Summary
A mathematical treatment is given of the instability of rotating cylindrical jets under the action of the inertial effects of the jet and its surface tension. Three types of jet are considered:
-
a.
the one whose liquid fills the space within a cylinder, called the solid jet,
-
b.
the one whose liquid fills the space between two cylinders, called the hollow jet and
-
c.
the one whose liquid fills the space on the outside of a cylinder, called the hollow, infinitely thick jet.
In general the viscosity of the liquid and the inertial effects of the surrounding air have been neglected except in two cases:
-
1.
For the non-rotating solid jet mentioned under a. the influence of the liquid viscosity was taken into account, while the inertial effects of the surrounding air were neglected, especially for rotationally symmetric perturbations and
-
2.
for the rotating solid jet the influence of the inertial effects of the surrounding air was taken into account while the liquid viscosity was neglected. Here the undisturbed velocity field for the air was put equal to zero or was chosen in such a way that the overall velocity field is continuous at the interface between liquid and air.
From the calculations the following conclusions may be drawn.
If the liquid viscosity and the inertial effects of the ambient air are neglected:
-
1.
The instability of the jets becomes greater if the ratio between their dynamic surface tension and the liquid density becomes greater.
-
2.
In some cases non-rotationally symmetric perturbations are more unstable than rotationally symmetric ones.
-
3.
The non-rotating jet is stable to perturbations whose wave number in tangential direction is a positive integer.
-
4.
The perturbations of the solid and the hollow, infinitely thick jet are unstable if the wave number in axial direction lies in a finite interval between zero and a certain critical value, or if the wave number in tangential direction takes the value zero or any integral value below a critical integer (only positive wave numbers are considered).
-
5.
The solid jet is the more unstable as it rotates faster; then the critical wave numbers in both axial and tangential directions are raised.
-
6.
The hollow, infinitely thick jet is the more stable as it rotates faster; then the critical wave numbers in both axial and the tangential direction are lowered.
If the inertial effects of the ambient air or the liquid viscosity are taken into account:
-
7.
The influence of the inertial effect of the ambient air on the solid jet may be neglected if the density of the air is small compared with the density of the liquid and if the velocity field is continuous at the interface.
-
8.
The instability range of the wave number in axial direction for solid (non-rotating) jets with low liquid viscosity is the same as that for solid jets with zero liquid viscosity for rotationally symmetric perturbations. The amplitude of the perturbations, however, grows somewhat more slowly with time than in the case of zero viscosity.
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Abbreviations
- A :
-
r \(\bar V\)
- b :
-
Im(ω)(ρr 30 /σ)1/2
- b*:
-
(ω−kW 0)(ρr 31 /σ)1/2
- I n (x):
-
modified Bessel function of the first kind of order n
- K n (x):
-
modified Bessel function of the third kind of order n
- K :
-
mean curvature of the boundary of the jet
- k :
-
wave number of the perturbations in axial direction
- l :
-
\(\sqrt {k^2 + (\omega - kW_0 )/iv}\), Re(l)≧0
- n :
-
wave number of the perturbations in tangential direction
- O:
-
Landau's order symbol
- P :
-
pressure
- p :
-
perturbation of the pressure
- q :
-
kr 0
- q*:
-
kr 1
- r :
-
radial co-ordinate of cylindrical co-ordinate system
- r 0 :
-
radius of undisturbed boundary of solid jet and hollow, infinitely thick jet
- r i :
-
i=1,2, radius of external and internal boundary of hollow jet
- s :
-
ρA 2/σr 0
- S*:
-
\(\sqrt {{{pA^2 } \mathord{\left/ {\vphantom {{pA^2 } {\sigma r1}}} \right. \kern-\nulldelimiterspace} {\sigma r1}}}\)
- t :
-
time
- U :
-
radial velocity component
- u :
-
perturbation of radial velocity component
- V :
-
tangential velocity component
- V :
-
velocity vector
- v :
-
perturbation of tangential velocity component
- W :
-
axial velocity component
- w :
-
perturbation of axial velocity component
- W 0=\(\bar W\) :
-
undisturbed axial velocity component
- y :
-
(ω−kW 0)r 20 /ν
- z :
-
axial co-ordinate of cylindrical co-ordinate system
- γ :
-
r 1/r 2
- δ :
-
perturbation of radial co-ordinate of the boundary of the solid jet and hollow, infinitely thick jet
- δ i :
-
i=1,2, perturbation of radial co-ordinate of the external and internal boundary of the hollow jet
- Δ :
-
Laplacian operator
- ν :
-
kinematic viscosity
- ρ :
-
density of fluid
- σ :
-
dynamic surface tension
- ϕ :
-
tangential co-ordinate of cylindrical co-ordinate system
- ω :
-
parameter first introduced in equation (2.11)
- C:
-
undisturbed value of C differentiation
- ()*:
-
amplitude function
- e :
-
external
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Weber, C., Z. angew. Math. Mech. 11 (1931) 136.
Rayleigh, Lord, Proc. Roy. Soc. London A93 (1917) 148.
Lin, C. C., The Theory of Hydrodynamic Stability, Cambridge, 1955.
Lamb, H., Hydrodynamics, 6th edition, Dover Publ., 1945.
Christiansen, R. M. and A. N. Hixson, Industr. Engng Chem. 49 (1957) 1017.
Christiansen, R. M., Doctoral Dissertation, Univ. of Pennsylvania, 1955.
Bateman, H., Higher Transcendental Functions II, compiled by A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, McGraw-Hill, 1953.
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Shell Internationale Research Maatschappij N.V.
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Ponstein, J. Instability of rotating cylindrical jets. Appl. sci. Res. 8, 425–456 (1959). https://doi.org/10.1007/BF00411768
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DOI: https://doi.org/10.1007/BF00411768