Relationship between drag coefficient and Reynolds number in the transition region

10 Relationship between drag coefficient and Reynolds number in the transition region 

Both equations (6.4) and (6.5) contain D and u and need to be expressed in terms of a single variable in order to determine Z) if m is known or m if Z) is known. Stokes neglected the terms due to inertia and obtained a very simple relationship between settling velocity and particle size for particles settling with low velocities. Several attempts at theoretical solutions for the relationship between and Re at higher velocities have been made. Oseen [29] partially allowed for inertial effects to obtain  

The equation is more complicated than Stokes, and in practice is found to be equally inaccurate, since the second term overcorrects Stokes equation and gives a value of C as much in excess as Stokes is too low. 

Proudman and Pearson [30] pointed out that Oseen s solution could only be used to justify Stokes law and not as a first order correction. They obtained a first order correction  

Goldstein [31] solved the equation without approximation and obtained the solution for Reynolds Number less than 2 in the form of the series  

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