Asymmetric fluid flow in helical pipes inspired by shark intestines

Significance

Moving flaps are commonly deployed to prevent fluid backflow in pipes. Within the body, these moving parts can fail, leading to digestive conditions like heartburn. Sharks and rays may have an advantage; their intestines contain helical structures that favor flow down the digestive tract, without moving parts. Here, we 3D print pipes with interior helical structures inspired by shark intestines. We identify the roles of the structure’s geometric parameters in generating asymmetric flow. The structures’ materials are as important as their shapes. When we print the structures in deformable materials, flow asymmetries are sevenfold larger than when we print them in rigid materials. Our geometrical designs and application of deformable materials pave the way for innovative designs of fluidic devices.

Abstract

Unlike human intestines, which are long, hollow tubes, the intestines of sharks and rays contain interior helical structures surrounding a cylindrical hole. One function of these structures may be to create asymmetric flow, favoring passage of fluid down the digestive tract, from anterior to posterior. Here, we design and 3D print biomimetic models of shark intestines, in both rigid and deformable materials. We use the rigid models to test which physical parameters of the interior helices (the pitch, the hole radius, the tilt angle, and the number of turns) yield the largest flow asymmetries. These asymmetries exceed those of traditional Tesla valves, structures specifically designed to create flow asymmetry without any moving parts. When we print the biomimetic models in elastomeric materials so that flow can couple to the structure’s shape, flow asymmetry is significantly amplified; it is sevenfold larger in deformable structures than in rigid structures. Last, we 3D-print deformable versions of the intestine of a dogfish shark, based on a tomogram of a biological sample. This biomimic produces flow asymmetry comparable to traditional Tesla valves. The ability to influence the direction of a flow through a structure has applications in biological tissues and artificial devices across many scales, from large industrial pipelines to small microfluidic devices.

Shark teeth typically elicit more attention than shark intestines (perhaps because the question of whether a shark will eat you elicits more anxiety than the question of whether it will digest you). Therefore, the beautiful intestinal structures of sharks and rays may come as a surprise. These structures consist of tubes with interior helices (1, 2) that are traditionally thought to confer an advantage by increasing surface area, thereby improving nutrient absorption (3, 4). Recently, an alternative function was proposed after the discovery of asymmetric flow in shark intestines. Researchers excised intestines from various shark species and measured flow rates for viscous fluids traveling from anterior to posterior, down the gastrointestinal tract, and in the reverse direction. Flow down the tract was faster than the reverse, meaning less peristaltic motion should be needed to push food through the intestines, increasing metabolic efficiency (5). This result is notable because asymmetric flow was achieved without the use of flaps like those found in valves of the human heart and stomach.

When a fluid conduit imposes asymmetric flow, it behaves like an electrical diode. The most famous fluidic diodes, Tesla valves, were invented over a century ago (6, 7). These quasi-two-dimensional (2D) elements generate vortices and high hydrodynamic drag in only one direction of flow. Structures resembling Tesla valves have been found in the lungs of birds (8) and have been incorporated into microfluidic circuits (9, 10). The discovery of asymmetric flow in shark intestines is exciting because similar helical structures are potentially scalable for large, 3D applications.

However, it is puzzling how helical shark intestines could operate as Tesla valves. The problem lies in the Reynolds number, the ratio between inertial and viscous forces in fluids. The Reynolds number is defined as $\text{Re}\equiv \frac{\mathit{uL}}{\nu }$, where u, L, and ν are the flow rate, characteristic length scale, and kinematic viscosity of the fluid, respectively. A disadvantage of Tesla valves is that flow asymmetry is high only when the Reynolds number is high (11–14), requiring high flow rates, large length scales, and/or low fluid viscosities. In contrast, food flowing through intestines is viscous and flows at low velocity, resulting in $\text{Re}\sim {10}^{-4}{10}^{-1}$ (15). In this low Reynolds number regime of fluid dynamics flow is reversible, and simple Tesla valves are inefficient (16, 17). Mathematically, asymmetric flow through rigid pipes at low Reynolds numbers should be impossible because it violates reciprocity (18). However, shark intestines are not rigid; they are soft tissues with mechanical stiffnesses on the order of kiloPascals (19).

Here, we hypothesize that flow-induced deformation of intestinal structures improves their performance as Tesla valves, enabling them to operate efficiently even at low Reynolds numbers. In other words, a deformable Tesla valve, whether composed of biological tissue or elastomeric materials, might generate high flow asymmetry that does not vanish in the limit of low Reynolds number. To test the effect of deformability on flow asymmetry, we 3D print biomimetic helical pipes of both soft and hard polymeric materials. Using well-defined test structures, we measure the influence of the interior helical pitch, hole radius, tilt angle, and pipe length on flow asymmetry (Fig. 1), and we fit our data to a phenomenological model. We then compare flow asymmetry in the biomimetic designs with direct replicas of a shark intestine.

Fig. 1.

Designing, printing, and evaluating biomimetic helical pipes. (A) Pipe with a 10 mm radius, a thick (2 mm) outer wall, and a thin (0.5 mm) inner helical sheet. The sheet begins at radius r_hole from the vertical axis. The interior helix is characterized by the pitch, p, the hole radius, r_hole, the tilt angle, α, and the number of turns, n_turns. The tilt angle defines the flow direction: up/down when the imaginary tip of the cone-like profile points upstream/downstream, respectively (shown in panel D). (B) Three helical pipe designs. (C) Left: Photo of a helical pipe printed in rigid material: side and top views. Right: 3D print of a transverse section of the same pipe with a view of the inner structure. (D) The total flow, Q_tot (gray), is split between a short reference pipe (Q₀, purple) and the helical pipe in the up ($Q_{\uparrow }$, red) or down ($Q_{\downarrow }$, blue) orientation.

Results

Our experiments address three questions: 1) Are tubes with interior helices good candidates for Tesla valves, 2) if so, which features of the design maximize flow asymmetry, and 3) can flow-induced deformation of the structures improve their flow asymmetry? Here, we designed structures inspired by shark intestines: an outer pipe with a sturdy cylindrical wall (2 mm thick) and an inner, thin helical sheet (0.5 mm thick) that is more likely to bend.

We printed helical pipes from two different materials, a thermoset rigid plastic that minimizes deformability (Fig. 1C) and a soft elastomer that maximizes deformability. Even though the elastomer is among the softest products commercially available for 3D printing (with stiffness of only 1 MPa; see SI Appendix, section 2), it is much stiffer than shark intestines (with stiffness ∼1 kPa) (19). Therefore, higher flow rates, Q, are required to deform elastomeric helices than to deform intestines. Here, we use flow rates of Q ≈ 100 cc/s, corresponding to Reynolds numbers $Re\sim {10}^4$. In this regime, flow can be turbulent, so flow asymmetry may arise in rigid pipes as well as in deformable pipes. Consequently, we use rigid pipes to test the effect of the pipe’s geometry, uncoupling it from the mechanical deformation. Then, we use soft pipes to measure how much the deformability enhances the asymmetry with respect to a rigid pipe of the same design.

Large Flow Asymmetries in Rigid, Helical Pipes.

The helical pipes in Fig. 1 are fully characterized by only a few geometrical parameters. These include the pitch (p, the vertical ascent with each revolution), the radius of the interior hole (r_hole), the tilt angle (α, which breaks up–down symmetry), the number of helical turns (n_turns), and the pipe radius (R, from the center to the outside edge). We fixed the pipe radius at $10\text{mm}$ and varied the rest of the parameters to generate a large family of structures (Fig. 1B and SI Appendix, Fig. S1). To determine how each parameter influences flow asymmetry, we first established a measurement method.

Asymmetric fluid flow is benchmarked by “diodicity,” Di (11, 12):

$$Di(Q)\equiv \frac{\mathrm{\Delta }{P}_{\uparrow }(Q)}{\mathrm{\Delta }{P}_{\downarrow }(Q)},$$ [1]

where $\mathrm{\Delta }{P}_{\downarrow }$ and $\mathrm{\Delta }{P}_{\uparrow }$ are pressure drops measured across a device in its forward and reverse directions, respectively. Since it is harder to push fluid through a Tesla valve in the reverse direction, $Di\ge 1$, where $Di=1$ indicates no asymmetry. Here, our device is a helical pipe, and we measure flow rates rather than pressure drops (Fig. 1D). The cone of the helix is oriented either down (the forward direction) or up (the reverse direction, Fig. 1D). We convert flow in the pipe, Q, into equivalent lengths of hollow tubes, ℓ, which correspond to pressure drops, ΔP (Fig. 2 and Materials and Methods).

Fig. 2.

Schematic of diodicity measurement. Flow meter values are translated into flow rates by the calibration curve in SI Appendix, Fig. S3A. Next, ratios of flow rates through both branches of the apparatus are translated into an effective length of hollow tubing, ℓ, using the calibration curve in SI Appendix, Fig. S4. This effective length is proportional to the pressure drop. The diodicity is the ratio between the pressure drop through the helical pipe oriented in the up direction to a helical pipe oriented in the down direction.

Limiting cases yield a qualitative understanding of how diodicity should vary with the four experimental parameters for helical pipes (p, r_hole α, and n_turns). These limits appear in the Insets of Fig. 3.

Fig. 3.

Rigid helical pipes result in large flow asymmetries. Effect of four parameters on flow diodicity; schematics illustrate each parameter. (A) Effect of pitch, p, on diodicity for pipes with r_hole = 3 mm, $tan(\alpha )$= 1.5, and n_turns = 7.5. (B) Effect of hole radius, r_hole, on diodicity for pipes with p = 7.5 mm, $tan(\alpha )$= 1.5, and n_turns = 7.5. (C) Effect of tilt angle, α, on diodicity for pipes with p = 7.5 mm, r_hole = 3 mm, and n_turns = 7.5. (D) Effect of number of turns, n_turns, on diodicity for pipes with p = 7.5 mm, r_hole = 3 mm, and $tan(\alpha )$= 1.5. Shaded regions on the graphs are only to guide the eye. Limiting cases for each parameter are illustrated in the Insets.

1. Pitch, p: As p → 0, the helical pipe resembles a cylindrical tube with thick walls, so $Di\to$ 1. At the other extreme, as $p\to \infty$, its effect on the flow diminishes, so $Di\to$ 1.

2. Hole radius, r_hole: As r_hole → 0, the helical pipe resembles a coiled, hollow tube, for which no flow asymmetry is expected, so $Di\to 1$. At the other extreme, as r_hole approaches the radius of the pipe, the pipe resembles a cylindrical tube. Therefore, as r_hole → 10 cm, $Di\to$ 1.

3. Angle, α: As α → 0, the helical pipe becomes symmetric. Therefore, as $tan(\alpha )\to 0$, $Di\to 1$. As the angle approaches 90°, the helical pipe resembles a cylindrical tube. Therefore, as $tan(\alpha )\to \infty$, $Di\to$ 1.

4. Number of turns, n_turns: As n_turns → 0, the inner helix vanishes, leaving a cylindrical tube, so $Di\to 1$. As n_turns increases, asymmetry is expected to increase, but its detailed behavior is unknown. For some designs of Tesla valves, diodicity approaches an asymptote as the number of identical elements in series increases (12, 20).

These limiting cases imply that flow asymmetry is maximized at intermediate values of pitch, r_hole, and angle, and reaches an asymptotic value as the number of turns increases. Because our system operates far from the regime of laminar flow, exact values of diodicities are difficult to predict. Therefore, we go beyond qualitative limits to quantitatively measure diodicities in rigid helical pipes.

Our most striking result for these rigid helical pipes is that nearly all values of the parameters tested induce large flow asymmetries (Fig. 3). The diodicity values we measure (in many cases, $2\le Di\le 3)$ are large compared to diodicities measured in traditional Tesla valves. The literature contains only a few experimentally measured Tesla valve diodicities, with reported values of $Di\sim 2$ (9, 13). In contrast, many numerical analyses have been conducted on Tesla valve designs; nearly all yield values of $1<Di<2$ (9, 11, 12). In one extreme case, a numerical shape optimization involving an intensive survey of designs and parameters found a highly efficient single-valve design with Di ∼ 2 to 4 (14).

In other words, biomimetic helical pipes (even rigid structures without flow-induced deformation) can perform better than most Tesla valves, comparable to heavily optimized designs. In Fig. 3, data in all four panels represent perturbations from an initial parameter set of p = 15 mm, r_hole = 3 mm, $tan(\alpha )$ = 1.5, and n_turns = 7.5. When we subsequently vary pitch and r_hole, clear maxima appear at p = ∼7.5 mm and at r_hole = 4 to 5 mm. As the number of turns increases, diodicity appears to plateau, consistent with asymptotic diodicities in microfluidic Tesla valves (12).

To understand how the design parameters affect diodicity within the range of values in Fig. 3, we developed a phenomenological model. First, we reduced the dimensionality of the problem by defining a helical length, $h\equiv \sqrt{p^2+{\pi }^2{r}_{\text{hole}}^2}$, where diodicity peaks at intermediate values of the dimensionless variable $h/R$ (SI Appendix, Fig. S8). Then, we tested a model that the effective length of a helical tube, ℓ, depends separately on three parameters: h, α, and n_turns, where $\ell =H(h)\times A(\alpha )\times N(n_{\text{turns}})$. We found good agreement between this model and the data over most parameter ranges in our experiments (SI Appendix, section 3 and Fig. S7).

Huge Flow Asymmetry in Deformable, Helical Pipes.

Deformation of helical pipes printed from soft elastomers amplifies flow asymmetry with respect to nearly all rigid pipes of the same design (Fig. 4). In the absence of a theoretical model, we fit diodicities in Fig. 4 to skewed Gaussians. The peaks of these Gaussians correspond to diodicities of ∼10 to 15, a flow asymmetry that is roughly sevenfold higher than in rigid helical pipes or in traditional Tesla valves (Fig. 4 A and E).

Fig. 4.

Deformable helical pipes result in high diodicities that vary with flow rate. (A) Ranges of measured diodicities of traditional Tesla valves (13) and of our rigid and soft helical pipes. (B) Effective lengths of a deformable pipe (p = 7.5 mm, r_hole = 3 mm, $tan(\alpha )$ = 1.5, and n_turns = 1.5) connected in the up (red) and down (blue) flow direction. Values for an equivalent rigid pipe are shown as red ($\ell =$1.0 m) and blue ($\ell =$0.75 m) horizontal lines. (C) Diodicity as a function of flow rate through the pipe in panel (B) (squares), through a deformable pipe with identical parameters (diamonds), and through a rigid pipe (horizontal green line). Scatter in the data of the two deformable pipes is indistinguishable. An optimal flow rate that yields the highest diodicity is estimated by fitting a skewed Gaussian (thick green line). (D) Skewed Gaussian fits (solid lines) of diodicity versus flow rate for deformable pipes with different numbers of turns. Large circles show the maximum diodicity for each fit. (E) Maximum diodicities from panel (D) (filled circles) and amplification of diodicity (unfilled circles, the maximum diodicity of the deformable pipe divided by the diodicity of the equivalent rigid pipe). The dotted line shows the average, 6.7-fold amplification. (F) Optimal flow rates from panel (D). At the highest number of turns, the highest flow rate is required to reach the maximum diodicity.

Deformable helical pipes do not generate a fixed value of diodicity, unlike rigid versions. Instead, $\mathit{Di}$ is a function of the flow rate, Q. This contrast between rigid and deformable pipes is analogous to the observation that an electrical resistor has a fixed resistance, whereas a diode does not. Flow rates corresponding to the highest diodicities increase with the number of turns of the helical pipe (Fig. 4F). This makes sense: Larger flow rates are needed to deform a larger number of helical blades. In contrast, the degree to which deformable pipes amplify diodicity relative to rigid pipes is independent of the number of turns. A separate observation is that higher diodicities are accompanied by higher scatter in the data, independent of whether helical pipes have the same parameters (Fig. 4C) or different numbers of helical turns (Fig. 4D).

The amplification of diodicity in deformable pipes relative to rigid pipes could arise from changes in the flow rate for helices oriented in the up direction, $Q_{\uparrow }$, helices in the down direction, $Q_{\downarrow }$, or both. Here, we find that nearly all amplification arises from changes in $Q_{\uparrow }$, resulting in high values of the effective length ${\ell }_{\uparrow }$, whereas ${\ell }_{\downarrow }$ is roughly constant (Fig. 4B). This large change in $Q_{\uparrow }$ implies that the interior blades of helical pipes deform more when the helix is oriented up. In contrast, if any deformation occurs when helical pipes are oriented down, it appears to slightly facilitate flow, resulting in lower values of ${\ell }_{\downarrow }$ than in rigid pipes (Fig. 4B).

To test that deformation of the inner helical sheet leads to higher diodicity, we simulated the deformation of a single-turn helix in analog scenarios of uniform load and gravity (SI Appendix, section 4 and Fig. S9). We find that most of the deformation is localized to the top and bottom edge of the helix (rather than uniformly changing a parameter such as the angle). Then, we 3D-printed the deformed configurations in rigid materials, locking in the deformation, and we measured flow through the resulting stiff structures. As in deformable pipes, rigid pipes with locked-in deformations produce large effective lengths when the helix is oriented upward, whereas deformations produce little change in effective length when the helix is oriented downward. As a result, the diodicity of the single-turn helical pipe with locked-in deformations is almost threefold larger than of the undeformed pipe.

Large Flow Asymmetries in Rigid Analogs of Shark Intestines.

The well-parameterized helical pipes in Fig. 1 were inspired by more complex structures in the intestines of sharks and rays. These structures vary from species to species (3–5). Here, we created 3D models of the spiral intestine of a dogfish shark (Centroscyllium nigrum), based on digital tomograms (5). We 3D-printed rigid versions of the model at three different lengths: the full model, the top two-thirds, and the top third (SI Appendix, Fig. S10). All three versions generated flow asymmetries in the range of ∼1.15 to 1.4, comparable with traditional Tesla valves (SI Appendix, section 5 and Fig. S10). Of course, sharks are not rigid. In vivo ultrasound reveals twisting, contraction waves, and undulatory waves in shark intestines (21). A fuller understanding of flow asymmetries in these structures will require ultrasoft materials coupled to biomimetic motions.

Discussion and Summary

This research began by posing three questions: Will helical pipes act as Tesla valves? What parameters might increase flow asymmetry? And will it be enhanced by flow-induced deformations of the inner helical structure? We found that rigid helical pipes impose high flow asymmetries, exceeding those of most traditional Tesla valves (and on par with highly optimized Tesla valves). We are unaware of any other experimentally measured structures that achieve such high flow asymmetries in the absence of moving parts. Optimization of the pipe’s design parameters would likely push the flow asymmetries to even higher values. An added bonus is that the pipes are three-dimensional, so they have the potential to accommodate larger fluid volumes than traditional quasi-two-dimensional Tesla valves, and they may find applications in larger commercial devices.

When we reproduced the helical pipes in deformable materials, we found very large flow asymmetries, roughly sevenfold higher than in rigid pipes or Tesla valves. Typically, interactions between elastic structures and fluids are considered from the standpoint of how changes in the structure’s shape affect flow in the surrounding fluid, resulting in mobility, as in Taylor swimmers (22, 23). Another common vantage is to consider how hydraulic flow induces large changes in a structure’s shape, as in soft robots (24, 25). In our system, the viewpoint is different: Fluid flow is affected by the deformation it imposes on the elastic structure.

In our experiments, we used one of the softest commercially available elastomers for 3D printing. Given that the field of 3D printing is quickly evolving (26), softer materials like hydrogels may soon be widely available (27, 28). However, an ongoing challenge is finding very soft materials that can withstand high deformations (29). As softer elastomeric materials are developed and integrated into helical pipe designs, we would expect flow asymmetries to arise at lower Reynolds numbers.

Materials and Methods

3D Printing Helical Pipes.

3D models of helical pipes with different values of pitch, hole radius, angle, and number of turns were generated using Wolfram Mathematica 13.1 and are available in Dryad (30). An adaptor (2 cm long and 4 mm thick) was added to the end of each pipe to connect it to the flow apparatus.

Models were printed at high spatial resolution and accuracy on a stereolithographic 3D printer (Fig. 1C and SI Appendix, Fig. S1). Specifically, models were 3D printed vertically on a Formlabs Form2 printer with a layer thickness of 0.1 mm, using Clear V4 (Formlabs) resin for rigid pipes and Elastic 50A (Formlabs; named for its hardness rating of 50 Shore A) for deformable pipes. Printed helical pipes were washed several times in isopropyl alcohol and dried in ambient conditions for a few hours. Finally, each end of the printed pipe was joined to short polyvinyl chloride (PVC) tubing (9.5 mm inner diameter, 2.5 cm length). The junction was sealed with epoxy on the interior surfaces and silicone sealant on the outside.

Measuring Diodicity.

In brief, we split the total fluid flow, Q_tot, into two branches with the same hydraulic resistance (SI Appendix, Fig. S3B). Each branch contained a flow meter (SI Appendix, Fig. S3A). We then affixed a helical pipe to one branch, while keeping the other branch empty, as a reference. We measured the flow rate through the reference pipe, Q₀, and the flow rate through the helical pipe, which is either $Q_{\uparrow }$ or down $Q_{\downarrow }$, depending on whether the helix was oriented up or down. Each helical pipe had a flow resistance equivalent to the resistance through a long span of tubing. The length, ℓ, of this tubing was proportional to the pressure drop $\mathrm{\Delta }P$.

In more detail, schematics of both branches of the flow apparatus are given in Fig. 1D and SI Appendix, Fig. S2, and the method of converting flow rates to pressure drops is given in Fig. 2 and SI Appendix. Both branches were equipped with a flow meter (DIGITEN G1/2” Water Flow Hall Sensor 1 to 30 L/min FL-408) in series with flexible PVC tubing (12.7 mm inner diameter, 30 cm length). Each flow meter was previously calibrated to maximize accuracy, as in SI Appendix, Fig. S3A. In each experiment, a helical pipe was attached to the right branch of the flow apparatus, and the left branch remained empty. Total flow was controlled by the output of a standard water faucet, and values of Q₀ (left arm) and $Q_{\downarrow }$/$Q_{\uparrow }$ (right arm) were recorded by hand. Flows through each branch corresponded to an effective length of tubing (SI Appendix, Fig. S4), which in turn corresponded to a pressure drop. The ratio between the pressure drops in both pipe orientations was the diodicity (see SI Appendix, section 1 for more information). For (only) rigid pipes, Q₀ depended linearly on both $Q_{\downarrow }$ and $Q_{\uparrow }$, with negligible offset. Therefore, we fit the flow in both directions with linear models, with no intercept terms (SI Appendix, Fig. S5) and used the slopes to estimate $\mathit{Di}$ (which is independent of Q).

Supplementary Material

Appendix 01 (PDF)

Dataset S01 (XLSX)

Dataset S02 (XLSX)

Dataset S03 (XLSX)

Data, Materials, and Software Availability

3D models, data tables, and code have been deposited in Dryad (30).