The Hydrodynamic Stability of Pendent Drop Under a Liquid Column

1996 ◽  
Vol 63 (1) ◽  
pp. 106-109
Author(s):  
S. P. Lin ◽  
Z. W. Zhou
Keyword(s):  
Hydrodynamic Stability ◽  
Density Ratio ◽  
Bond Number ◽  
Vertical Tube ◽  
Liquid Column ◽  
Maximum Height ◽  
Pendent Drop

The hydrodynamic stability of a liquid column resting on a gas in a vertical tube with its upper end closed is analyzed. The maximum height above which the interface is unstable is given as a function of the Bond number and the density ratio. The instability is shown to be monotonic, i.e., nonoscillatory.

Linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian fluid

2021 ◽  
Vol 927 ◽  
Author(s):  
Romain Canu ◽  
Marie-Charlotte Renoult
Keyword(s):  
Magnetic Field ◽  
Stability Analysis ◽  
Dispersion Relation ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Density Ratio ◽  
Bond Number ◽  
Wire Radius ◽  
Azimuthal Magnetic Field ◽  
Relative Magnetic Permeability

We performed a linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian non-magnetic fluid in an azimuthal magnetic field. A wire is used at the centre of the ferrofluid cylinder to create this magnetic field. Isothermal conditions are considered and gravity is ignored. An axisymmetric perturbation is imposed at the interface between the two fluids and a dispersion relation is obtained allowing us to predict whether the flow is stable or unstable with respect to this perturbation. This relation is dependent on the Ohnesorge number of the ferrofluid, the dynamic viscosity ratio, the density ratio, the magnetic Bond number, the relative magnetic permeability and the dimensionless wire radius. Solutions to this dispersion relation are compared with experimental data from Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and, more recently, Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502). A better agreement than the inviscid theory and the theory that only takes into account the viscosity of the ferrofluid is shown with the data of Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and those of Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502) for small wavenumbers.

scholarly journals Lattice Boltzmann simulations of incompressible liquid–gas systems on partial wetting surfaces

2011 ◽  
Vol 369 (1945) ◽  
pp. 2510-2518 ◽  
Author(s):  
Ching-Hsiang Shih ◽  
Cheng-Long Wu ◽  
Li-Chen Chang ◽  
Chao-An Lin
Keyword(s):  
Lattice Boltzmann ◽  
Density Ratio ◽  
Three Dimensional ◽  
Bond Number ◽  
Two Phase ◽  
Droplet Shape ◽  
Gas Density ◽  
Partial Wetting ◽  
Lattice Boltzmann Simulations ◽  
Large Density Ratio

A three-dimensional Lattice Boltzmann two-phase model capable of dealing with large liquid and gas density ratios and with a partial wetting surface is introduced. This is based on a high density ratio model combined with a partial wetting boundary method. The predicted three-dimensional droplets at different partial wetting conditions at equilibrium are in good agreement with analytical solutions. Despite the large density ratio, the spurious velocity around the interface is not substantial, and is rather insensitive to the examined liquid and gas density and viscosity ratios. The influence of the gravitational force on the droplet shape is also examined through the variations of the Bond number, where the droplet shape migrates from spherical to flattened interface in tandem with the increase of the Bond number. The predicted interfaces under constant Bond number are also validated against measurements with good agreements.

scholarly journals On super free fall

2013 ◽  
Vol 723 ◽  
pp. 653-664 ◽  
Author(s):  
A. Torres ◽  
A. Medina ◽  
F. J. Higuera ◽  
P. D. Weidman
Keyword(s):  
Governing Equation ◽  
Free Fall ◽  
Vertical Tube ◽  
Liquid Column ◽  
Experimental Measurements ◽  
Centre Of Mass ◽  
Interface Motion ◽  
Inviscid Liquid ◽  
Conical Tube ◽  
End Point

AbstractVillermaux & Pomeau (J. Fluid Mech., vol. 642, 2010, p. 147) analysed the motion of the interface of an inviscid liquid column released from rest in a vertical tube whose area expands gradually downwards, with application to an inverted conical container for which experimental measurements were carried out. An error in the analysis is found and corrected in the present investigation, which provides the new governing equation for the super-accelerated interface motion down gradually varying tubes in general, and integrated results for interface trajectories, velocities and accelerations down a conical tube in particular. Interestingly, the error does not affect any of the conclusions given in the 2010 paper. Further new results are reported here such as the equation governing the centre of mass and proof that the end point acceleration is exactly that of gravity.

The development of a bubble rising in a viscous liquid

1999 ◽  
Vol 387 ◽  
pp. 61-96 ◽  
Author(s):  
LI CHEN ◽  
SURESH V. GARIMELLA ◽  
JOHN A. REIZES ◽  
EDDIE LEONARDI
Keyword(s):  
Control Volume ◽  
Density Ratio ◽  
Three Dimensional ◽  
Vertical Cylinder ◽  
Bond Number ◽  
Vof Method ◽  
Bubble Deformation ◽  
Toroidal Bubble ◽  
Bubble Rising ◽  
Three Dimensional Coordinate

The rise and deformation of a gas bubble in an otherwise stationary liquid contained in a closed, right vertical cylinder is investigated using a modified volume-of-fluid (VOF) method incorporating surface tension stresses. Starting from a perfectly spherical bubble which is initially at rest, the upward motion of the bubble in a gravitational field is studied by tracking the liquid–gas interface. The gas in the bubble can be treated as incompressible. The problem is simulated using primitive variables in a control-volume formulation in conjunction with a pressure–velocity coupling based on the SIMPLE algorithm. The modified VOF method used in this study is able to identify and physically treat features such as bubble deformation, cusp formation, breakup and joining. Results in a two-dimensional as well as a three-dimensional coordinate framework are presented. The bubble deformation and its motion are characterized by the Reynolds number, the Bond number, the density ratio, and the viscosity ratio. The effects of these parameters on the bubble rise are demonstrated. Physical mechanisms are discussed for the computational results obtained, especially the formation of a toroidal bubble. The results agree with experiments reported in the literature.

scholarly journals On the hydrodynamic stability of two viscous incompressible fluids in parallel uniform shearing motion

1957 ◽  
Vol 2 (4) ◽  
pp. 343-370 ◽  
Author(s):  
Saul Feldman
Keyword(s):  
Liquid Film ◽  
Hydrodynamic Stability ◽  
Solid Wall ◽  
Density Ratio ◽  
Reynolds Numbers ◽  
Incompressible Fluids ◽  
Present Problem ◽  
Neutral Stability ◽  
Infinite Plane ◽  
Linear Ordinary Differential Equations

A new problem in hydrodynamic stability is investigated. Given two contiguous viscous incompressible fluids, the fluid on one side of the plane interface being bounded by a solid wall and that on the other side being unbounded, the problem is to determine the hydrodynamic stability when the fluids are in steady unidirectional motion, parallel to the interface, with uniform rate of shear in each fluid. The mathematical analysis, based on small disturbance theory, leads to a characteristic value problem in a system of two linear ordinary differential equations. The essential dimensionless parameters that appear in the present problem are the viscosity ratiom, the density ratior, the Froude numberF, and the Weber numberW, as well as the parameters α,R(which is proportional here to the flow rate of the inner fluid) andc, that occur in the study of hydrodynamic stability of a single fluid. The results obtained are presented graphically for most fluid combinations of possible interest. The neutral stability curve in the (α,R)-plane is single-looped, as in the boundary layer case. The calculated critical Reynolds numbers are higher than the values observed in liquid film cooling experiments. (In these experiments, the outer fluid is usually a turbulent gas, in which the thickness of the laminar sublayer is of the same order of magnitude as the liquid film thickness.) General agreement between the theoretical and experimental values exists for all critical quantities except the Reynolds number. Gravity and surface tension are found here to have a destabilizing effect on the flow, in agreement with experimental evidence. Semi-infinite plane Couette flow is a special case of the present problem and the known stability of this flow is recovered. The linear velocity profile of two adjacent fluids with the same viscosity, but different densities, is shown to be unstable for high enough Reynolds numbers. The Reynolds stress distribution for a neutral oscillation in the general case is discussed qualitatively.

Transition scenario of a sphere freely falling in a vertical tube

2012 ◽  
Vol 711 ◽  
pp. 40-60 ◽  
Author(s):  
Thibaut Deloze ◽  
Yannick Hoarau ◽  
Jan Dušek
Keyword(s):  
Density Ratio ◽  
Vertical Tube ◽  
Falling Sphere ◽  
Unsteady Wake ◽  
Planar Trajectory ◽  
Freely Moving ◽  
Wall Interaction ◽  
Density Ratios ◽  
Two Stages ◽  
Transition Scenario

AbstractThe paper presents the results of direct numerical simulations of the fall of a single freely moving sphere in a vertical circular tube. Most results are obtained for the solid–fluid density ratio ${\rho }_{s} / \rho = 2$. The parametric investigation is carried out depending on the Galileo number defined in Jenny, Dušek & Bouchet J. Fluid Mech., vol. 508, 2004, pp. 201–239. A qualitatively new scenario is found, as compared to that of an unconfined sphere. The primary bifurcation making the sphere deviate from a vertical fall along the tube axis at a constant velocity is of Hopf type. It sets in at a Galileo number (between 155 and 160) similar to that for an unconfined sphere. We find evidence for two stages of the primary regime: a planar trajectory at $G= 160$ and a helical one (at $G= 165$ and 170). At these Galileo numbers, the regime is perfectly periodic, with a slow period corresponding to a Strouhal number only slightly above 0.01. The dynamics is identified as a periodic wake–wall interaction. The helical regime is found to give way directly to chaos between $G= 170$ and $G= 180$. This transition is associated with the onset of vortex shedding in the wake of the falling sphere and with a complex interaction between the unsteady wake and the wall marked by intermittent wake extinction. The effect of density ratio is partly investigated at $G= 250$ by considering three density ratios: 2, 3 and 5. A significant change of behaviour is found between the ratios 3 and 5.

Upward versus downward non-Boussinesq turbulent fountains

2019 ◽  
Vol 867 ◽  
pp. 374-391 ◽  
Author(s):  
Samuel Vaux ◽  
Rabah Mehaddi ◽  
Olivier Vauquelin ◽  
Fabien Candelier
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Density Ratio ◽  
Large Eddy Simulations ◽  
Linear Scaling ◽  
Power Exponent ◽  
Maximum Height ◽  
Source Density ◽  
Wide Range ◽  
Large Eddy

Turbulent miscible fountains discharged vertically from a round source into quiescent uniform unbounded environments of density $\unicode[STIX]{x1D70C}_{0}$ are investigated numerically using large-eddy simulations. Both upward and downward fountains are considered. The numerical simulations cover a wide range of the density ratio $\unicode[STIX]{x1D70C}_{i}/\unicode[STIX]{x1D70C}_{0}$, where $\unicode[STIX]{x1D70C}_{i}$ is the source density of the released fluid. These simulations are used to evaluate how the initial maximum height $H_{i}$ and the steady state height $H_{ss}$ of the fountains are affected by large density contrasts, i.e. in the general non-Boussinesq case. For both upward and downward non-Boussinesq fountains, the ratio $\unicode[STIX]{x1D706}=H_{i}/H_{ss}$ remains close to $1.45$, as usually observed for Boussinesq fountains. However the Froude (linear) scaling originally introduced by Turner (J. Fluid Mech., vol. 26 (4), 1966, pp. 779–792) for Boussinesq fountains is no longer valid to determine the steady fountain height. The ratio between $H_{ss}$ and the height predicted by the Turner’s relation turns out to be proportional to $(\unicode[STIX]{x1D70C}_{i}/\unicode[STIX]{x1D70C}_{0})^{n}$. Remarkably, it is found that the power exponent $n$ differs following the direction in which the buoyant fluid is released ($n=1/2$ for downward fountains and $n=3/4$ for upward fountains). This new result demonstrates that for non-Boussinesq turbulent fountains the configurations heavy/light and light/heavy are not equivalent. Finally, scalings are proposed for fountains, regardless of the direction (upwards and downwards) and of the density difference (Boussinesq and non-Boussinesq).

scholarly journals Helical modes in combined Rayleigh–Taylor and Kelvin–Helmholtz instability of a cylindrical interface

2016 ◽  
Vol 8 (4) ◽  
pp. 219-234 ◽  
Author(s):  
M Vadivukkarasan ◽  
Mahesh V Panchagnula
Keyword(s):  
Short Wavelength ◽  
Weber Number ◽  
Scaling Law ◽  
Density Ratio ◽  
Bond Number ◽  
Length Scale ◽  
Neutral Stability ◽  
Mean Flow ◽  
Helmholtz Instability ◽  
Regime Map

The effect of competing Rayleigh–Taylor and Kelvin–Helmholtz mechanisms of instability applied to a cylindrical two-fluid interface is discussed. A three-dimensional temporal linear stability model for the instability growth is developed based on the frozen time approximation. The fluids are assumed to be inviscid and incompressible. From the governing equations and the boundary conditions, a dispersion relation is derived and analyzed for instability. Four different regimes have been shown to be possible, based on the most unstable axial and circumferential wavenumbers. The four modes are the Taylor mode, the sinuous mode, the flute mode and long and short wavelength helical modes. The effect of Bond number, Weber number, and density ratio are investigated in the context of the mode chosen. It is found that Bond number is the primary determinant of the neutral stability while Weber number plays a key role in identifying the instability mode that is manifest. A regime map is presented to delineate the modes realized for a given set of flow parameter values. From this regime map, a short wavelength helical mode is identified which is shown to result only when both the Rayleigh–Taylor and Kelvin–Helmholtz instability mechanisms are active. A scaling law for the magnitude of the wavenumber vector as a function of Bond number and Weber number are also developed. A length scale is defined to characterize the interface distortion. Using this length scale, the set of conditions where the interface exhibits a maximum in surface area creation is identified. With the objective of achieving the smallest characteristic length scale of interface distortion, a criterion to optimally budget mean flow energy is also proposed.

Numerical computation of solitary waves in a two-layer fluid

2011 ◽  
Vol 688 ◽  
pp. 528-550 ◽  
Author(s):  
H. C. Woolfenden ◽  
E. I. Pǎrǎu
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Froude Number ◽  
Solitary Waves ◽  
Integral Formula ◽  
Density Ratio ◽  
Bond Number ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Damped Oscillations

AbstractWe consider steady two-dimensional flow in a two-layer fluid under the effects of gravity and surface tension. The upper fluid is bounded above by a free surface and the lower fluid is bounded below by a rigid bottom. We assume the fluids to be inviscid and the flow to be irrotational in each layer. Solitary wave solutions are found to the fully nonlinear problem using a boundary integral method based on the Cauchy integral formula. The behaviour of the solitary waves on the interface and free surface is determined by the density ratio of the two fluids, the fluid depth ratio, the Froude number and the Bond numbers. The dispersion relation obtained for the linearized equations demonstrates the presence of two modes: a ‘slow’ mode and a ‘fast’ mode. When a sufficiently strong surface tension is present only on the free surface, there is a region, or ‘gap’, between the two modes where no linear periodic waves are found. In-phase and out-of-phase solitary waves are computed in this spectral gap. Damped oscillations appear in the tails of the solitary waves when the value of the free-surface Bond number is either sufficiently small or large. The out-of-phase waves broaden as the Froude number tends towards a critical value. When surface tension is present on both surfaces, out-of-phase solitary waves are computed. Damped oscillations occur in the tails of the waves when the interfacial Bond number is sufficiently small. Oppositely oriented solitary waves are shown to coexist for identical parameter values.

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