscoelastic nature of a complex fluid is expected to influence the − Δ relations [6] of such flows through non-uniform geometries. Specifically, flow-induced deformation causes the streamlines of the flow to curve, induces elastic stresses, and changes the flow resistance. Viscoelastic corrections to the −Δ relation would thus be expected. Importantly, however, at low Deborah number ≪ 1 (i.e., for weakly viscoelastic flow), most viscoelastic constitutive equations' predictions reduce to those of the Oldroyd-B model [7, 36]. Recently, employing lubrication theory with the structural deformation of a thick plate and performing a perturbation expansion in ≪ 1, Boyko and Christov [11] developed a theory for the − Δ relation under the Oldroyd-B model, taking into account the flow-induced deformation, and quantifying how both fluid viscoelasticity and channel compliance reduce the pressure drop.

On the other hand, to the best of our knowledge, there is no quantitative comparison between these theoretical predictions and experimental data for constant-viscosity viscoelastic (i.e., Boger [37, 38]) fluids. We find that the lack of prior experiments is partly due to challenges in the preparation of stable working fluids, as well as the challenges of accurately measuring the pressure drop change in weakly viscoelastic flows. In this study, we overcome these challenges and provide an experimental demonstration of the coupled effects of fluid viscoelasticity and wall compliance, guided by quantitative comparisons between theory and experiments. We design an experimental platform to measure the pressure drop, for an imposed steady flow rate, in the canonical microfluidic configuration of a three-dimensional (3D) rectangular channel with a deformable top wall.

^∗ jiefeng@illinois.edu

FIG. 1. Illustration of the configurations used for experiments and modeling. (a) Image of the experimental system and (b) schematic of the 3D rectangular channel geometry with a top deformable wall, showing the mathematical notation. The geometry consists of a compliant section of length ℓ, with a pressure port placed along the stream-wise direction to measure theΔ across ℓ between the pressure port and the outlet, which is open to the atmosphere. The initially rectangular cross-section has a width, , and undeformed height, ℎ0, with ℎ⁰ ≪ ≪ ℓ (see Table I). The thickness of the top deformable wall is, and is the vertical displacement of the fluid–solid interface. The steady imposed flow rate is .

II. EXPERIMENTAL SETUP

A. Design and fabrication of the deformable microchannel

We follow our previous work on shear-thinning [23] in designing and fabricating the microfluidic device used in the experiments. We designed the device mold using a Computer Aided Design software (Fusion 360, Autodesk) and fabricated it using a 3D printer (ELEGOO Mars Resin 3D Printer) based on the required (slender, ℎ⁰ ≪ ℓ, and shallow, ℎ⁰ ≪ ) channel dimensions (Fig. 1). We prepared the polydimethylsiloxane elastomer (PDMS, Sylgard 184, Dow Corning) with a mixing ratio of 10:1 (w/w) between the silicone elastomer base and the curing agent. Then, we poured the PDMS mixture onto the 3D printed mold and degassed the mixture under vacuum for two hours to eliminate small air bubbles. After curing the degassed mixture in an oven at 90◦C for 12 hours, we carefully removed the PDMS channel from the mold and punched holes to provide a flow inlet and a pressure sensor port (diameter = 1.07 ± 0.05 mm). Next, we treated both the PDMS channel and thin PDMS film (GASKET-500PK, SiMPore) with a 4.5 MHz hand-held corona treater (BD-20AC, Electro-Technic Products) for 30 seconds and brought them together in conformal contact for bonding. To restrict the wall deformation to the region of interest, we further attached a rigid glass slide on top of the thin PDMS film using oxygen plasma bonding, leaving only the deformable region (length ℓ, see Fig. 1) exposed. We then verified the dimensions of the fabricated channel through microscope visualization and summarized the dimensions in Table I. In addition, the Young's modulus and the Poisson's ratio of the thin PDMS were measured using a dynamic mechanical analyzer (DMA 850, TA Instruments) at 25◦C and are summarized in Table I.

B. Preparation and characterization of the working fluids

An aqueous polymer solution with constant shear viscosity ⁰ (i.e., negligible shear-thinning effects), yet exhibiting viscoelasticity (i.e., a Boger fluid) was prepared by dissolving 300 ppm of 6 MDa polyacrylamide (PAA, Polysciences), in 89 wt% glycerol (Gly, Fisher Scientific), 10 wt% deionized (DI) water, and 1 wt% NaCl (Sigma-Aldrich) [39, 40]. The Boger fluid samples were prepared by gradually dissolving the appropriate mass of powder into DI water in a cylindrical beaker, followed by continuous stirring for 24 hours to ensure complete dissolution and homogeneity. Then, we diluted the solution with 89 wt% glycerol and added 1 wt% NaCl, continuously stirring the mixture for another 24 hours to achieve a uniform composition. We estimate the polymer overlap concentration through the relation ^∗ ≈ (/)/, where is the molecular weight of the polymer, = 4³ /3 is the volume occupied by a single polymer molecule, and and are Avogadro's number and the radius of gyration, respectively [39, 40]. ∗ defines the upper limit below which a polymer solution is considered dilute. Above this threshold, polymer chains start to enter a semi-dilute regime to overlap, interact, and may thereafter form entangled networks. The calculation yields an overlapping concentration as ^∗ ≈ 700 ppm [41, 42] and therefore, our experiments utilized a dilute polymer solution at / ^∗ ≈ 0.4. For the reference Newtonian fluid as a baseline for comparison, we used an aqueous solution containing 92.1 wt% glycerol in DI water (Gly-Aq). We performed rheological measurements of the total viscosity ⁰ and solvent viscosity of the 300 ppm PAA solution, using a controlledstress rheometer (DHR-3, TA Instruments) equipped with a cone–plate geometry (diameter = 40 mm, cone angle = 1◦ ) at a controlled temperature of 25◦C. We observed that the 300 ppm PAA solution shows constant viscosity ⁰ in the range of shear rates over three orders of magnitude (1 s−¹ ≤ ¤ ≤ 10³ s −1 ), similar to that of the Gly-Aq solution in Fig. 2(a).

Next, we conducted capillary breakup extensional experiments using dripping experiments [43–45] with a custom-built setup at room temperature = (25 ± 0.5) ◦C. In the dripping experiments, the polymer solution is gradually pumped through a vertical needle with a low flow rate (≈ 5 µL/min). A fluid filament forms and subsequently thins due to the action of the capillary force. The time evolution of the minimum neck diameter of the liquid bridge is recorded. The Bond number = ² / (representing the relative importance of gravitational and capillary effects), where is the surface tension, is such that ≪ 1 for the entire process, meaning that the gravitational force is negligible in the experiments. Fig. 2(b) shows the transient evolution of for the 300 ppm PAA solution, which is initially similar to that of the Newtonian fluid, where follows a power-law relation with time. After an initial inertial and/or viscous regime, an elastic regime emerges, wherein the polymer chains stretch and generate elastic stresses [46]. In this regime, a cylindrical filament thins exponentially [47, 48] and it appears as a linear region in a semi-logarithmic plot when the shifted time is plotted along the abscissa, as shown in Fig. 2(b), where represents the time corresponding to the transition from the Newtonian to the elasto-capillary regime.

TABLE I. Physical dimensions of the slender (ℎ⁰ ≪ ℓ) and shallow (ℎ⁰ ≪ ) initially rectangular channel and the elastic properties of the deformable wall used in the experiments for measuring the pressure drop Δ as a function of the flow rate .

ℎ0	𝑤	ℓ	𝑡	𝐸	𝜈
(mm)	(mm)	(mm)	(mm)	(MPa)	(–)
0.455	2.9	26.0	0.5	1.0	0.48
± 0.005	± 0.06	± 0.2	± 0.005	± 0.05	± 0.1

TABLE II. Rheological properties and composition of all the test liquids.

Name	Composition (% w/w)	𝜂0 (Pa s)	𝜂𝑠 (Pa s)	𝜆 (ms)	𝑛
300 ppm PAA	PAA-0.03 Gly-89 DI water-10 NaCl-1	0.22 ±0.003	0.12 ±0.004	96.5	0.97 ±0.01
Gly-Aq	Gly-92.1 DI water-7.9	0.22 ±0.001	–	–	1.0

FIG. 2. (a) Experimental data for the shear viscosities cross several orders of magnitude of the shear rate ¤ for the Gly-Aq (◦), 300 ppm PAA (□) solutions and solvent of 300 ppm PAA (△), showing negligible shear thinning. (b) Transient evolution of the minimum neck diameter for the 300 ppm PAA solution obtained from the custombuilt dripping rheometry experiments, showing how the extensional relaxation time is obtained in the exponential thinning regime; here, ≈ 35 ms.

This behavior is consistent with the Oldroyd-B model, which predicts [45, 46, 49] ∝ exp −( − )/3 , from which the extensional relaxation time of the viscoelastic liquid was determined by fitting [46]. The rheological properties and composition of all the test fluids are summarized in Table II.

C. Pressure drop measurement

To generate a steady flow in experiments, we employed a syringe pump (11 Pico Plus Elite, Harvard Apparatus) connected to the channel via Teflon tubing. The syringe pump delivered the test fluid at a constant volumetric flow rate . The flow rate range was set as 0.5 mL/min ≤ ≤ 3.5 mL/min to maintain the viscous flow regime, with low Reynolds number = /(0) ≃ 10−⁴ , where = 1240 and 1190 kg/m³ for the Gly-Aq and 300 ppm PAA solutions, respectively. The differential pressure drop Δ along the streamwise length ℓ was measured using a digital pressure transducer (PX409- 015GUSBH, OMEGA) connected to a digital transducer application (OMEGA) that continuously acquired data at a sampling frequency of 1000 Hz. The pressure port in the channel was positioned such that the measured length was matched with the length of the deformable portion ℓ, as shown in Fig. 1.

III. FLOW RATE–PRESSURE DROP RELATION FOR A BOGER FLUID IN A DEFORMABLE MICROCHANNEL

The rheological characterization thus leads us to believe that the Oldroyd-B model [34, 35] suitably captures the behavior of the 300 ppm PAA solution as a constant-viscosity polymeric liquid solution with a well-defined elastic relaxation time. Turning to the theory of viscoelastic flow (under the Oldroyd-B model) through a compliant microchannel [11], three key dimensionless numbers govern the flow-induced deformation of a slender and shallow deformable channel:

$$\tilde{\beta} = \frac{\eta_p}{\eta_0}, \qquad De = \frac{\lambda}{t_r}, \qquad \alpha = \frac{u_c}{h_0}.$$ (1)

In Eq. (1), ˜ is the ratio of the polymeric viscosity = 0− to the total viscosity ⁰ of the Boger fluid. Viscoelastic effects in the flow are quantified by , which is the ratio between the relaxation time of the fluid to the residence time = ℓ/ in the channel [11, 13, 16] based on the characteristic axial velocity ∼ /(ℎ0). The theory assumes that viscoelastic effects are weak, ≪ 1. The compliance number quantifies the coupling between the fluid flow and the structural deformation [20, 50], defined as the ratio of the characteristic deformation scale = / (set by the balance of the fluid's flow-rate-based pressure scale ∼ 0ℓ/(ℎ 3 0 ) and the bending stiffness of the wall ∼ 2 ³ /[(1 − ) 4 ]) to the initial channel height ℎ0.

Boyko and Christov [11] derived a set of coupled, nonlinear ordinary differential equations for the Newtonian pressure distribution in the deformable channel, ⁰ (), and its () correction, ¹ (). Under the non-dimensionalization,

$$P(Z) = \frac{p(z)}{p_c}, \qquad Z = \frac{z}{\ell}, \qquad X = \frac{x}{w}, \tag{2}$$

and setting the outlet pressure to gauge, the governing equations are

$$\frac{\mathrm{d}P_0}{\mathrm{d}Z} = -\frac{12}{H_e[P_0]^3}, \qquad P_0(1) = 0, \tag{3}$$

$$\frac{dP_1}{dZ} = \tilde{\beta} \frac{S[P_0]}{H_e[P_0]^3}, \qquad P_1(1) = 0, \tag{4}$$

where the "effective" deformed channel height [50] is

$$H_e[P_0] = \left[1 + 3\alpha I_1 P_0 + 3\alpha^2 I_2 P_0^2 + \alpha^3 I_3 P_0^3\right]^{1/3}, \quad (5)$$

and, for convenience, we have defined

$$S[P_0] = \frac{dP_0}{dZ} \frac{36\alpha}{H_e[P_0]^6} \left[ I_1 + 4\alpha I_2 P_0 + 6\alpha^2 I_3 P_0^2 + 4\alpha^3 I_4 P_0^3 + \alpha^4 I_5 P_0^4 \right]$$

$$- \frac{dH_e}{dZ} \frac{108}{H_e[P_0]^7} \left[ 1 + 5\alpha I_1 P_0 + 10\alpha^2 I_2 P_0^2 + 10\alpha^3 I_3 P_0^3 + 5\alpha^4 I_4 P_0^4 + \alpha^5 I_5 P_0^5 \right]. \quad (6)$$

Note that S [0] and [0] are solely functions of ⁰ = ⁰ (), but they depend on the parameters and I . Here, I = ∫ ⁺1/² −1/2 () d, = 1, 2, 3, . . . are calculated from the crosssectional shape function of the wall displacement (induced by bending under the hydrodynamic pressure load) under the Reissner–Mindlin thick-plate theory [51]:

$$F(X) = \left(\frac{1}{4} - X^2\right) \left[\frac{2(b/w)^2}{(1-\nu)} + \left(\frac{1}{4} - X^2\right)\right]. \tag{7}$$

The expressions for I = I(/, ) are lengthy but straightforward to calculate; thus, we do not include them here. Although Eq. (3) with from Eq. (5) can be integrated analytically, it yields an implicit solution [11]. Therefore, we solve it numerically using Matlab's ode45 (with absolute and relative tolerances of 10−12). Then, we use cumtrapz with Δ = 2 × 10−³ to numerically evaluate ¹ () from Eq. (4)–(5) using the numerical solution for 0. Finally, Δ = ⁰ (0) + ¹ (0) is the dimensionless total pressure drop, taking into account flow-induced deformation and viscoelastic effects.

IV. QUANTITATIVE COMPARISON BETWEEN EXPERIMENTS AND THEORY AND DISCUSSION

Before comparing the theory to the experiments in a deformable channel, we validated the experimental setup using

FIG. 3. Comparison between theory, i.e., Δ = 120ℓ/(ℎ³ 0 ) (dashed line) which holds for both Newtonian and Boger fluids in straight channels, and the experimental measurements for the flow rate–pressure drop relation in the rigid channel. Symbols represent experimental data for 300 ppm PAA (□) and Gly-Aq (◦) solutions.

FIG. 4. Comparison between theory, i.e., Δ = ⁰ (0) (dash-dotted line) and Δ = ⁰ (0) + ¹ (0) (solid line) computed from Eqs. (3)– (7), and the experimental measurements (300 ppm PAA (□) and Gly-Aq (◦) solutions) for the flow rate–pressure drop relation in the compliant channel.

a rigid channel of identical dimensions. The rigid channel served as a control to ensure the accuracy and reliability of the pressure measurement system, flow rate control, and fluid injection protocols, as well as to demonstrate that there are no viscoelastic effects on the pressure drop of Boger fluids in a uniform conduit. We performed experiments with both 300 ppm PAA and Gly-Aq solutions, covering the range of flow rates used in subsequent compliant channel experiments. Fig. 3 shows that the measured pressure drops across the rigid channel fall right on the theoretical line given by Δ = 120ℓ/(ℎ³ 0 ), confirming that the experimental setup operated correctly under well-defined conditions and reproduces the textbook result.

Fig. 4 compares the theoretical predictions and experimental measurements of the −Δ relationship for the 300 ppm PAA and the Gly-Aq solution in the rectangular channel with a deformable top wall, over the parameter ranges 0.09 ≤ ≤ 0.56 and 0.02 ≤ ≤ 0.17. The solid curves in Fig. 4 are the predicted − Δ relations for the deformable channel. The theoretical prediction agrees well with the experimental data for both the Boger and Newtonian fluids. Thus, the experiments have successfully verified the theory of the − Δ relations in a weakly viscoelastic flow of a Boger fluid in compliant channels, making Eqs. (3)–(7) ready for use in applications.

For the Boger fluid, the theory starts to underestimate the pressure drop at flow rates ≳ 3.25 mL/min ( ≳ 0.16). Of course, the theory is only valid for ≪ 1. One reason that the pressure drop of the Boger fluid in the compliant channel deviates from theoretical predictions may be because the lubrication-based theory posits that viscous stresses are dominant. However, the viscoelastic effects in the flow become more significant at larger , and elastic normal stresses may also be generated, exerting additional forces on the deformable wall and enhancing the wall deflection, thus resulting in a higher pressure drop than predicted by theory [6, 8]. Furthermore, we note that for ≳ 3.25 mL/min, the residence time is estimated as = ℓ/(/ℎ0) ≲ 700 ms, which becomes comparable to the relaxation time on the order of 100 ms of the polymeric liquid. In this case, there may not be sufficient time for the elastic stresses accumulated during the flow to fully relax, and the viscoelastic fluid exhibits predominantly elastic behaviors rather than viscous behaviors, as the polymers remain stretched throughout the length of the channel [6, 8]. Consequently, the fluid retains the memory of its deformation history, leading to additional elastic normal stresses that resist the flow. These stresses lead to a larger pressure drop compared to what would be expected if the polymers had time to relax.

V. CONCLUSION

In this work, we investigated the interplay between purely viscoelastic rheology and wall compliance to provide a comprehensive experimental–theoretical understanding of the low-Reynolds-number flow of Boger fluid in a rectangular channel with a deformable top wall—a prototypical geometry in microfluidic applications. We showed good agreement between theory and experiments on the steady flow rate–pressure drop relation. This quantitative comparison holds fundamental significance in soft hydraulics and non-Newtonian fluid mechanics because it clearly quantifies the viscoelastic contribution, which is only present in the deformed channels, despite the intrinsic strong coupling between wall compliance and fluid viscoelasticity. A decade ago, Del Giudice et al. [24] argued that microrheometry results are affected by channel deformation due to the inherent compliance of PDMS-based devices. In this study, we have precisely quantified how much the pressure drop is affected and rationalized the measurements with theory. Furthermore, the experimental validation provides a reliable assessment of the adequacy of the Oldroyd-B model for describing such flows of Boger fluids. Importantly, in this work, we performed independent rheological characterization of all fluids, leading to no fitting parameters in the theory, with each fluid, wall, and geometric quantity being experimentally characterized.

An intriguing avenue for future research is to investigate how the design of the pre-inlet and post-outlet sections connected to the compliant channel affects the experimental pressure drop by influencing the elastic stresses generated by the flow, as well as via flow-induced deformation of the test section. In addition, oscillatory flows are particularly relevant in biological and industrial applications [52, 53], including harnessing viscoelastic effects in the flow [54]. Therefore, oscillatory flows of Boger may be the next step in the present experimental– theoretical framework for soft hydraulics with complex fluids.

ACKNOWLEDGMENTS

We acknowledge the use of facilities and instrumentation at the Materials Research Laboratory Central Research Facilities, University of Illinois, for the rheological experiments. S.C. and J.F. acknowledge partial support by the National Science Foundation (NSF) under grant No. CBET 2323045. We thank Nan Hu and Jeffrey S. Guasto for helpful discussions on Boger fluid preparation, Anxu Huang on microfluidic design, and Evgeniy Boyko for many fruitful discussions on viscoelastic fluid flow in deformable conduits.

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