# Brownian Force: Analyzing Particle Diffusion Using CFD Tools

### Key Takeaways

• Brownian force develops due to the random movement of a suspended particle in the fluid.

• The diffusive nature of Brownian motion can be explained with the Stokes-Einstein equation for low-Reynolds number fluids.

• By simulating particle behavior at a microscopic level, it is possible to make predictions of fluid behavior in complex real-world systems.

Many fluid flow problems deal with the study of the randomness that flow exhibits and the force associated with it. On a larger scale, this randomness influences the behavior of an object in contact; for instance, an airfoil passing through the wind or water flowing through a rotor blade. Looking at a more microscopic level, fluid particles may exhibit randomness due to particle suspension and molecular collision. This is the Brownian force principle.

In computational fluid dynamics, this principle can be applied to simulate how the particle suspension and its randomness affect the flow within a system; for instance, how smoke particles may diffuse through the atmospheric gas or how suspended particles may affect the flow dynamics of water.

Lets us take a deeper look into Brownian force and understand its importance for flow simulation.

## Understanding the Nature of Brownian Force

Brownian force is the force that the fluid exerts on the suspended particle due to its molecular collision. This force is the result of the random motion of the suspended particles, also known as Brownian motion. The Brownian force is stochastic in nature, which means the particle’s random motion causes the direction and magnitude of the force to change constantly. A thermally excited molecule is enough to initiate this random motion and no active force is required.

In many cases, this force is diffusive in nature. The diffusion theory can be explained with the help of the Stokes-Einstein equation for a spherical particle in contact with the fluid. Numerically, this is expressed as:

Note that:

D is the diffusion coefficient

KB is the Boltzmann constant

T is the absolute temperature

η is the dynamic viscosity

r is the radius of the particle

It is important to note that the equation is assumed for low-Reynolds number fluid, i.e., the equation is more effective for smaller particles with low velocity. The equation indicates the dependency of the diffusion coefficient on the temperature and viscosity of the fluid particle and the size of the suspended particle. This understanding is crucial for the simulation of Brownian force in the field of microfluidic system design or diffusion analysis.

## Numerical Simulation of the Brownian Diffusion of Particles

In fluid system design, Brownian diffusion analysis is critical to understanding processes such as mixing and the related diffusion principle. The analysis also allows for the optimization of the system when dealing with suspended particles and the consequential force. Using CFD tools for diffusion analysis facilitates simulation of the randomness due to Brownian force and numerically analyzes the particle behavior. This can be done in many ways.

The Direct Numerical Simulation (DNS) technique is one way to evaluate the Brownian force of the particle. The DNS approach uses the Navier-Stokes equation to compute the motion in both fluid and suspended particles. Using a high-resolution grid, it is possible to simulate this motion, particle behavior, boundary layer conditions, and their effect on the fluid system on a smaller scale.  While Large Eddy Simulation (LES) may deliver a more accurate result for this analysis, the approach is best suited for large-scale simulations. The hybrid method, which combines the DNS and LES methods, is better suited for simulating Brownian diffusion, as it provides a balanced approach to ensuring computational accuracy for large or small-scale simulations.

## Effective Simulation of Brownian Force With CFD Tools

The analysis of particle diffusion on a microscopic level is a complex subject that cannot be handled efficiently without modern CFD tools. For particle fluctuation in random motion, the CFD simulation application can solve the associated Navier-Stokes equation. It is also possible to model the thermal profile of Brownian force and the turbulence associated with particle diffusion. With the right tools and techniques, Brownian force analysis can be conducted with a high level of accuracy and efficiency.

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# About the Author

With an industry-leading meshing approach and a robust host of solver and post-processing capabilities, Cadence Fidelity provides a comprehensive Computational Fluid Dynamics (CFD) workflow for applications including propulsion, aerodynamics, hydrodynamics, and combustion.

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