Volumetric effects in viscous flows in circular and annular tubes with wavy walls

This paper demonstrates that the common practice of maintaining a constant mean radius in wavy-walled tube models inadvertently increases interior volume, leading to significant discrepancies (up to 50%) in flow rate and hydraulic resistance compared to constant-volume models for both steady and peristaltic viscous flows, and provides a scaling law to relate these two cases.

The Gist

Imagine you have a garden hose. Now, imagine someone pinches and squeezes that hose in a wavy pattern, like a snake, along its entire length. This is what scientists call a "wavy wall."

For decades, researchers studying how fluids (like water or blood) move through these wavy tubes have been making a subtle but huge mistake. They assumed that when they squeezed the hose to make waves, the average size of the hose stayed the same.

This paper, by Yisen Guo and John Thomas, points out a simple geometric truth: If you squeeze a tube to make waves, you actually make the inside of the tube bigger.

Here is the breakdown of their discovery using everyday analogies.

The "Balloon" Mistake

Think of a straight, smooth balloon. Now, imagine you push your fingers into the sides of the balloon to make a wavy pattern, but you keep the average width of the balloon the same.

• The Old Way of Thinking: Scientists assumed that if you push the sides in, the space inside stays the same size.
• The Reality: When you push the sides in to make a wave, the "hills" of the wave bulge out more than the "valleys" go in. Because of the way circles work, the total volume of air inside that wavy balloon is actually larger than the smooth one.

The authors say that previous studies ignored this extra space. They treated the wavy tube as if it had the same amount of room as the smooth tube, but with a bumpy floor. In reality, the wavy tube is a bigger room with a bumpy floor.

Why Does This Matter? (The Traffic Jam Analogy)

Imagine a highway (the tube) where cars (the fluid) are driving.

1. The "Constant Radius" Scenario (The Old Way): You build a highway with hills and valleys, but you pretend the total number of lanes is the same as a flat highway. Because the road is bumpy, cars slow down in the narrow spots (the valleys). This creates a "traffic jam," increasing the resistance to flow.
2. The "Constant Volume" Scenario (The New Way): You realize that building those hills actually added extra lanes to the highway. Now, even though there are still narrow spots, there is more total space for the cars to spread out.

The Result:

• If you keep the average size of the tube the same (Old Way), the fluid flows faster because the tube is secretly bigger.
• If you keep the total volume the same (New Way), you have to make the tube slightly narrower to compensate for the extra space the waves create. This makes the flow much slower and the "traffic jam" (resistance) much worse.

The authors found that for waves that are just 20% the size of the tube's width, the difference in how fast the fluid flows can be 10%. For bigger waves, the difference can be 50%. That is a massive difference in engineering and biology.

The "Peristaltic Pump" (The Squeeze Tube)

The paper also looks at "peristaltic pumping." This is how your body moves food through your gut or how a toothpaste tube works. You squeeze the tube in a wave that moves forward, pushing the fluid along.

• The Old View: Scientists calculated how much toothpaste comes out based on the wave moving, ignoring the fact that the wave itself changes the size of the tube.
• The New View: Because the wave actually expands the tube's volume, it acts like a bigger pump.
• The Shock: If you keep the tube's volume constant (by making the tube narrower to start with), the pump is much less efficient. But if you let the tube expand (the usual way), the pump moves 50% more fluid at maximum capacity.

The "Brain" Connection

Why do the authors care about this? They are studying the brain.

Inside your brain, there are tiny gaps between blood vessels and brain tissue called "perivascular spaces." These gaps act like tiny pipes that wash away waste (like the brain's garbage disposal system). The walls of these gaps wiggle because your heart beats and your arteries pulse.

If scientists use the "Old Way" of calculating flow, they might think the brain's cleaning system is working better than it actually is, or they might misunderstand how blockages affect the brain. By correcting for the volume change, they can get a much more accurate picture of how the brain cleans itself.

The Bottom Line

The paper is a reminder that in physics and engineering, geometry matters.

• The Mistake: Assuming a wavy tube is the same size as a straight one.
• The Truth: A wavy tube is a bigger container.
• The Fix: If you want to compare apples to apples, you must either account for the extra space the waves create, or shrink the tube slightly to keep the total space the same.

Ignoring this simple fact has led to errors in predicting how blood flows, how drugs move through the body, and how industrial pipes work. The authors provide a new "rule of thumb" (a scaling law) so scientists can easily correct their calculations and get the right answer.

Technical Summary

Here is a detailed technical summary of the paper "Volumetric effects in viscous flows in circular and annular tubes with wavy walls" by Yisen Guo and John H. Thomas.

1. Problem Statement

The paper addresses a fundamental geometric oversight in the study of viscous flows within tubes with sinusoidally wavy walls (both circular and annular).

• The Conventional Approach: Most existing literature models a wavy wall by defining the radius as $r(z) = r_0 + b \sin(2\pi z/\lambda)$, where $r_0$ is a fixed mean radius and $b$ is the wave amplitude.
• The Oversight: This formulation inadvertently increases the total internal volume of the tube as the wave amplitude ($b$) increases. Specifically, the volume increases by $\pi b^2 \lambda / 2$ per wavelength.
• The Gap: Previous studies have generally ignored this volume change, assuming that keeping the mean radius constant is sufficient. The authors argue that for significant wave amplitudes (or even small ones in specific contexts like peristalsis), this volume change significantly alters hydraulic resistance, flow rates, and flow topology, leading to potentially erroneous conclusions if not accounted for.

2. Methodology

The authors employ a combination of analytical scaling and numerical simulation to compare two distinct modeling scenarios:

1. Constant Mean Radius (CMR): The standard approach where $r_0$ is fixed, and volume increases with amplitude.
2. Constant Volume (CV): A corrected approach where the mean radius is reduced ($r^*$) as amplitude increases to maintain a constant total internal volume ($V = \pi r_0^2 \lambda$).

Key Techniques:

• Governing Equations: Steady, incompressible, laminar Navier-Stokes equations are solved numerically using the Finite Element Method (COMSOL Multiphysics).
• Flow Types Analyzed:
  • Pressure-Driven Flow: Steady flow driven by an axial pressure gradient in stationary wavy tubes.
  • Peristaltic Pumping: Flow driven by a propagating sinusoidal wall wave ($r(z,t) = r_0 + b \sin[2\pi(z-ct)/\lambda]$).
• Geometries: Both circular tubes (open) and annular tubes (concentric, with wavy inner or outer boundaries).
• Dimensional Analysis: The authors derive a scaling law to relate the CV and CMR cases, allowing results from one to predict the other.

3. Key Contributions

• Identification of the Volume Effect: The paper formally identifies and quantifies the "volume-change effect," demonstrating that the standard CMR assumption artificially inflates the flow capacity of wavy tubes.
• Scaling Law Derivation: A rigorous dimensional analysis is presented (Eq. 13-14) showing that to achieve dynamic similarity between the CV and CMR cases, the pressure drop in the CV case must be scaled by a factor of $(r_0/r^*)^2$. This allows researchers to convert existing CMR data to CV conditions.
• Annular Tube Extension: The analysis is extended to annular tubes, revealing that the sign of the volume change depends on which wall is wavy:
  • Wavy Outer Wall: Volume increases (similar to circular tubes).
  • Wavy Inner Wall: Volume decreases as amplitude increases.

4. Key Results

A. Pressure-Driven Flow (Stationary Walls)

• Hydraulic Resistance:
  • In the Constant Volume case, hydraulic resistance is systematically higher than in the Constant Mean Radius case because the tube is effectively narrower on average.
  • For wave amplitudes as small as 20% of the mean radius ($b/r_0 = 0.2$), the difference in flow rate/resistance can reach 10%.
  • At larger amplitudes ($b/r_0 \approx 0.6$), the difference can exceed 50%.
• Flow Topology (Recirculation):
  • The formation of recirculating eddies in the widened sections of the tube is highly sensitive to the volume constraint.
  • Qualitative Difference: The paper highlights a case ($Re_0=100, \lambda/r_0=10, b/r_0=0.6$) where a recirculating eddy forms in the CMR case but does not form in the CV case. This indicates that the volume constraint can fundamentally alter the flow regime, not just the magnitude.
• Annular Tubes:
  • For a wavy outer wall, the CV case yields higher resistance (underestimated by CMR).
  • For a wavy inner wall, the volume decreases with amplitude. Consequently, the CV case yields lower resistance than the CMR case. The CMR approach can overestimate resistance by up to 340% in this specific configuration.

B. Peristaltic Pumping (Moving Walls)

• Significance: The volume change is critical here because the flow rate scales with $(b/r_0)^2$, the same scaling as the relative volume change.
• Maximum Flow Rate:
  • In the extreme case where the tube is pinched off ($b=r_0$), the CMR case predicts a maximum flow rate of $Q_{max} = 1.5 \pi r_0^2 c$.
  • The CV case predicts $Q_{max} = \pi r_0^2 c$.
  • Result: The CMR assumption overestimates the maximum pumping capacity by 50% compared to the constant-volume scenario.

5. Significance and Applications

• Correction of Literature: The paper suggests that many previous studies on wavy tube flows may have overestimated flow rates and underestimated hydraulic resistance due to the unaccounted volume increase.
• Biological Relevance (Glymphatic System): The authors explicitly link these findings to cerebrospinal fluid (CSF) flow in perivascular spaces (PVS) surrounding brain arteries.
  • Arterial pulsations drive CSF flow.
  • Experimental data suggests blood volume is tightly regulated (constant volume).
  • Using the standard CMR model for PVS flow could underestimate hydraulic resistance by ~20% (for 15% wall thickness and 10% amplitude), leading to inaccurate models of waste clearance in the brain.
• Design Implications: For engineering applications involving corrugated pipes or microfluidic devices with wavy walls, maintaining a constant volume constraint is essential for accurate prediction of pressure drops and pumping efficiency.

Conclusion

Guo and Thomas demonstrate that the "constant mean radius" assumption is geometrically inconsistent with a constant-volume system for wavy tubes. By introducing a "constant volume" framework and a scaling law to bridge the two, they provide a more physically accurate method for analyzing viscous flows. The results show that ignoring volumetric changes leads to significant quantitative errors (up to 50%) and potential qualitative errors (flow topology changes), particularly in peristaltic pumping and annular geometries.