fluid flow control

One of the main tasks of any science is not only to obtain knowledge regarding a particular process/phenomenon, but also to gain control over it, which can be extremely useful for the development of technologies used in various sectors of human life. Most often, in order to gain control over the process, we need to create systems that manipulate and modulate it. Sometimes these systems are quite complex, although the result of their operation may seem quite simple. For example, force a liquid to flow exclusively in one direction. The systems use valves to do this, but in the wildlife world there is a much more effective and simpler alternative: shark intestines. Scientists from the University of Washington (Seattle, USA) have developed a system that mimics the intestines of sharks, which forces the liquid inside it to move in a precisely specified direction. What is the secret of shark entrails, how did scientists recreate them, and where can their creation be used in practice? We will find answers to these questions in the scientists' report.

Basis of the study

People associate sharks with a hot temper, insatiable hunger and sharp teeth. Only the latter is true, and even then not in all species. Without being ichthyologists, many of us can name a shark's distinctive features or two, from the fin on its back to the multiple rows of serrated teeth (again, these characteristics are not universal across all species). But few people know that the intestines of sharks and rays are no less unique, both visually and functionally.

Pacific spiny shark (Squalus suckleyi) and CT scan of its spiral intestine.

As seen in the CT scan above, the structure of the shark's intestine consists of tubes with internal coils, which are traditionally thought to provide the benefit of increasing surface area, thereby improving nutrient absorption. An alternative function has recently been proposed following the discovery of asymmetric flow in the shark gut. The researchers cut out the intestines of different species of sharks and measured the flow rate of viscous fluids moving from front to back, down the gastrointestinal tract, and back again. The flow down the tract was faster than the flow back up, meaning that fewer peristaltic movements should have been required to push food through the intestines, increasing metabolic efficiency. This result is remarkable because the asymmetrical flow was achieved without the use of leaflets like those found in the valves of the human heart and stomach.

A detailed story about Nikola Tesla's valve, invented by him in 1916.

When a fluid channel imposes asymmetrical flow, it behaves like an electrical diode. The most famous liquid diodes, Tesla valves, were invented more than a century ago. These quasi-two-dimensional elements generate vortices and high hydrodynamic drag in only one direction of flow. Structures resembling Tesla valves have been discovered in the lungs of birds and have been incorporated into microfluidic circuits. The discovery of asymmetric flow in the shark gut is exciting because such helical structures are potentially scalable to large 3D applications.

However, it is not clear how a shark's spiral guts could work like Tesla valves. The problem lies in the Reynolds number, the relationship between inertial and viscous forces in fluids. The Reynolds number is defined as Re = uL/ν, where u, L and ν are the flow velocity, characteristic length scale and kinematic viscosity of the fluid, respectively. A disadvantage of Tesla valves is that flow asymmetry is only high at high Reynolds numbers (11–14), which requires high flow rates, large length scales, and/or low fluid viscosity. In contrast, food flowing through a shark's gut is viscous and flows at low speed, resulting in Re ~ 10^–4 – 10^–1. In this low Reynolds number fluid dynamics regime, flow is reversible and simple Tesla valves are ineffective. Mathematically, asymmetric flow through rigid pipes at low Reynolds numbers should be impossible because it violates the principle of reciprocity. However, the shark's intestines are not hard; they are soft tissues with mechanical rigidity on the order of kilopascals.

Image #1

In the paper we're looking at today, the scientists suggested that flow-induced deformation of intestinal structures improves their performance as Tesla valves, allowing them to operate effectively even at low Reynolds numbers. In other words, a deformable Tesla valve, whether composed of biological tissue or elastomeric materials, can generate high flow asymmetry that does not disappear in the low Reynolds number limit. To test the effect of deformability on flow asymmetry, the scientists 3D printed biomimetic spiral tubes from both soft and hard polymer materials. Using well-defined test structures, the scientists measured the effects of internal helix pitch, orifice radius, pitch angle, and pipe length on flow asymmetry (diagrams above). The flow asymmetry in the biomimetic designs was then compared with direct replicas of the shark intestine.

Research results

Scientists note that their experiments are aimed at finding answers to three main questions:

• are tubes with internal spirals good candidates for Tesla valves?
• If so, what design features maximize flow asymmetry?
• Can flow-induced deformation of structures improve their flow asymmetry?

To achieve this, a structure inspired by the intestines of a shark was designed: an outer tube with a strong cylindrical wall (2mm thick) and an inner thin spiral sheet (0.5mm thick) that is more likely to bend.

The spiral tubes were printed from two different materials: a thermoset, rigid plastic that minimizes deformation (1C), and a soft elastomer that maximizes deformability. Although the elastomer is one of the softest products available for 3D printing (with a stiffness of only 1 MPa), it is much stiffer than shark gut (with a stiffness of 1 kPa). Therefore, deformation of elastomeric coils requires higher flow rates (Q) than intestinal deformation. The study used a flow velocity of Q ≈ 100 cm³/s, which corresponds to the Reynolds number Re ~ 10⁴. In this regime, the flow can be turbulent, so flow asymmetries can occur in both rigid and deformable pipes. Consequently, rigid pipes were used to test the effect of pipe geometry, separating it from mechanical deformation. The soft pipes were then used to measure how much deformability enhanced the asymmetry relative to a rigid pipe of the same design.

The spiral pipes in the diagram above are fully characterized by just a few geometric parameters. These include pitch (p, vertical rise with each revolution), radius of the internal hole (r_hole), tilt angle (α; which breaks up-down symmetry), number of spiral turns (n_turns) and pipe radius (R, from center to outer edge). The pipe radius was fixed at 10 mm, while other parameters were varied to create a wide variety of structures (1B). To determine how each parameter affects flow asymmetry, the first step was to establish a measurement method.

Asymmetrical fluid flow is characterized by “diodicity” (D_i):

D_i (Q) = ∆P_↑ (Q)/∆P_↓ (Q),

where ∆P_↓ and ∆P_↑ are the pressure drops measured across the device in the forward and reverse directions, respectively. Since it is more difficult to push fluid through the Tesla valve in the opposite direction, D_i ≥ 1, where D_i = 1 indicates no asymmetry. In these experiments, the device is a spiral pipe, so the flow rate was measured, not pressure drops (1D). The cone of the spiral is oriented either downward (forward direction) or upward (reverse direction; 1D). The scientists converted the flow in the pipe (Q) into equivalent lengths of hollow tubes (l), which correspond to pressure differences (P; diagram below).

Image #2

Limiting cases provide qualitative insight into how diodeity should vary with four experimental parameters for spiral tubes (p, r_holeα and n_turns). These limits are shown in the insets of image No. 3:

• Step (p): When p → 0, the spiral pipe resembles a cylindrical pipe with thick walls, so D_i → 1. At the other pole, as p → ∞, its influence on the flow decreases, so D_i → 1;
• Hole radius (r_hole): At r_hole → 0 spiral pipe resembles a spiral hollow pipe for which no flow asymmetry is expected, so D_i → 1. At the other pole, when r_hole approaches the radius of the pipe, the pipe resembles a cylindrical pipe. Therefore, when r_hole → 10 cm, D_i → 1;
• Corner (α),: When α → 0, the spiral pipe becomes symmetrical. Therefore, as tan(α) → 0, D_i → 1. When the angle approaches 90°, the spiral pipe resembles a cylindrical pipe. Therefore, as tan(α) → ∞, D_i → 1;
• Number of turns (n_turns): When n_turns → 0 the inner helix disappears, leaving a cylindrical tube, so D_i → 1. As n increases_turns asymmetry is expected to increase, but its detailed behavior is unknown. For some Tesla valve designs, diodicity approaches an asymptote as the number of identical elements in the series increases.

These limiting cases imply that flow asymmetry is greatest at intermediate values ​​of the pitch, r

_hole

and angle and reaches an asymptotic value as the number of turns increases. Since the system under study operates far from the laminar flow regime, the exact diodicity values ​​are difficult to predict. Therefore, scientists went beyond qualitative limitations to quantify diodes in rigid spiral tubes.

Image #3

According to the scientists, the most striking result for these rigid spiral pipes is that almost all values ​​of the tested parameters produce large flow asymmetries (graphs above). Measured diode values ​​(in many cases 2 ≤ D_i ≤ 3) are large compared to diodes measured in traditional Tesla valves. The literature contains only a few experimentally measured Tesla valve diodes with recorded D values_i ~ 2. In contrast, there have been many numerical analyzes of Tesla valve designs, almost all of which have returned 1 < D_i < 2. In one extreme case, numerical shape optimization involving intensive design and parameter exploration revealed a highly efficient single valve design with D_i ~2 to 4.

In other words, biomimetic spiral tubes (even rigid structures without flow-induced deformation) can perform better than most Tesla valves, which is comparable to highly optimized designs. The data above in all four graphs represent perturbations from the initial set of parameters p = 15 mm, r_hole = 3 mm, tan(α) = 1.5 and n_turns = 7.5. When the step and r changed successively_holeclear maxima appeared at p = 7.5 mm and at r_hole = 4–5 mm. As the number of turns increases, the diodicity appears to plateau, which is consistent with the asymptotic diodicity in microfluidic valves.

To understand how design parameters affect diode across the range of values ​​in the graphs above, the scientists developed a phenomenological model. First, the dimension of the problem was reduced by defining the spiral length h = √ p² + π²²_holewhere diodeity reaches its peak at intermediate values ​​of the dimensionless variable h/R. A model was then tested in which the effective length of the spiral tube (l) depends separately on three parameters: h, α and n_turnswhere l = H(h) ∙ A(α) ∙ N(n_turns). Good agreement was found between this model and the data in most parameter ranges in the experiments performed.

Image #4

The deformation of spiral tubes printed with soft elastomers increases the flow asymmetry relative to almost all rigid tubes of the same design (graphs above). In the absence of a theoretical model, the scientists fitted the diodicities in the graphs above to distorted Gaussians. The peaks of these Gaussians correspond to diodes of 10 to 15, a flow asymmetry approximately seven times higher than in rigid spiral tubes or traditional Tesla valves (4A And 4E).

Deformable spiral tubes do not generate a fixed diode value, unlike their rigid equivalents. Instead D_i is a function of flow speed (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. The flow rates corresponding to the highest diodes increase with the number of turns of the spiral pipe (4F). The logic of this is quite simple: to deform a larger number of spiral blades, higher flow rates are required. In contrast, the extent to which deformable pipes enhance diodeity of relatively rigid pipes is independent of the number of turns. A separate observation is that higher diodes are accompanied by greater scatter in the data, regardless of whether the spiral tubes have the same parameters (4C) or a different number of spiral turns (4D).

Increased diodeity in deformable pipes relative to rigid pipes can arise from changes in flow velocity for up-direction spirals (Q↑), down-direction spirals (Q↓), or both. The scientists found that almost all of the gain arises from changes in Q↑, resulting in high values ​​of the effective length l↑, while l↓ is approximately constant (4B). This large change in Q↑ implies that the inner blades of spiral tubes deform more when the spiral is oriented upward. In contrast, if any deformation occurs when the spiral pipes are oriented downward, it appears to facilitate flow slightly, resulting in lower l↓ values ​​than in rigid pipes (4B).

To test that deformation of the inner helical sheet results in higher diodeity, the scientists simulated the deformation of a single-turn helix under analog uniform load and gravity scenarios. It was found that most of the deformation is localized at the upper and lower edges of the spiral (rather than uniformly changing a parameter such as angle). The deformed configurations were then created using a 3D printer in rigid materials with deformation fixed. The scientists measured the flow through the resulting rigid structures. As in deformable pipes, rigid pipes with fixed deformations produce large effective lengths when the helix is ​​oriented upward, whereas deformations produce a small change in effective length when the helix is ​​oriented downward. As a result, the diodeity of a single-turn spiral pipe with fixed deformations is almost three times greater than that of an undeformed pipe.

The black dog shark (Centroscyllium nigrum) lives in the eastern Pacific Ocean at depths ranging from 269 to 1143 m.

The spiral tubes shown in image #1 were inspired by the more complex structures in the intestines of sharks and rays. These structures vary from species to species. Scientists have created 3D models of the dog shark's spiral intestine (Centroscyllium nigrum) based on digital tomograms. Rigid versions of the model were 3D printed in three different lengths: the full model, the top two-thirds, and the top third. All three versions generated flow asymmetries ranging from 1.15 to 1.4, which is comparable to traditional Tesla valves. Of course, sharks are not tough. In vivo ultrasound reveals twisting, contraction waves and undulations in sharks' intestines. A more complete understanding of flow asymmetries in these structures will require ultrasoft materials coupled to biomimetic movements.

For a more detailed look at the nuances of the study, I recommend taking a look at scientists' report And additional materials to him.

Epilogue

In the work we reviewed today, scientists decided to study the spiral-shaped intestines of sharks and compare it with Tesla valves. In the study, the scientists attempted to answer a number of questions: would spiral tubes act like Tesla valves, what parameters could increase flow asymmetry, and would it be amplified by flow-induced deformations of the internal spiral structure?

It was found that the rigid spiral tubes impose high flow asymmetry, exceeding that of most traditional Tesla valves. The scientists note that they are not aware of any other experimentally measured structures that achieve such high flow asymmetry in the absence of moving parts. Optimization of pipe design parameters is likely to push flow asymmetry to even higher values. An added bonus is that the tubes are three-dimensional, so they have the potential to accommodate larger volumes of fluid than traditional quasi-two-dimensional Tesla valves, and they could find use in larger commercial applications.

When the scientists replicated spiral tubes in deformable materials, they found very large flow asymmetries, about seven times higher than in rigid tubes or Tesla valves. Typically, interactions between elastic structures and fluids are considered in terms of how changes in the shape of the structure affect the flow in the surrounding fluid, resulting in mobility. Another common benefit is to consider how hydraulic flow causes large changes in the shape of the structure, as in soft robots. In this work, a different idea was expressed: the flow of liquid depends on the deformation that it imposes on the elastic structure.

During the experiments, the scientists used one of the softest commercially available elastomers for 3D printing. Given that the field of 3D printing is rapidly evolving, softer materials such as hydrogels may soon become widely available. However, a constant challenge is finding very soft materials that can withstand high deformations. As softer elastomeric materials are developed and integrated into spiral tube designs, scientists believe flow asymmetries will occur at lower Reynolds numbers.

The study's authors believe that their work to achieve maximum control over fluid flows will be extremely useful in many industries, ranging from medicine to soft robotics. However, the insides are much softer than even the softest synthetic materials available today. Therefore, there is room for improvement of this technology.

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