It is convenient to adopt cylindrical coordinates, , , , whose symmetry axis coincides with the common axis of the two shells. Thus, the inner and outer shells correspond to and , respectively. Suppose that the flow velocity within the fluid is written

(10.30) |

where is the angular velocity profile. Application of the no slip condition at the two shells leads to the boundary conditions

(10.31) | ||

(10.32) |

It follows from Section 1.19 that and

(10.33) | ||

(10.34) |

Hence, Equation (10.2) yields

(10.35) |

and

assuming that . The solution of Equation (10.36) that satisfies the boundary conditions is

(10.37) |

It can be seen that this angular velocity profile is a combination of the solid body rotation profile , and the irrotational rotation profile .

From Section 1.19, the only non-zero component of the viscous stress tensor within the fluid is

(10.38) |

Thus, the viscous torque (acting in the -direction) per unit height (in the -direction) exerted on the inner cylinder is

(10.39) |

Likewise, the torque per unit height exerted on the outer cylinder is

(10.40) |

As expected, these two torques are equal and opposite, and act to make the two cylinders rotate at the same angular velocity (in which case, the fluid between them rotates as a solid body).