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In addition to enforcing the zero velocity condition at the wall, these Eckman layers can also control long-range properties of the flow. A classical illustration is given by the everyday experience of how a cup of tea returns to rest after stirring. We expect decay of the rigid body motion through dissipative effects, which reach out from the stationary sides of the cup over a diffusion timescale , where here is the cup radius. However, it so happens that this process takes much longer than the observed spin-down time and to obtain the correct rate of decay we must consider the Eckman layer which has formed at the bottom of the cup, with thickness (where is the Eckman Number ). Within the rotating core of the fluid inward pressure balances centrifugal forces, yet in the Eckman layer at the bottom of the cup fluid moves more slowly (due to the no-slip condition) leading to lower centrifugal forces. As pressure in constant through the boundary layer, an excess pressure gradient drives an inwards boundary-layer flow (this produces the spiral patterns often observed in tea-leaves) and fluid is ultimately ejected from the boundary layer into the core region. This causes columns of rotating fluid in the inviscid core flow to shorten and widen, and hence rotate at a slower rate in order to conserve angular momentum. It is through this mechanism by which a cup of stirred tea returns to rest. Note that a complete description for the flow requires us to take account of the Stewartson boundary layers, of thickness , located along the side walls of the cup which act as a conduit for fluid to return to the Eckman layer at the cup bottom and hence complete the circuit. In fact, embedded within this layer lies a thinner transition region, of thickness , where vertical flow velocities are reduced to zero. In a highly non-linear regime where the change of rotation rate is substantial, resulting in a non-negligible Rossby Number , the Stewartson layers can become detached from the side walls and propogate into the core flow. References
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