B Model Functions for Size Distributions

Note that we could not have derived (2.1.2) by letting C equal zero in (2. A.8) since the viscosity /x was assumed to be nonzero to arrive at (2. A.8) in the first place. 

In the same way that (2.A.7) was derived from the 0-component equation of (2.A.2), two other differential equations for the flow field in a vortex motion can be derived from the r- and the 2 -component equations. They are, respectively  

Equation (2.A.12) is the balance between the centrifugal force (or the mass times the centripetal acceleration) and the pressure force, all on a per unit volume basis. It shows, as we also saw on basis of heuristic arguments in the main text, that the pressure in a vortex flow increases towards the periphery and more so the stronger the tangential velocity. The radial pressure distribution can be obtained by integrating the right-hand side over r. 

Equation (2.A.13) simply says that the axial pressure distribution is the hydrostatic pressure, which in gas cyclones is not very interesting, since the fluid density is low. 

This completes the derivation of the basic equations for swirling flows from the Navier-Stokes equations. When deriving flow equations, particularly in cylindrical coordinates, this method is safer than using heuristic arguments. 

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