Combined Effect of Viscosity, Surface Tension and Compressibility on Rayleigh-Taylor Bubble Growth Between Two Fluids

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
Sourav Roy ◽  
L. K. Mandal ◽  
Manoranjan Khan ◽  
M. R. Gupta
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Bubble Growth ◽  
Critical Value ◽  
Combined Effect ◽  
Bubble Velocity ◽  
Wave Number ◽  
Damping Factor ◽  
Perturbation Wave ◽  
Damped Oscillation

The combined effect of viscosity, surface tension, and the compressibility on the nonlinear growth rate of Rayleigh-Taylor (RT) instability has been investigated. For the incompressible case, it is seen that both viscosity and surface tension have a retarding effect on RT bubble growth for the interface perturbation wave number having a value less than three times of a critical value (kc=(ρh-ρl)g/T, T is the surface tension). For the value of wave number greater than three times of the critical value, the RT induced unstable interface is stabilized through damped nonlinear oscillation. In the absence of surface tension and viscosity, the compressibility has both a stabilizing and destabilizing effect on RTI bubble growth. The presence of surface tension and viscosity reduces the growth rate. Above a certain wave number, the perturbed interface exhibits damped oscillation. The damping factor increases with increasing kinematic viscosity of the heavier fluid and the saturation value of the damped oscillation depends on the surface tension of the perturbed fluid interface and interface perturbation wave number. An approximate expression for asymptotic bubble velocity considering only the lighter fluid as a compressible one is presented here. The numerical results describing the dynamics of the bubble are represented in diagrams.

Numerical Simulation of Cavitation Bubble Growth within a Droplet

2015 ◽  
Vol 32 (2) ◽  
pp. 211-217 ◽  
Author(s):  
M. Lü ◽  
Z. Ning ◽  
K. Yan ◽  
J. Fu ◽  
C.-H. Sun
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Bubble Growth ◽  
Cavitation Bubble ◽  
Control Mechanism ◽  
Inertial Force ◽  
Viscous Force ◽  
Two Phase ◽  
Second Stage ◽  
Third Stage

ABSTRACTCavitation bubbles, which always exist in the diesel jet leaving the nozzle and in diesel droplets breaking up from the jet as a result of supercavitation of the diesel within the injection nozzle, increase the instability of jet and droplets in part due to the two-phase mixture, while the mechanism of this effect is still unclear. Cavitation bubble expansion within the diesel droplet has been simulated numerically based on the volume of fluid (VOF) method, and the control mechanism of bubble growth process is analyzed by Rayleigh-Plesset equation. The process of bubble growth is divided into three parts, including surface tension controlled domain, comprehensive competition controlled domain and inertial force controlled domain. During the first stage, cavitation bubble growth is controlled by the surface tension, and the decrease of the surface tension leads to the increase of the bubble growth rate. During the second stage, the bubble growth rate is controlled by the comprehensive competition of the surface tension, the inertial force and the viscous force. During the third stage, the process of bubble growth is majorly controlled by the inertial force.

Pearling instability of a cylindrical vesicle

2014 ◽  
Vol 743 ◽  
pp. 262-279 ◽  
Author(s):  
G. Boedec ◽  
M. Jaeger ◽  
M. Leonetti
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Asymptotic Behaviour ◽  
Critical Value ◽  
Instability Growth Rate ◽  
Instability Growth

AbstractA cylindrical vesicle under tension can undergo a pearling instability, characterized by the growth of a sinusoidal perturbation which evolves towards a collection of quasi-spherical bulbs connected by thin tethers, like pearls on a necklace. This is reminiscent of the well-known Rayleigh–Plateau instability, where surface tension drives the amplification of sinusoidal perturbations of a cylinder of fluid. We calculate the growth rate of perturbations for a cylindrical vesicle under tension, considering the effect of both inner and outer fluids, with different viscosities. We show that this situation differs strongly from the classical Rayleigh–Plateau case in the sense that, first, the tension must be above a critical value for the instability to develop and, second, even in the strong tension limit, the surface preservation constraint imposed by the presence of the membrane leads to a different asymptotic behaviour. The results differ from previous studies on pearling due to the consideration of variations of tension, which are shown to enhance the pearling instability growth rate, and lower the wavenumber of the fastest growing mode.

Linear Stability Analysis of an Electrified Viscoelastic Liquid Jet

2012 ◽  
Vol 134 (7) ◽  
Author(s):  
Li-jun Yang ◽  
Yu-xin Liu ◽  
Qing-fei Fu
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Euler Number ◽  
Critical Value ◽  
Linear Instability ◽  
Viscoelastic Liquid ◽  
Liquid Jet ◽  
Axisymmetric Mode ◽  
Linear Viscoelastic ◽  
Jet Velocity

A linear instability analysis method has been used to investigate the breakup of an electrified viscoelastic liquid jet. The liquid is assumed to be a dilute polymer solution modeled by the linear viscoelastic constitutive equation. As for its electric properties, the liquid is assumed to be of perfect electrical conductivity. The axisymmetric and nonaxisymmetric disturbance wave growth rate has been worked out by solving the dispersion equation of an electrified viscoelastic liquid jet, which was obtained by combining the linear instability model of an electrified Newtonian liquid jet with the linear viscoelastic model. The maximum growth rate and corresponding dominant wavenumbers have been observed. The electrical Euler number, non-Newtonian rheological parameters and some flow parameters have been tested for their influence on the instability of the electrified viscoelastic liquid jet. The results show that the disturbance growth rate of electrified viscoelastic liquid jets is higher than that of Newtonian ones for axisymmetric mode disturbance and almost the same for the nonaxisymmetric mode. The growth rate of the axisymmetric mode is greater than that of the nonaxisymmetric mode for large wavenumbers, and the trend is opposite in the small wavenumber range. The ratio of gas to liquid density, electrical Euler number, and elasticity number can accelerate the breakup of the electrified viscoelastic liquid jet for both modes. The increase of the time constant ratio, zero shear viscosity, and jet radius can decrease the growth rate of the axisymmetric mode; however, their effects on the nonaxisymmetric mode are different. As for the effect of surface tension and jet velocity, there is a critical value. The variation trend is opposite when the surface tension or jet velocity is larger or smaller than the critical value.

Saturation mechanism and generated viscosity in gravito-turbulent accretion disks

2020 ◽  
Vol 640 ◽  
pp. A53
Author(s):  
L. Löhnert ◽  
S. Krätschmer ◽  
A. G. Peeters
Keyword(s):  
Growth Rate ◽  
Numerical Simulations ◽  
Accretion Disks ◽  
Gravitational Instability ◽  
Turbulent Viscosity ◽  
Radial Distance ◽  
Maximum Growth ◽  
Wave Number ◽  
Radiative Cooling ◽  
Mixing Length

Here, we address the turbulent dynamics of the gravitational instability in accretion disks, retaining both radiative cooling and irradiation. Due to radiative cooling, the disk is unstable for all values of the Toomre parameter, and an accurate estimate of the maximum growth rate is derived analytically. A detailed study of the turbulent spectra shows a rapid decay with an azimuthal wave number stronger than ky−3, whereas the spectrum is more broad in the radial direction and shows a scaling in the range kx−3 to kx−2. The radial component of the radial velocity profile consists of a superposition of shocks of different heights, and is similar to that found in Burgers’ turbulence. Assuming saturation occurs through nonlinear wave steepening leading to shock formation, we developed a mixing-length model in which the typical length scale is related to the average radial distance between shocks. Furthermore, since the numerical simulations show that linear drive is necessary in order to sustain turbulence, we used the growth rate of the most unstable mode to estimate the typical timescale. The mixing-length model that was obtained agrees well with numerical simulations. The model gives an analytic expression for the turbulent viscosity as a function of the Toomre parameter and cooling time. It predicts that relevant values of α = 10−3 can be obtained in disks that have a Toomre parameter as high as Q ≈ 10.

scholarly journals A new upper bound for the largest growth rate of linear Rayleigh–Taylor instability

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Changsheng Dou ◽  
Jialiang Wang ◽  
Weiwei Wang
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Upper Bound ◽  
Viscous Fluids ◽  
Instability Condition ◽  
Interface Surface ◽  
Incompressible Viscous Fluids ◽  
Rayleigh Taylor Instability ◽  
Surface Tensor ◽  
Rt Instability

AbstractWe investigate the effect of (interface) surface tensor on the linear Rayleigh–Taylor (RT) instability in stratified incompressible viscous fluids. The existence of linear RT instability solutions with largest growth rate Λ is proved under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold $\vartheta _{\mathrm{c}}$ ϑ c ) by the modified variational method of PDEs. Moreover, we find a new upper bound for Λ. In particular, we directly observe from the upper bound that Λ decreasingly converges to zero as ϑ goes from zero to the threshold $\vartheta _{\mathrm{c}}$ ϑ c .

scholarly journals The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell

2017 ◽  
Vol 473 (2201) ◽  
pp. 20170050 ◽  
Author(s):  
Christopher C. Green ◽  
Christopher J. Lustri ◽  
Scott W. McCue
Keyword(s):  
Surface Tension ◽  
Numerical Solutions ◽  
Bubble Velocity ◽  
Number Of Solutions ◽  
Branch Number ◽  
Single Solution ◽  
Countably Infinite ◽  
Prime Function ◽  
Hele Shaw Cell ◽  
Effect Of Surface

New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.

Instability waves on the air–sea interface

1993 ◽  
Vol 248 ◽  
pp. 363-381 ◽  
Author(s):  
G. H. Wheless ◽  
G. T. Csanady
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Surface Velocity ◽  
Instability Waves ◽  
Interface Surface ◽  
Fluid Properties ◽  
Compound Matrix Method ◽  
Order Of Magnitude

We used a compound matrix method to integrate the Orr–Sommerfeld equation in an investigation of short instability waves (λ < 6 cm) on the coupled shear flow at the air–sea interface under suddenly imposed wind (a gust model). The method is robust and fast, so that the effects of external variables on growth rate could easily be explored. As expected from past theoretical studies, the growth rate proved sensitive to air and water viscosity, and to the curvature of the air velocity profile very close to the interface. Surface tension had less influence, growth rate increasing somewhat with decreasing surface tension. Maximum growth rate and minimum wave speed nearly coincided for some combinations of fluid properties, but not for others.The most important new finding is that, contrary to some past order of magnitude estimates made on theoretical grounds, the eigenfunctions at these short wavelengths are confined to a distance of the order of the viscous wave boundary-layer thickness from the interface. Correspondingly, the perturbation vorticity is high, the streamwise surface velocity perturbation in typical cases being five times the orbital velocity of free waves on an undisturbed water surface. The instability waves should therefore be thought of as fundamentally different flow structures from free waves: given their high vorticity, they are akin to incipient turbulent eddies. They may also be expected to break at a much lower steepness than free waves.

scholarly journals Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes

1966 ◽  
Vol 25 (4) ◽  
pp. 821-837 ◽  
Author(s):  
E. E. Zukoski
Keyword(s):  
Experimental Study ◽  
Surface Tension ◽  
Inclination Angle ◽  
Bubble Velocity ◽  
Vertical Tubes

An experimental study has been made of the motion of long bubbles in closed tubes. The influence of viscosity and surface tension on the bubble velocity is clarified. A correlation of bubble velocities in vertical tubes is suggested and is shown to be useful for the whole range of parameters investigated. In addition, the effect of tube inclination angle on bubble velocity is presented, and certain features of the flow are described qualitatively.

scholarly journals Kelvin-Helmholtz discontinuity in two superposed viscous conducting fluids in a horizontal magnetic field

2008 ◽  
Vol 12 (3) ◽  
pp. 103-110 ◽  
Author(s):  
Aiyub Khan ◽  
Neha Sharma ◽  
P.K. Bhatia
Keyword(s):  
Magnetic Field ◽  
Surface Tension ◽  
Growth Rate ◽  
Unstable Mode ◽  
Two Dimensional ◽  
Horizontal Magnetic Field ◽  
Streaming Velocity ◽  
The Stability ◽  
Conducting Fluids ◽  
Streaming Motion

The Kelvin-Helmholtz discontinuity in two superposed viscous conducting fluids has been investigated in the taking account of effects of surface tension, when the whole system is immersed in a uniform horizontal magnetic field. The streaming motion is assumed to be two-dimensional. The stability analysis has been carried out for two highly viscous fluid of uniform densities. The dispersion relation has been derived and solved numerically. It is found that the effect of viscosity, porosity and surface tension have stabilizing influence on the growth rate of the unstable mode, while streaming velocity has a destabilizing influence on the system.

scholarly journals Qualitative Analysis of the Effect of Weeds Removal in Paddy Ecosystems in Fallow Season

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Leru Zhou ◽  
Zhigang Liu ◽  
Tiejun Zhou
Keyword(s):  
Growth Rate ◽  
Equilibrium State ◽  
Excretion Rate ◽  
Critical Value ◽  
Inorganic Fertilizer ◽  
Paddy Fields ◽  
Asymptotically Stable ◽  
Intrinsic Growth Rate ◽  
Stable Conditions ◽  
Coexistence Equilibrium

In the paper, we introduce a differential equations model of paddy ecosystems in the fallow season to study the effect of weeds removal from the paddy fields. We found that there is an unstable equilibrium of the extinction of weeds and herbivores in the system. When the intensity of weeds removal meets certain conditions and the intrinsic growth rate of herbivores is higher than their excretion rate, there is a coexistence equilibrium state in the system. By linearizing the system and using the Routh–Hurwitz criterion, we obtained the local asymptotically stable conditions of the coexistence equilibrium state. The critical value formula of the Hopf bifurcation is presented too. The model demonstrates that weeds removal from paddy fields could largely reduce the weeds biomass in the equilibrium state, but it also decreases the herbivore biomass, which probably reduces the content of inorganic fertilizer in the soil. We found a particular intensity of weeds removal that could result in the minimum content of inorganic fertilizer, suggesting weeds removal should be kept away from this intensity.

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