Rotating Shaft in Infinite Media

For a nondimensional oscillation period of tp = 0.1, Fig. 4.15 shows the circumferential velocity profiles at four instants in the period. The wall velocity follows the specified rotation rate exactly, which it must by boundary-condition specification. The center velocity r — 0 is constrained by boundary condition to be exactly zero, incenter = 0. The interior velocities are seen to lag the wall velocity, owing to fluid inertia and the time required for the wall s influence to be diffused inward by fluid shearing action. 

An alternative approach is to derive scale factors from the structure of the differential equations, seeking parameter-free equations. Starting from Eq. 4.102, the following 

In these variables there is only one parameter, r,. Therefore, for a given value of the parameter, the equation can be solved once and for all. Furthermore the entire family of solutions can be determined as a function of the single parameter. Consequently the nondimensional formulation has offered a potentially enormous reduction of the problem. In Chapter 6 we make extensive use of this form of nondimensionalization for problems on semi-infinite domains. 

One must be aware of the practical limitations that are inherent in the one-dimensional analysis of a problem like this one. If the analysis is carried to an infinite amount of time, the solution of Eq. 4.119 would predict that the fluid rotation is induced in an infinite extent of space surrounding the shaft. It is obvious that while such a shaft can produce motion in the nearby fluid, it cannot ultimately bring the entire atmosphere up to a solid-body rotation. After a certain amount of time, as the rotating fluid expands outward, the one-dimensionality must be interruped by encounter with surfaces or by fluid instability. 

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