Analytical Solutions for Transport Processes by Günter Brenn

Author:Günter Brenn
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

(6.53)

for a jet with radius R, which Keller et al. rewrite into the form

(6.54)

Here, the definitions

were used, where We (denoted in [16]) is the Weber number, a known parameter for the physical state of the jet. The analysis of the spatial instability of the liquid jet is now a matter of solving Eq. (6.54) for the complex wave number kR, in which the equation is transcendental. The number of roots of the equation in this sense is infinite, while the corresponding equation for the real wavenumber had one root for the disturbance growth rate only [16]. Keller et al. analysed the equation by investigating its asymptotic behaviour for very small and very large Weber numbers, and by solving the equation numerically for the non-dimensional wavenumber kR with . The results reproduced here in Fig. 6.7 are all located in the fourth quadrant of the complex kR plane. In view of the formulation (6.52) of the velocity potential of the jet, this indicates spatial growth of disturbances for all wave numbers. The asymptotic behaviour of the dispersion relation and its numerical results show that, for large We numbers, the dispersion relation of Rayleigh for the growth with time is restored, while, for small We, the results may be very different. The good agreement between Rayleigh’s result for the wavenumber of dominant growth rate and the experiment may now be explained by the usually high jet We numbers in the experiment. In this case, the spatial instability may be just unimportant. Keller et al. investigated this behaviour for We numbers up to 100. For much lower values, the spatial instability may be quite important for the jet breakup.

Fig. 6.7Complex kR plane for a spatially unstable jet with the Weber number and the non-dimensional frequency as the parameters [16]

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