Liquid jets viscous

A more involved version of the quasi-one-dimensional equations of the dynamics of thin liquid jets was proposed in [37, 38] where radial inertia in the jet cross-section was accounted for. The final version of these equations for a Newtonian viscous jet with a straight axis derived in [37-39] has the form... 

On the other hand, capillary breakup of sufficiently viscous liquid jets is a longwave phenomenon, and its description in the framework of the quasi-one-dimensional equations of the dynamics of liquid jets is sufficiently accurate. The effect of the viscosity on the capillary breakup of highly-viscous liquid jets was studied numerically by Yarin [29]. The initial perturbation of the jet surface was imposed as a harmonic... 

The effects of such physical properties as hquid density, viscosity and surface tension on the capillary jet breakup in the case of Newtonian viscous hquids are discussed in the previous sections of this chapter. In many applications non-Newtonian liquid jet flows are used, which demonstrate very peculiar deviations from the Newtonian behavior. This section is devoted to the discussion of the dominating effects of rheological properties on jet breakup. 

Bogy, D. B. Use of one-dimensional Cosserat theory to study instability of a viscous liquid jet. Pkys. Fluids 21, 190-197 (1978). 

Leib, S. J. Goldstein, M. E. Convective and absolute instability of a viscous liquid jet. Phys. Fluids 29,952-954 (1986). 

Given its efficiency, BEM has been applied to a variety of problems involving large deformations of a free surface. Several solutions have been developed for problems related to the nonlinear evolution of water waves, [9-11] and for problems related to nonlinear deformations of both viscous and inviscid drops [12,13]. BEM has been applied to several applications of creeping (Stokes) flows in liquid columns [14-16] and in annular layers [16]. Inviscid solutions have also been obtained for both infinite [16] and finite-length [13] liquid jet problems, as well as for dripping flows [13], fountains [13], and fluid sloshing problems [17]. 

The dispersion relation of an incompressible viscous Newtraiian liquid jet in an inviscid ambient medium was derived by Rayleigh [3], and, more widely recognized in the literature, by Weber [4], relating to experiments by Haenlein [5]. The dispersion relation for this case reads... 

As the most general case of a liquid jet in a fluid ambient medium, the instability of a viscous column of liquid in another immiscible incompressible viscous Newtonian host medium was analyzed by Tomotika [6] as a generahzation of the works by Rayleigh [3] and Weber [4]. It has the form of a determinant that equals zero and reads... 

J. H. Hilbing and S. D. Heister, Nonlinear simulation of a high-speed, viscous, liquid jet. Atomization Sprays, 8, 155-178, 1997. 

Thus, the jet must have a smaller diameter than the tube in order for momentum to be conserved. This result is valid when the liquid s momentum is dominant. At very low Reynolds numbers, viscous stresses are dominant and the velocity profile starts to change even before the exit plane in this case the jet diameter is slightly larger than the tube diameter. 

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