On the development of OpenFOAM solvers for simulating MHD micropolar fluid flows with or without the effect of micromagnetorotation

This paper presents the development and validation of two new OpenFOAM solvers, epotMicropolarFoam and epotMMRFoam, which simulate magnetohydrodynamic micropolar flows and demonstrate that incorporating micromagnetorotation significantly reduces fluid velocity and microrotation while suppressing vortex cores, offering critical insights for biomedical applications like targeted drug delivery.

The Gist

Imagine your bloodstream isn't just a river of red liquid, but a busy highway filled with tiny, spinning cars (your red blood cells). Usually, these cars spin a little bit as they move, and they can also be influenced by magnets.

This paper is about building a super-advanced traffic simulator (using a free software called OpenFOAM) to understand exactly what happens when you put a giant magnet near these spinning blood cars.

Here is the breakdown of the research in simple terms:

1. The Problem: The "Spinning Car" Theory

Most computer models treat blood like water—a simple, smooth fluid. But blood is actually full of tiny particles (cells) that can spin independently. Scientists call this a "Micropolar Fluid."

Furthermore, these blood cells contain iron (hemoglobin), which means they act like tiny magnets. When you apply a strong magnetic field (like in an MRI machine), two things happen:

• The Lorentz Force: The magnetic field tries to push the moving liquid sideways (like wind pushing a sail).
• The "Micromagnetorotation" (MMR): This is the star of the show. Because the blood cells are spinning, the magnetic field grabs them and tries to stop their spin and align them all in the same direction. It's like a magnet grabbing a spinning top and forcing it to stand perfectly still and point North.

The Big Mistake: Most previous computer models ignored this "spinning stop" effect (MMR). They only looked at the wind pushing the sail. This paper says, "Hey, we need to account for the spinning top stopping, or our predictions will be wrong!"

2. The Solution: Two New "Traffic Cop" Programs

The researchers built two new tools (solvers) inside the OpenFOAM software to fix this:

• Tool A (epotMicropolarFoam): This is the "Standard Magnet" simulator. It knows blood has spinning cells and that magnets can push the flow, but it ignores the effect of the magnet stopping the spin.
• Tool B (epotMMRFoam): This is the "Super Magnet" simulator. It includes everything Tool A does, PLUS the "Micromagnetorotation" (MMR). It calculates how the magnet grabs the spinning cells and forces them to stop spinning.

3. The Test Drive: From Straight Roads to Traffic Jams

To make sure their tools worked, they ran three different tests:

• Test 1: The Straight Highway (Poiseuille Flow): They simulated blood flowing through a straight, narrow tube.

  • Result: When they turned on the "Super Magnet" (MMR), the blood slowed down significantly (up to 40% slower), and the cells stopped spinning almost entirely (99.9% reduction). Without the MMR, the magnet barely did anything.
  • Analogy: Imagine a highway where the cars are spinning. If you turn on a giant magnet, the cars stop spinning and line up perfectly. This alignment creates friction, slowing the whole traffic jam down.

• Test 2: The 3D Artery: They simulated blood flowing through a realistic 3D artery.

  • Result: Same story. The "Super Magnet" (MMR) made the blood flow much slower and stopped the cells from spinning. The "Standard Magnet" (without MMR) barely changed anything.

• Test 3: The Swollen Road (Aneurysm): An aneurysm is a weak spot in a blood vessel that bulges out like a balloon. This is dangerous because the blood swirls and creates eddies (whirlpools) inside the bulge, which can lead to clots or rupture.

  • The Magic: When they used the "Super Magnet" (MMR), the magnetic field acted like a stabilizer. It suppressed the swirling whirlpools inside the aneurysm. The blood became calmer and more stable.
  • Analogy: Think of the aneurysm as a whirlpool in a river. The magnetic field acts like a giant hand smoothing out the water, stopping the dangerous swirls.

4. Why Does This Matter?

You might ask, "Why do we care if blood slows down a bit?"

• Medical Safety: If you are in an MRI machine (which uses huge magnets), this research helps doctors understand why some people feel dizzy or nauseous. It's not just the magnet pushing the blood; it's the magnet stopping the blood cells from spinning, which changes how the blood flows.
• Targeted Medicine: Imagine using magnets to steer drug-carrying blood cells to a specific tumor. If we ignore the "spinning stop" effect, our steering will be off. These new tools help us navigate more accurately.
• Treating Aneurysms: The finding that magnets can calm down the swirling blood in an aneurysm suggests a potential new way to treat weak blood vessels without surgery.

The Bottom Line

The researchers built two new computer programs. One is okay, but the other is great because it remembers that blood cells are tiny spinning magnets.

They proved that when you put a strong magnet near blood, it doesn't just push the liquid; it freezes the spin of the cells. This "freezing" slows the blood down and smooths out dangerous swirls. Ignoring this effect is like trying to drive a car without knowing the brakes exist—you'll never get the full picture of how the vehicle behaves.

This work opens the door to better medical treatments, safer MRI scans, and smarter ways to deliver drugs inside the human body.

Technical Summary

1. Problem Statement

The paper addresses a critical gap in computational fluid dynamics (CFD) regarding the simulation of magnetohydrodynamic (MHD) micropolar fluids, specifically biological fluids like blood.

• The Gap: While micropolar fluid theory (which accounts for the micro-rotation of particles like erythrocytes) and MHD theory are well-established, existing open-source tools lack the capability to fully couple these phenomena while including Micromagnetorotation (MMR).
• The Specific Issue: In magnetic fields, ferromagnetic particles (e.g., hemoglobin in red blood cells) experience a magnetic torque due to the misalignment between their magnetization and the external field. This torque, known as MMR, significantly alters the internal angular momentum and flow dynamics. Previous studies often neglected MMR, assuming the Lorentz force was the dominant magnetic effect. However, due to blood's low electrical conductivity, the Lorentz force is weak, making the neglect of MMR a significant oversight in accurate modeling.
• Objective: To develop and validate open-source numerical solvers capable of simulating MHD micropolar flows with and without the MMR effect to better understand blood flow under magnetic fields (relevant for MRI, magnetic hyperthermia, and drug delivery).

2. Methodology

The authors developed two transient solvers within the OpenFOAM (Open Source Field Operation and Manipulation) framework, utilizing the Finite Volume Method (FVM) and the PISO (Pressure-Implicit with Splitting of Operators) algorithm for pressure-velocity coupling.

A. Mathematical Model

The solvers are based on the governing equations derived by Shizawa and Tanahashi, which extend the Navier-Stokes equations:

1. Linear Momentum Equation: Includes the standard pressure and viscous terms, plus a source term for the difference between microrotation and fluid vorticity ($2\mu_r \nabla \times (\omega - \Omega)$), the magnetic body force, and the Lorentz force.
2. Internal Angular Momentum Equation: Solves for the microrotation vector ($\omega$). It includes diffusion, internal torque, and the crucial magnetic torque term ($M \times H$) representing MMR.
3. MHD Equations: Uses the low-magnetic-Reynolds-number approximation ($Re_m \ll 1$). This ignores the induced magnetic field and solves for the electric potential ($\phi$) to reconstruct the current density ($j$) and Lorentz force.
4. Constitutive Magnetization Equation: Solves for the magnetization vector ($M$) based on the Shizawa-Tanahashi model, which relates magnetization to the applied field and the local microrotation.

B. Solver Development

• epotMicropolarFoam: Simulates MHD micropolar flows without the MMR term. It couples the micropolar formulation (from micropolarFoam) with the electric-potential MHD approach (from epotFoam).
• epotMMRFoam: An extension of the first solver that includes the MMR term in the angular momentum equation and solves the constitutive magnetization equation. This captures the magnetic torque aligning particles with the field.

C. Validation and Simulation Cases

The solvers were validated and applied to three distinct scenarios:

1. 2D Planar Poiseuille Flow: Analytical validation against known solutions for blood flow with varying hematocrit ($\phi = 25\%, 45\%$) and magnetic field strengths ($1T, 3T, 8T$).
2. 3D Artery Flow: Simulation of flow in a cylindrical artery ($D=3mm$) to observe MMR effects in a realistic vascular geometry.
3. 2D Aneurysm Flow: Simulation of flow through fusiform aneurysms (166% and 200% dilation) to study the stabilizing effects of MMR on recirculation zones and vortices.

3. Key Contributions

• Novel Solvers: The creation of epotMicropolarFoam and epotMMRFoam, which are the first open-source OpenFOAM solvers capable of fully coupled MHD micropolar flow simulations including the MMR term and constitutive magnetization.
• Implementation of MMR: Successful integration of the Shizawa-Tanahashi magnetization model and the magnetic torque term into the OpenFOAM C++ codebase.
• Comprehensive Validation: Rigorous validation against analytical solutions showing high accuracy (errors $< 2\%$) for both velocity and microrotation fields.
• Biomedical Insight: Demonstration that MMR is the dominant magnetic mechanism in blood flow, far outweighing the Lorentz force, and significantly alters flow dynamics in complex geometries.

4. Key Results

• Validation Accuracy: The solvers achieved excellent agreement with analytical solutions. Errors for velocity and microrotation were consistently below 2% (often $< 0.6\%$) across various hematocrit levels and magnetic field intensities.
• Impact of MMR vs. Lorentz Force:
  • Without MMR: The magnetic field had a negligible effect on blood flow (velocity reduction $< 2\%$) due to blood's low electrical conductivity.
  • With MMR: The magnetic field caused drastic changes.
    • Velocity Reduction: Up to 40% reduction in flow velocity at high magnetic fields ($8T$) and high hematocrit ($45\%$).
    • Microrotation Suppression: Up to 99.9% reduction in microrotation. This indicates that erythrocytes align with the magnetic field, effectively "freezing" their internal rotation.
• Aneurysm Dynamics:
  • In the absence of MMR, magnetic fields did not significantly alter the large recirculation zones (vortices) within aneurysms.
  • With MMR: The magnetic torque suppressed the formation of recirculation cores. The vortices became tighter, less diffusive, and more localized. This highlights a stabilizing and shear-dampening effect of MMR, which could be crucial for preventing aneurysm rupture or thrombosis.
• Hematocrit Dependence: The effects of MMR were more pronounced at higher hematocrit levels ($45\%$) because a higher concentration of magnetic particles (erythrocytes) increases the cumulative magnetic torque.

5. Significance and Future Perspectives

• Biomedical Applications: The developed solvers provide a powerful tool for designing and optimizing biomedical technologies such as magnetic hyperthermia, targeted drug delivery, and magnetic separation of blood components. Understanding how magnetic fields suppress micro-rotation and alter flow resistance is vital for these applications.
• Clinical Relevance: The findings explain experimental observations (e.g., vertigo or metallic taste in MRI) that cannot be attributed solely to the Lorentz force, suggesting MMR-induced changes in erythrocyte rotation and flow resistance are the primary mechanisms.
• Limitations & Future Work:
  • Current simulations are restricted to laminar, steady-state flows; future work will address pulsatile (physiological) flows.
  • The model assumes rigid particles; future iterations will incorporate erythrocyte deformability and aggregation.
  • The magnetic body force term ($M \cdot \nabla H$) was neglected; future studies may include this for flows with high concentrations of magnetic contrast agents.
  • Expansion to patient-specific geometries and non-Newtonian viscosity models is planned.

In conclusion, this paper successfully bridges the gap between theoretical micropolar MHD models and practical CFD applications, demonstrating that Micromagnetorotation is a dominant factor in magnetic blood flow simulations that cannot be ignored.