Consider a sphere of radius *R*, moving with velocity
in a
quiescent fluid . Referring all coordinates and velocities to a frame
which moves with velocity
relative to the fluid transforms the problem into one of a
resting sphere in a fluid which, at large distances from the sphere, moves
with constant velocity
.

Our velocity field will be stationary. The Stokes equations then read

with boundary conditions

= | (5.18) | ||

= | (5.19) |

As a first step towards the solution, we take the divergence of Eq. (5.16). Making use of the incompressibility equation (5.17) we then get

Now, notice that the flow field will be linear in . From Eq. (5.16) it follows that will then also be linear in . Moreover must be invariant under a rotation of both the coordinate frame and . Therefore we write

(5.21) |

Introducing this into Eq. (5.20) we get

= | 0 | ||

= | 0 | (5.22) |

Solving this under the condition , we get

Eq. (5.16) now reads

Now again, we notice that must be linear in . Moreover it must transform like and under a joint rotation of and the coordinate frame. Therefore

Introducing this expression into Eq. (5.24), we get

= | |||

= | (5.26) |

from which

The boundary conditions for these equations are

We first solve Eq. (5.28) under the condition Eq. (5.30). Solutions to the homogeneous equation are

Equation (5.27) now reads

(5.32) |

Solutions to the homogeneous equation are -1+

Finally we have to fulfill the incompressibility equation

= | |||

= | (5.34) |

Introducing

We now introduce *F*_{1} and *F*_{2} from Eqs. (5.33) and (5.31), together with
into Eq. (5.25), and add
to get the flow field around a sphere
at the origin, moving with velocity
in a quiescent fluid.

One easily checks the boundary conditions

= | (5.36) | ||

= | (5.37) |

We shall now use this flow field to calculate the force exerted by the fluid on the particle.

According to Eq. (5.8) the force exerted by the fluid on the
sphere is given by

where is a surface element on the sphere. The stress tensor is given by Eq. (5.13), which in the case of an incompressible fluid reads

(5.39) |

Using the flow field given in Eq. (5.35) and the pressure given in Eq. (5.23), where , we calculate

= | |||

= | (5.40) |

Introducing this into Eq. (5.38) and performing the integral we obtain

(5.41) |

is known as the Stokes friction .