# How to calculate the viscous damping coefficient of a viscous layer between an inner sphere and an enclosing outer sphere?

In this article by Rahn and Barba, a flat-spin transition manoeuvre is investigated. For this it is assumed that a rigid spacecraft contains a spherical, dissipative fuel slug of inertia $$\boldsymbol{J}$$, which is surrounded by a viscous layer. The coupled equations of motion are given as:

$$\mu \boldsymbol{\sigma} + {\boldsymbol{T}} = \left( \boldsymbol{I} - \boldsymbol{J} \right) \dot{\boldsymbol{\omega}} + \boldsymbol{\omega} \times \boldsymbol{I} \boldsymbol{\omega}$$

$$\dot{\boldsymbol{\sigma}} = -\dot{\boldsymbol{\omega}} - \boldsymbol{J}^{-1}\left(\mu \boldsymbol{\sigma}\right) - \boldsymbol{\omega} \times \boldsymbol{\sigma}$$

where $$\boldsymbol{I}$$ is the inertia tensor of the spacecraft including the slug, $$\boldsymbol{T}$$ is the torque about the principal axes, $$\boldsymbol{\omega}$$ is the angular velocity of the spacecraft, $$\boldsymbol{\sigma}$$ is the angular velocity of the slug relative to the spacecraft, and finally, $$\mu$$ is the so-called viscous damping coefficient of the fuel slug in [Nms]. This is also the unit of angular impulse.

NB: I'm not sure as to whether this definition is actually accurate, as in for example this article, the viscous damping coefficient is measured in [Ns/m].

In the article, $$\mu=30$$ [Nms]. However, no explanation is provided about how this coefficient was determined. Hopefully someone here has a clue.

Question: How does one calculate the viscous damping coefficient of a viscous layer between an inner solid sphere and an enclosing outer hollow sphere (see cross-section below)?