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The dynamics of propagation and disintegration of laminar liquid jets moving in air has been investigated theoretically. It is assumed that the jet is thin, i.e. the ratio of the characteristic transverse size to the longitudinal one is small. It is assumed also that the lateral surface of the jet is free of shearing forces and is ‘almost free’ of normal ones in the sense that the normal tractions other than isotropic pressure are small in comparison with the internal stresses acting in the jet cross-section.Asymptotic quasi-one-dimensional equations of the continuity, momentum and moment of momentum of liquid motion in the jet have been derived. These equations were used as a basis for studying the process of growth of long-wave bending (transverse) disturbances of high-velocity jets of circular cross-section during their motion through air. The instability condition has been obtained and the growth rate of small bending disturbances of the jet has been found; the evolution of the jet shape at the stage of finite disturbances is investigated.

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Herschel–Bulkley fluids are materials that behave as rigid solids when the local stress τ is lower than a finite yield stress τ0, and flow as nonlinearly viscous fluids for τ>τ0. The flow domain then is characterized by two distinct areas, τ<τ0 and τ>τ0. The surface τ=τ0 is known as the yield surface. In this paper, by using analytic solutions for antiplane shear flow in a wedge between two rigid walls, we discuss the ability of regularized Herschel–Bulkley models such as the Papanastasiou, the bi-viscosity and the Bercovier and Engelman models in determining the topography of the yield surface. Results are shown for different flow parameters and compared to the exact solutions. It is concluded that regularized models with a proper choice of the regularizing parameters can be used to both predict the bulk flow and describe the unyielded zones. The Papanastasiou model predicts well the yield surface, while both the Papanastasiou and the bi-viscosity models predict well the stress field away from τ=τ0. The Bercovier and Engelman model is equivalent to the Papanastasiou model provided a proper choice of the regularization parameter δ is made. It is also demonstrated that in some cases the yield surface can be effectively recovered using an extrapolation procedure based upon an analytical representation of the solution.