Periodically forced Hopf bifurcation in annular liquid jets with mass transfer

摘要

The leading-order fluid dynamics equations of thin, annular liquid jets at high Reynolds numbers are obtained by means of perturbation methods and employed to determine numerically the nonlinear dynamics of these jets with mass transfer, when these jets are forced by periodic body forces near the solubility ratio at which a Hopf bifurcation occurs in the absence of forcing, for both Henry's and Sievert's solubility laws. It is shown that the nonlinear dynamics associated with Sievert's law is much richer than that with Henry's law. It is also shown that, for values of the bifurcation parameter smaller than the critical one and Henry's law, the jet's dynamics are periodic with a frequency equal to that of the excitation and undergo period-doubling as the amplitude of the forcing is increased, whereas, for values of the bifurcation parameter larger than the critical one, the dynamics undergo a first transition to quasiperiodic motion followed by a still further transition to chaos as the amplitude of the excitation is increased. For Sievert's law, it is found that, if the value of the bifurcation parameter is equal to that at which the Hopf bifurcation occurs, an increase in the forcing amplitude results in quasiperiodic motions whose amplitude increases as the forcing amplitude is increased, whereas, for values of the bifurcation parameter larger than the critical one, the flow bifurcates to period-four solutions and quasiperiodic motions upon increasing both the amplitude and frequency of the excitation. For Sievert's law and values of the bifurcation parameter smaller than the critical one, it is shown that an increase in the forcing frequency first results in flows characterized by several frequencies and broad spectra and that further increases in the forcing amplitude yield broader spectra, whereas increases in the forcing frequency result first in quasiperiodic motions and then frequency locking phenomena. It is also shown that periodically forced Hopf bifurcations in annular liquid jets may result in peak transfer rates which are about twice as large as those that occur at the Hopf bifurcation without forcing, even for very small excitation amplitudes.