Efficient Numerical Methods for Maxey-Riley Equations with Basset History Term

Core Concepts

Efficient numerical methods are developed to solve the Maxey-Riley equations with a Basset history term, providing insights into particle motion in fluid.

Abstract

The content discusses numerical methods for solving the Maxey-Riley equations (MRE) with a focus on the Basset history term. It compares different approaches, including finite difference schemes and direct integration methods, highlighting challenges and solutions in modeling particle motion in fluids.
Abstract:

MRE model particle motion in fluid with wake effects.
Prasath et al. propose a reformulation using fractional derivatives.
A numerical approach based on finite differences is developed.
Introduction:

MRE used to study various phenomena involving inertial particles.
Historical background of MRE development discussed.
Different numerical approaches proposed by researchers outlined.
Data Extraction:

"R = 1 + 2β" - Parameters definition.
"S = a^2 / (3Tν)" - Parameters definition.
"lim t→0 q(0, t) = v0 − u0" - Boundary condition description.
Quotes:

"Particles do not alter the fluid."
"Neglecting Basset history term leads to inaccuracies."

Stats

R = 1 + 2β
S = a^2 / (3Tν)
lim t→0 q(0, t) = v0 − u0

Quotes

Particles do not alter the fluid.
Neglecting Basset history term leads to inaccuracies.

Key Insights Distilled From

Efficient numerical methods for the Maxey-Riley equations with Basset history term

by Julio Urizar... at arxiv.org 03-21-2024

https://arxiv.org/pdf/2403.13515.pdf

Deeper Inquiries

How do different numerical methods impact computational efficiency

The different numerical methods discussed in the context impact computational efficiency in various ways.

Finite Difference Methods: These methods, such as second-order and fourth-order finite difference schemes, are efficient for solving partial differential equations on unbounded domains. They discretize the spatial domain with a quasi-uniform grid and use techniques like implicit-explicit Runge-Kutta methods to integrate the semi-discrete problem in time. While they provide accurate solutions, their computational efficiency can be affected by the need for high-resolution grids to capture complex flow dynamics accurately.

Direct Integration Methods: Direct integration methods proposed by Daitche rely on linear multistep methods to solve the Maxey-Riley equations efficiently without approximations of the kernels. These methods are computationally fast but may require significant memory storage for long simulations due to storing all time steps.

Reformulated Approaches: Reformulated approaches like Prasath et al.'s method reformulate the Maxey-Riley equations as a diffusive heat equation posed on a semi-infinite pseudo-space with a nonlinear boundary condition. This reformulation allows standard numerical techniques for partial differential equations to be applied, enhancing computational efficiency while eliminating memory-intensive aspects associated with integral terms.

What are the implications of neglecting the Basset history term

Neglecting the Basset history term in models like the Maxey-Riley equations can have several implications:

Inaccurate Results: Neglecting this term can lead to noticeable inaccuracies in modeled trajectories, especially for small Stokes numbers where its effects become more pronounced.

Loss of Physical Realism: The Basset history term accounts for past trajectory effects on particle motion due to fluid interactions, which is crucial for capturing realistic particle behavior.

Memory Intensive Solutions: Dealing with integral terms requires storing all previously computed steps during straightforward numerical integration, leading to memory-intensive computations that may not be feasible or efficient for long simulations.

Model Validity Concerns: Neglecting important physical phenomena represented by the Basset history term could compromise model validity and hinder accurate predictions of particle motion in fluid dynamics scenarios.

How can these numerical approaches be extended to other complex fluid dynamics problems

These numerical approaches developed for solving complex problems like those involving inertial particles in fluid dynamics can be extended to other challenging fluid dynamics problems through similar methodologies:

Reformulation Techniques:

Utilize reformulation strategies that transform integro-differential equations into simpler forms amenable to standard numerical techniques.
Explore connections between fractional derivatives and integral terms present in governing equations.

Numerical Algorithms:

Develop finite difference schemes tailored towards specific problem characteristics (e.g., unbounded domains) using advanced discretization techniques.
Implement direct integration methods based on linear multistep algorithms optimized for accuracy and efficiency.

Open-source Implementations:

Provide accessible code repositories or software tools implementing these advanced numerical algorithms allowing researchers across disciplines easy access and application.

By applying these principles and adapting them creatively based on specific problem requirements, researchers can tackle diverse challenges within fluid dynamics effectively using advanced numerical methodologies originally designed for inertial particle motion studies like those described above from Prasath et al., Daitche's work, etc..