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Journal ArticleDOI

Extreme singular events associated with inertial-viscous cusp formation in fluids

12 Jun 2020-Physics of Fluids (AIP Publishing LLCAIP Publishing)-Vol. 32, Iss: 6, pp 062104
TL;DR: In this article, the collapse of a free surface wave depression cavity can lead to inertial-viscous cusp formation at local Re > 1 and Ca > 1, which gives rise to extreme events, i.e., very high-velocity surface jets.
Abstract: Cusp singularities in fluids have been experimentally demonstrated in the past only at a low Reynolds number, Re ≪ 1, and large capillary number, Ca ≫ 1, in Newtonian or non-Newtonian fluids. Here, we show that the collapse of a free surface wave depression cavity can lead to inertial-viscous cusp formation at local Re > 1 and Ca > 1, which gives rise to extreme events, i.e., very high-velocity surface jets. The cavities are generated in a cylindrical container (2R = 10 cm), partially filled with glycerine–water solution, by parametrically forcing the axi-symmetric wave mode beyond the breaking limit. By varying the forcing amplitude and the fluid viscosity, parabolic or cusp singularities manifest, depending on the last stable wave amplitude b that determines the cavity shape. Cusp formation in collapse without bubble pinch-off, leading to very high-velocity surface jets, is obtained when b is close to the singular wave amplitude bs and Ca > 1. The free surface shape is self-similar, changing from an inertial to a viscous regime when the singularity is approached. At cusp singularity, the cavity shape takes the form of (z − Z0)/R ∼ −(r/R)2/3, where Z0 is the final cavity depth. Cavity collapse with bubble pinch-off, which occurs when b > bs, also exhibits a cusp singularity when bs 1, but surface jet velocities are much less because about half of the wave energy is lost.
Citations
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Journal ArticleDOI
03 Dec 2020
TL;DR: In low-viscosity fluids, an inertial-capillary transition occurs (exponent changes from 1/2 to 2/3), while it is inertialviscous in viscous fluids at a late stage of collapse as discussed by the authors.
Abstract: Viscosity and surface tension of fluids regulates the collapse dynamics of a large cavity where collapse is initially inertial with the minimum radius ${r}_{m}\ensuremath{\sim}({t}_{0}\ensuremath{-}t{)}^{1/2}$, where ${t}_{0}\ensuremath{-}t$ is the time remaining for collapse. In low-viscosity fluids, an inertial-capillary transition occurs (exponent changes from 1/2 to 2/3), while it is inertial-viscous (exponent changes from 1/2 to 1) in viscous fluids at a late stage of collapse. The inertial-viscous transition occurs when the local capillary number is greater than 1 and the local Ohnesorge number is greater than 0.1.

3 citations

Journal ArticleDOI
TL;DR: In this paper , the effect of the contact angle on the generation position and focusing efficiency of annular focused jets between parallel plates was investigated, and a new calculation method for the jet focusing efficiency was proposed.
Abstract: Focused jets have been widely studied owing to the abundance of attractive flow phenomena and industrial applications, whereas annular focused jets are less studied. This study combines experiments, numerical simulations, and analytical modeling to investigate the effect of the contact angle on the generation position and focusing efficiency of annular focused jets between parallel plates. In the experiment, a pulsed laser generates a cavitation bubble inside the droplet, and the rapidly expanding cavitation bubble drives an annular-focused jet on the droplet surface. Changing the plate wettability creates different contact angles and droplet surface shapes between the droplet and plates, which modulates the position and focusing efficiency of the annular jet. Based on the jet singularity theory and by neglecting gravity, the derived formula for the jet position offset is found to depend only on the contact angle, which is in good agreement with the experimental and numerical simulation results. Combined with numerical simulations to analyze the flow characteristics of the droplets between the parallel plates, a new calculation method for the jet focusing efficiency is proposed. Interestingly, when the liquid surface radius is small, the focusing efficiency can be improved by adjusting the contact angle to make the jet position closer to the flat plate, whereas the same operation reduces the focusing efficiency when the radius is large. The study of annular jets can expand the scope of traditional jet research and has the potential to provide new approaches for applications such as high-throughput inkjet printing and liquid transfer.

3 citations

Journal ArticleDOI
TL;DR: The effect of fluid depth on the collapse of large cavities generated by over-driven axisymmetric gravity waves in a 10 cm diameter cylindrical container has been studied in this paper.
Abstract: The effect of fluid depth on the collapse of large cavities generated by over-driven axisymmetric gravity waves in a 10 cm diameter cylindrical container has been studied. At a large fluid depth in a viscous glycerine–water solution, the collapse of the cavities is inertia dominant at the initial phase with the time-dependent cavity radius (rm) obeying rm ∝ τ1/2; τ = t − t0 being the time remaining for collapse, with t0 being the time at collapse. However, enhanced damping at a low liquid depth turns the late stage of the transition into the viscous regime (rm ∝ τ) at some critical depth beyond which a singular collapse (transition from non-pinch-off and pinch-off collapse) is impossible. At a shallow depth, the change in cavity radius follows a flip of the power law, i.e., rm ∝ τ at the initial stage of collapse followed by a transition to rm ∝ τ1/2, suggesting a viscous–inertial transition. For fluids with relatively lower viscosity but similar surface tension, here water, a smoother cavity with damped parasitic waves at a small liquid depth collapses at a smaller radius. The surface jet velocity due to the collapse of the cavity monotonically decreases with the decrease in the depth, whereas in the case of water, it increases with the depth reaching a maximum at a critical depth followed by a decrease again. The self-similarity, exhibited by the cavity up to the critical depth, is lost due to the axial movement restriction by the bottom wall.

3 citations

Journal ArticleDOI
TL;DR: In this paper , the authors used X-ray phase-contrast imaging and direct numerical simulations based on the Volume-of-Fluid method to study the mechanisms underlying the bubble entrainment in a piezo-acoustic printhead.
Abstract: The oscillatory flows present in an inkjet printhead can lead to strong deformations of the air-liquid interface at the nozzle exit. Such deformations may lead to an inward directed air jet with bubble pinch-off and the subsequent entrainment of an air bubble, which is highly detrimental to the stability of inkjet printing. Understanding the mechanisms of bubble entrainment is therefore crucial in improving print stability. In the present work, we use ultrafast X-ray phase-contrast imaging and direct numerical simulations based on the Volume-of-Fluid method to study the mechanisms underlying the bubble entrainment in a piezo-acoustic printhead. We first demonstrate good agreement between experiments and numerics. We then show the different classes of bubble pinch-off obtained in experiments, and that those were also captured numerically. The numerical results are then used to show that the baroclinic torque, which is generated at the gas-liquid interface due to the misalignment of density and pressure gradients, results in a flow-focusing effect that drives the formation of the air jet from which a bubble can pinch-off.

1 citations

References
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Journal ArticleDOI
TL;DR: The physical and chemical condition of emulsions of two fluids which do not mix has been the subject of many studies, but very little seems to be known about the mechanics of the stirring processes which are used in making them.
Abstract: The physical and chemical condition of emulsions of two fluids which do not mix has been the subject of many studies, but very little seems to be known about the mechanics of the stirring processes which are used in making them. The conditions which govern the breaking up of a jet of one fluid projected into another have been studied by Rayleigh and others, but most of these studies have been concerned with the effect of surface tension or dynamical forces in making a cylindrical thread unstable so that it breaks into drops. The mode of formation of the cylindrical thread has not been discussed. As a rule in experimental work it has been formed by projecting one liquid into the other under pressure through a hole. It seems that studies of this kind which neglect the disruptive effect of the viscous drag of one fluid on the other, though interesting in themselves, tell us very little about the manner in which two liquids can be stirred together to form an emulsion. When one liquid is at rest in another liquid of the same density it assumes the form of a spherical drop. Any movement of the out er fluid (apart from pure rotation or translation) will distort the drop owing to the dynamical and viscous forces which then act on its surface. Surface tension, however, will tend to keep the drop spherical. When the drop is very small, or the liquid very viscous, the stresses due to inertia will be small compared with those due to viscosity.

2,250 citations

Journal ArticleDOI
TL;DR: In this article, the free-surface motion following a small air bubble burst at an equilibrium position at an air/water interface is modelled numerically using a boundary integral method.
Abstract: When a small air bubble bursts from an equilibrium position at an air/water interface, a complex motion ensues resulting in the production of a high-speed liquid jet. This free-surface motion following the burst is modelled numerically using a boundary integral method. Jet formation and liquid entrainment rates from jet breakup into drops are calculated and compared with existing experimental evidence. In order to investigate viscous effects, a boundary layer is included in the calculations by employing a time-stepping technique which allows the boundary mesh to remain orthogonal to the surface. This allows an approximation of the vorticity development in the region of boundary-layer separation during jet formation. Calculated values of pressure and energy dissipation rates in the fluid indicate a violent motion, particularly for smaller bubbles. This has important implications for the biological industry where animal cells in bioreactors have been found to be killed by the presence of small bubbles.

292 citations

Journal ArticleDOI
27 Jan 2000-Nature
TL;DR: This paper reports a theoretical and experimental study of the generation of a singularity by inertial focusing, in which no break-up of the fluid surface occurs, and predicts that the surface profiles should be describable by a single universal exponent.
Abstract: Finite-time singularities—local divergences in the amplitude or gradient of a physical observable at a particular time—occur in a diverse range of physical systems. Examples include singularities capable of damaging optical fibres and lasers in nonlinear optical systems1, and gravitational singularities2 associated with black holes. In fluid systems, the formation of finite-time singularities cause spray and air-bubble entrainment3, processes which influence air–sea interaction on a global scale4,5. Singularities driven by surface tension have been studied in the break-up of pendant drops6,7,8,9 and liquid sheets10,11,12. Here we report a theoretical and experimental study of the generation of a singularity by inertial focusing, in which no break-up of the fluid surface occurs. Inertial forces cause a collapse of the surface that leads to jet formation; our analysis, which includes surface tension effects, predicts that the surface profiles should be describable by a single universal exponent. These theoretical predictions correlate closely with our experimental measurements of a collapsing surface singularity. The solution can be generalized to apply to a broad class of singular phenomena.

232 citations

Journal ArticleDOI
TL;DR: In this article, the authors provided an analysis of the flow in the neighbourhood of the cusp, via an idealized problem which is solved completely: the cylinders are represented by a vortex dipole and the solution is obtained by complex variable techniques.
Abstract: When two cylinders are counter-rotated at low Reynolds number about parallel horizontal axes below the free surface of a viscous fluid, the rotation being such as to induce convergence of the flow on the free surface, then above a certain critical angular velocity Ωc, the free surface dips downwards and a cusp forms. This paper provides an analysis of the flow in the neighbourhood of the cusp, via an idealized problem which is solved completely: the cylinders are represented by a vortex dipole and the solution is obtained by complex variable techniques. Surface tension effects are included, but gravity is neglected. The solution is analytic for finite capillary number [Cscr ], but the radius of curvature on the line of symmetry on the free surface is proportional to exp (−32π[Cscr ]) and is extremely small for [Cscr ] [gsim ] 0.25, implying (in a real fluid) the formation of a cusp. The equation of the free surface is cubic in (x, y) with coefficients depending on [Cscr ], and with a cusp singularity when [Cscr ] = ∞.The influence of gravity is considered through a stability analysis of the free surface subjected to converging uniform strain, and a necessary condition for the development of a finite-amplitude disturbance of the free surface is obtained.An experiment was carried out using the counter-rotating cylinders as described above, over a range of capillary numbers from zero to 60; the resulting photographs of a cross-section of the free surface are shown in figure 13. For Ω Ωc, the downward-pointing cusp forms, and its structure shows good agreement with the foregoing theory.

182 citations