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Experiments on the stability of viscous flow between rotating cylinders III. Enhancement of stability by modulation J. By R. D onnelly Institute for the Study of Metals and Department of Physics The University of Chicago, Chicago 37, Illinois (<Communicated by S. Chandrasekhar, —Received 10 January 1964) The onset of instability in Couette flow can be inhibited by modulating the rate of rotation of the inner cylinder. The origin of the inhibition has been shown experimentally to be due to the viscous wave propagated across the annulus by the modulated cylinder.
132 R. J. Donnelly generates an analog voltage proportional to the number being printed. This voltage was then recorded as a function of time. A typical modulation waveform is shown in figure 2. The principal distortion is a tendency to spend a longer time on the lower than the upper half of the cycle—here this is reduced to less than 1 %. Favourable checks have also been made comparing the mean speed obtained by integration of the data on the printed tape and by averaging the maximum and minimum speeds: in the example of figure 2, the integrated average is 4-498 rad/s and the average of the peaks is 4-51 rad/s. 3 time (sec) Figure 2. Reproduction of a chart record showing a modulation cycle of period 46-1 s and modulation AQ/S2 = 0-085.
Stability of viscous flow between rotating cylinders. 133 part IV, this does not produce a change in ion current. However, if is raised until flmax = Q + All = Hc, where Qc is the (unmodulated) critical angular velocity, for the onset of instability, then a distinct change in signal is recorded. Indeed can be roughly located in this way as shown in figure 3. A typical wave form of the viscous wave without integration is shown in figure 4.
134 R. J. Donnelly not amplify according to the law discussed by Donnelly (1963). We take as our criterion for the onset of instability that speed at which the signal amplitude starts to increase with increasing Q.
Stability of viscous flow between rotating cylinders. Ill 135 in general the degree of enhancement increases as increases until the period becomes so long that it becomes difficult to make observations of the vortex cells at all. Figure 5 shows the presence of transient vortices especially well in the example for P = 20 s, AQ/Q = 0-19.
136 R. J. Donnelly speed of the inner cylinder. A characteristic length for the viscous wave will be S = ^l(2v/o)). At high frequencies, 8 d, whereas at low frequencies there will be no phase shift across the annulus. The period 40 s corresponds to = 2-7; the reason for this exact value must come from a complete theory of the effect.
Stability of viscous flow between rotating cylinders. Ill 137 If the period of otpimum enhancement is governed by the degree of penetration of the viscous wave across the annulus and the shift in critical velocity is governed by the condition QC + AQ « Qc, then one might suppose that a universal curve could be drawn by plotting (f% —Oc)/AQ as a function of *J(P/d2). The results of such a scaling are shown in figure 11. It can be seen that in spite of the scatter, P (sec) Figure 8. Plot of the critical angular velocity with modulation for 7j= 0-9623 (d = 0-75 mm). AQ/D = 10%.
138 R. J. Donnelly P (sec) Figure 10. Plot of the critical angular velocity as a function of period of modulation P for rj = 0-90 (d = 2 mm). Here we could not take data at sufficiently large values of P to locate the maximum in the curve.