Slip flows of a Bingham fluid in curved channels

Motivated by varicose vein sclerotherapy, this study analyzes pressure-driven Bingham fluid flows in curved channels with Navier slip, revealing that while wall slip reduces stagnant foam regions in high-curvature areas, it also narrows the unyielded plug, potentially compromising the foam's efficacy in displacing blood.

The Gist

Imagine you are trying to push a thick, sticky blob of toothpaste through a garden hose. But this isn't just any toothpaste; it's a special kind that acts like a solid until you push it hard enough, at which point it suddenly turns into a liquid and flows. In the scientific world, this is called a Bingham fluid.

Now, imagine that garden hose isn't straight. It's bent, twisted, and curvy, just like the veins in your legs. This is the real-world problem the authors of this paper are solving: How does this special "toothpaste" (which is actually a medical foam used to treat varicose veins) flow through a curved, bumpy tube when the walls of the tube are slippery?

Here is the breakdown of their discovery, using some everyday analogies.

1. The Setup: The "Sticky" Foam and the "Slippery" Vein

Doctors treat varicose veins by injecting a foam (a mix of gas bubbles and liquid) to push the stagnant blood out and collapse the vein.

• The Fluid: Think of the foam as a crowd of people holding hands. If you push gently, they stand still (they are "unyielded" or solid). If you push hard enough, they break their grip and start running (they "yield" and flow).
• The Vein: Veins aren't perfect tubes; they bulge and curve.
• The Slip: Usually, we assume fluids stick to the walls of a pipe (like tape on a wall). But foams are different. The bubbles near the wall can slide over it like ice skates on ice. This is called slip.

2. The Straight Tube vs. The Curved Tube

The researchers first looked at a straight pipe, then a curved one.

• In a Straight Pipe: If you add slip (make the walls slippery), the whole crowd of people moves faster. However, the "solid" core in the middle (the plug) stays the same size. It's like everyone on a moving walkway at the airport just walking a bit faster, but the group in the middle doesn't get smaller.
• In a Curved Pipe: This is where things get interesting. When the pipe bends, the "solid" core in the middle gets squished and pushed toward the inside of the curve.
  • The Analogy: Imagine a group of people running around a corner. The people on the outside have to run a longer path, so they get stretched out. The people on the inside have a shorter path. The "solid" crowd gets squeezed toward the inner wall.
  • The Slip Effect: When you add slip to a curved pipe, this squeezing effect gets worse. The solid core gets even smaller and moves even closer to the inner wall.

3. The "Dead Zones" (The Traffic Jams)

In very curvy sections of the tube, the fluid can get stuck in "dead zones" where it stops moving entirely, even though the rest of the fluid is flowing.

• The Analogy: Think of a traffic jam in a sharp turn. The cars on the very inside of the turn might get stuck in a pocket where no one can move.
• The Good News: The researchers found that slip is a hero here. Even a tiny bit of slip at the wall is enough to melt these "dead zones." The slippery wall helps the fluid keep moving, preventing it from getting stuck in the corners.

4. The Medical Dilemma: A Double-Edged Sword

This is the most important part for the doctors treating patients. The results show that slip has two opposite effects:

The Bad News (The Narrowing):
Because slip makes the "solid" plug of foam smaller in curved veins, the foam might not cover the whole width of the vein.

• The Consequence: If the foam plug is too narrow, it might not push all the blood out effectively. It's like trying to sweep a floor with a broom that is too thin; you'll leave patches of dirt (blood) behind. If the plug gets too small, it might even touch the inner wall and stop the flow entirely.

The Good News (The Dead Zones):
As mentioned, slip prevents the foam from getting stuck in the tightest curves.

• The Consequence: This ensures the foam keeps moving and doesn't leave "pockets" of unmoving foam that aren't doing their job of collapsing the vein.

5. The Takeaway for Doctors

The paper suggests that when treating varicose veins, doctors need to be very careful about the shape of the vein and the "stickiness" of the foam.

• Keep it Straight: If the vein is very curvy, the foam plug shrinks. Doctors might need to inject with more pressure or use a foam that is "stiffer" (has a higher yield stress) to keep the plug wide enough to push the blood out.
• Watch the Slip: If the vein walls are very slippery, the foam plug will shrink even more. To fix this, doctors might need to adjust the foam's recipe (make the bubbles smaller or change the liquid content) to make it "stickier" so it stays wide enough to do its job.

In summary: Slip makes the foam flow faster and prevents it from getting stuck in corners, which is good. But it also shrinks the main "pushing" part of the foam, which can make the treatment less effective at clearing out the blood. It's a balancing act between keeping the foam moving and keeping it wide enough to do the work.

Technical Summary

1. Problem Statement

The study investigates the pressure-driven flow of Bingham fluids (yield-stress fluids) through channels with varying curvatures, specifically incorporating Navier wall slip. The primary motivation is the medical application of foam sclerotherapy for treating varicose veins.

• Context: In this procedure, an aqueous foam is injected into a vein to displace stagnant blood and deliver a sclerosant. The foam behaves as a yield-stress fluid; it flows only when shear stress exceeds a critical yield stress ($\tau_y$).
• Challenge: Varicose veins often exhibit significant curvature and bulges. Previous models assumed a "no-slip" boundary condition, which is unrealistic for foams in veins. Foams can slip at walls due to lubrication layers, bubble deformation, or wetting films.
• Objective: To determine how wall slip affects the velocity profiles, the position and width of the unyielded "plug" region, and the formation of "dead zones" (static fluid regions) in both uniformly curved and non-uniformly curved (sinusoidal) channels.

2. Methodology

The authors employ a combination of semi-analytical derivation and numerical simulation using the Finite Element Method (FEM).

A. Mathematical Model

• Governing Equations: The flow is modeled using the Stokes equations (low Reynolds number, $Re \sim 10^{-5}$) coupled with the Bingham constitutive model.
  • Fluid flows when $|\tau| > \tau_y$.
  • Fluid is static (plug) when $|\tau| \le \tau_y$.
• Boundary Conditions:
  • Navier Slip Law: The slip velocity at the wall is proportional to the wall shear stress: $u_{slip} = \beta |\tau_{wall}|$, where $\beta$ is the slip length.
  • Pressure Gradient: A constant pressure gradient ($G$) is assumed, consistent with syringe-driven injection.
• Non-dimensionalization: Variables are scaled by channel width ($W$), viscosity ($\mu$), and pressure gradient. A Bingham number ($B$) is defined based on the pressure gradient: $B = 2\tau_y / (GW)$.

B. Analytical Approach (Uniform Curvature)

• Straight Channel: The authors first derive a closed-form solution for a straight channel with slip. They confirm that while slip increases flow rate and velocity magnitude, it does not alter the plug width or position in a straight channel under a fixed pressure gradient.
• Uniformly Curved Channel: They extend the derivation to a channel formed by a section of an annulus (constant curvature $\kappa$).
  • The stress distribution is non-linear ($1/r^2$ dependence).
  • The solution involves solving for yield surface positions ($\hat{r}_i, \hat{r}_o$) using a matching condition that results in a transcendental equation, solved numerically (semi-analytic) using root-finding routines.

C. Numerical Approach (Non-Uniform Curvature)

• Geometry: Two complex geometries are analyzed:
  1. Transition Region: A straight channel abruptly joining a curved channel.
  2. Sinusoidal Channel: A channel with continuously varying curvature (wavy walls).
• Solver: The FreeFem++ software is used with the augmented Lagrangian method and Papanastasiou regularization to handle the yield stress singularity.
• Validation: The numerical results for the transition region are validated against the semi-analytical solutions for the straight and curved limits.

3. Key Contributions

1. Extension of Closed-Form Solutions: The paper provides the first closed-form (semi-analytic) expressions for Bingham fluid flow in curved channels incorporating Navier slip, extending previous no-slip work by Roberts and Cox.
2. Quantification of Slip Effects: It rigorously quantifies how slip length ($\beta$) interacts with channel curvature to modify the unyielded plug region, a factor previously ignored in foam flow modeling.
3. Dead Zone Elimination: The study demonstrates numerically that even small amounts of wall slip can rapidly eliminate "dead zones" (static fluid pockets) that form in highly curved regions of non-uniform channels.
4. Clinical Insight: It bridges the gap between rheological theory and clinical practice by analyzing how foam parameters (liquid fraction, bubble size) and vein geometry affect treatment efficacy.

4. Key Results

Straight vs. Curved Channels

• Straight Channel: Slip increases the overall flow rate and velocity but does not change the plug width or its position relative to the channel center.
• Curved Channel:
  • Plug Shift: Increasing slip length moves the unyielded plug toward the inner wall of the curve.
  • Plug Narrowing: Slip causes a slight decrease in plug width.
  • Critical Slip: For high Bingham numbers and high curvature, increasing slip can cause the inner yield surface to touch the inner wall, effectively stopping the flow in the outer region.

Non-Uniform Geometries (Transition & Sinusoidal)

• Transition Region: In the junction between straight and curved sections, the plug is suppressed (narrowed) more significantly as slip increases. The plug area in the transition zone drops to approximately 70% of the theoretical value expected from an abrupt change.
• Sinusoidal Channels:
  • Dead Zones: In no-slip scenarios, high curvature at the apex of the sine wave creates "dead zones" where fluid remains static.
  • Slip Effect: Introducing slip drastically reduces the area of these dead zones. For moderate to high slip lengths, dead zones disappear entirely, even for high Bingham numbers.
  • Yielding: Slip enhances fluid yielding in curved regions due to increased shear stresses near the walls.

Flow Rate and Pressure

• Under a fixed pressure gradient, increasing slip length increases the flow rate linearly.
• To maintain a unit pressure gradient in sinusoidal channels, the required inlet pressure must be corrected for the increased path length and channel narrowing, a factor that depends on the Bingham number and slip length.

5. Significance and Implications

For Foam Sclerotherapy

The results present a dual-edged sword regarding wall slip in varicose vein treatment:

• Negative Consequence (Reduced Efficacy): Slip narrows the foam plug. Since the plug is responsible for displacing blood, a narrower plug means less of the vein's cross-section is swept clean. If the plug shifts too far inward or narrows too much, the displacement efficiency drops, potentially leaving blood behind.
• Positive Consequence (Dead Zone Elimination): Slip prevents the formation of static "dead zones" in highly curved or bulging sections of the vein. Dead zones trap foam without contributing to blood displacement; eliminating them ensures the foam moves continuously, though the overall displacement volume might be reduced due to the narrower plug.

Clinical Recommendations

• Vein Geometry: Procedures should aim to keep the vein as straight as possible to maximize plug width and displacement efficiency.
• Foam Formulation: If a patient's vein geometry suggests high slip (e.g., smooth walls or specific foam properties), clinicians should adjust the foam parameters:
  • Reduce Liquid Fraction: This increases the yield stress ($\tau_y$), widening the plug to counteract the narrowing effect of slip.
  • Increase Injection Pressure: A higher pressure gradient can increase shear stresses, promoting yielding and maintaining flow.
• Future Work: The authors suggest investigating transient flows (ramp-up of injection), the effect of gravity/buoyancy, and experimental validation using biomimetic veins.

In conclusion, the paper establishes that wall slip is a critical parameter in yield-stress fluid flows through curved geometries. While it improves flow continuity by eliminating stagnant regions, it simultaneously reduces the displacement efficiency of the foam plug, necessitating careful optimization of foam rheology and injection pressure in medical applications.