Two Dimensional Three Sphere Swimmer with Navier-Stokes Equation

1. Introduction

In this example, we study the two-dimensional three-sphere swimmer using the Navier-Stokes equation. Unlike the Stokes equation where the Reynolds number \(Re = 0\), we consider low Reynolds number in the equation of Navier-Stokes and check results.

2. 2D-Three sphere swimmer

The model we treat here was studied with the Stokes equation in 3-spheres-2D.

3. Geometry

The geometry is mensioned in 3-spheres-2D.

4. Inpute parameters

Name

Description

values

Unit

\(R\)

spheres radius

\(10^{-3}\)

\(m\)

\(D\)

arm length

\(10^{-2}\)

\(m\)

\(\varepsilon\)

relative displacement of the spheres

4 \(\times 10^{-3}\)

\(m\)

\(L\)

length of the channel

50 \(\times 10^{-3}\)

\(m\)

\(l\)

width of the channel

20 \(\times 10^{-3}\)

\(m\)

5. Materials

Name

Description

values

Unit

\(\rho_{fluid}\)

fluid density

1.025\(\times 10^{-3}\)

\(kg/m^3\)

\(\rho_{spheres}\)

spheres density

1.025\(\times 10^{-3}\)

\(kg/m^3\)

\(\mu\)

fluid viscosity

\(10^{-3}\)

\(N.s/m^2\)

6. Run simulations

The command line to run the simulations is :

mpirun -np 16 feelpp_toolbox_fluid --config-file three_sphere_2D.cfg

7. Results

In the figure below, we present the dimensionless displacement \(\frac{\Delta}{R}\) of the three-sphere swimmer.

Figure NS

  • The dimensionless net displacement found here after each period is \(\Delta /R = 0.26\)

  • Note that the net-displacement found using the Stokes equation (3-spheres-2D) is \(\Delta /R = 0.29\)

  • It is clear that the net-displacement depends on the dimensions of the swimmer. In other words, the net-displacement of the swimmer depends on the Reynolds number \(Re\).

References on Swimming