The Couette or Concentric Cylinder

If the length of the bob is L and the radii of the cylinders are R and R0 for the inner and outer cylinders respectively, the local change in tangential velocity across a small liquid element dr at r from the centre line of the bob is the local shear rate = rdco/dr and the shear stress in the element is a = rj rdco/dr for our Newtonian liquid. The moment of the drag force is the sum of the stress in each element multiplied by the area of the cylindrical surface times the distance r, i.e. 

So to calculate the angular velocity of the outer cylinder, we have to sum all the contributions from the static cylinder to the moving one  

We can rearrange this in terms of the viscosity to give the Margules equation  

We can now calculate the local shear rate in this Newtonian liquid at any point in the gap. The stress at r is 

This equation enables us to calculate the shear rate at each cylinder surface and the mean value. 

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