# Solution of Navier-Stokes eq for a single particle?

• DarkLindt
In summary, the conversation is about the solution of the Navier-Stokes equation for a single particle in fluid dynamics. Equation (13) describes the velocity distribution around a single bead subject to a force. The individual is seeking resources and citations for the derivation of this equation, and a solution is found in another paper.
DarkLindt
Solution of Navier-Stokes eq for a single particle?

Hi!

I'm reading this paper on fluid dynamics:
http://jcp.aip.org/resource/1/jcpsa6/v50/i11/p4831_s1
Its equation (13) is the velocity distribution around a single bead of radius a subjecting to force fi in solution. (the subscript i is irrelevant here). The bead is located at the origin and $\mathbf\rho^{'}$ is the coordinate for an arbitrary point in space.

Equation (13):
$\mathbf u_{i}(\mathbf\rho^{'}) = (8\pi\eta a)^{-1}\left [ \left ( \frac{a}{\rho ^{'}}+ \frac{1}{3}\frac{a^3}{\rho ^{'3}} \right ) \mathbf f_i + \left ( \frac{a}{\rho ^{'3}}- \frac{a^3}{\rho ^{'5}} \right ) \mathbf f_i \cdot \mathbf\rho^{'}\mathbf\rho^{'} \right ]$

There is not even citation for this equation, it looks like some textbook solution of the Navier-Stokes equation for this simple system. I just want to know how this can be derived? Could anyone provide me some resource to look at?

Thanks sooo much!

Last edited by a moderator:

Ahhh! I found a derivation from another paper: http://jcp.aip.org/resource/1/jcpsa6/v53/i1/p436_s1
The derivation are equation (1)-(7). Equation (7) is equivalent to equation (13) from the post above.
As for the derivation of equation (2) in this paper, one needs to refer to earlier texts of fluid dynamics.

Last edited by a moderator:

## 1. What is the Navier-Stokes equation?

The Navier-Stokes equation is a mathematical equation that describes the motion of a fluid in terms of its velocity, pressure, density, and viscosity. It is named after the scientists who first derived it, Claude-Louis Navier and George Gabriel Stokes.

## 2. What is the significance of solving the Navier-Stokes equation for a single particle?

Solving the Navier-Stokes equation for a single particle allows scientists to understand the behavior and movement of individual particles within a fluid. This is important in fields such as fluid mechanics and aerodynamics.

## 3. How is the Navier-Stokes equation typically solved for a single particle?

The Navier-Stokes equation is typically solved using numerical methods, such as finite difference or finite element methods. These methods involve discretizing the equations and solving them iteratively to obtain a numerical solution.

## 4. What factors affect the solution of the Navier-Stokes equation for a single particle?

The solution of the Navier-Stokes equation for a single particle is affected by various factors, including the fluid's viscosity, density, and velocity, as well as the particle's size and shape. Other external factors, such as temperature and pressure, may also affect the solution.

## 5. What are some applications of the solution of the Navier-Stokes equation for a single particle?

The solution of the Navier-Stokes equation for a single particle has a wide range of applications, including predicting the flow of air and water around objects, designing aircraft and cars, and understanding the behavior of fluids in industrial processes. It is also used in weather forecasting and in the study of ocean currents and atmospheric patterns.

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