Function Of Bond Number Research Articles

Correlations and Asymptotic Behaviors of the Shape Parameters of Floating Bubbles Using an Improved Numerical Procedure

An improved numerical procedure is used to present the correlations between the shape parameters and Bond numbers of floating bubbles for a wider range of Bond numbers (5×10−5<Bo<5000) than the previously reported range of Bond numbers (0.003<Bo<241), and their asymptotic relations as Bo → 0 and Bo → ∞. The proposed method is proven to be more precise and robust than the conventional methods in comparison with previous numerical and experimental results. In addition, the profile of floating bubbles and the related parameters are presented for a wide range of bubble sizes. The shape parameters are divided into three distinct Bond number regions, and are fitted with a fifth-order polynomial as a function of Bond number on a log-log scale for each region. The parameters show two asymptotes, which can be expressed in a simple power law. In addition, the dimensionless maximum depth of the floating bubble is obtained as H=0.7291015 when Bo=4.755563. These correlations and asymptotic relations are expected to assist in the development of scale models of dynamic bubble-related phenomena such as bubble bursting.

Enhancement of electric field on bubble dispersion characteristics in leaky-dielectric liquid medium

Enhancement of the electric field on mixture has attached wide interests owing to its advantages in the terrestrial and space applications. In this paper, the dispersion characteristics of bubbles in leaky-dielectric liquid were investigated with a novel designed capillary electrode in the presence of a non-uniform electric field, employing high-speed photography and with a focus on the electrohydrodynamic (EHD) effects. The results show that the bubble dispersion is associated with three typical patterns in the manipulated parameter regions, including the dripping, mixed, and spraying patterns. With the increase of electrical Bond number (BoE), bubble size, characterized by the dimensionless bubble diameter ζ, is decreased significantly, while the evolution process of bubbles becomes faster and more complex. In particular, the size of the generated micro/nanobubbles in the spraying pattern is basically below 20 μm. The rise of the dimensionless flow rate (Q/Q0) increases ζ. It is found that the transition of bubble dispersion patterns depends strongly on the BoE, and the rise in Q/Q0 tends to slow down the transition process. In addition, an empirical correlation of ζ as a function of BoE and Q/Q0 is established for the dripping pattern of bubbles based on the present experiment data, which is believed to be of significance for the modeling of bubble dispersion under electric field. The bubble shapes, characterized by aspect ratios, show a diversity of distribution in the cross-section close to the capillary orifice, while terminally tending to follow the conventional formulation in the literature as the bubbles develop in the vertical direction.

Theoretical study of the complexassion of bivalent ions: Mg2+, Zn2+, Cu2+ and Fe2+ by 4-hydroxy-3-(1-phenylpropyl)-2H-chromen-2-one(HPCO), 4-hydroxy-3-(3-oxo-1-phenylbutyl)-2H-chromen-2-one(HOCO) and 4-hydroy-6-(1-phenylbutan-2-yl)-3-(1-phenylpropyl)-2H-pyran-2-one(HPPO) in water and in ethanol

Trace elements ensure the proper functioning of enzymes. Their theoretical study of complexassion shows that: Iron II is not chelated. $$\hbox {Mg}^{2+}$$ , $$\hbox {Zn}^{2+}$$ and $$\hbox {Cu}^{2+}$$ are chelated by HPCO and HOCO. And HPPO chelates only the $$\hbox {Mg}^{2+}$$ and $$\hbox {Zn}^{2+}$$ . Some of the chelates have become diphenols or are assisting in the modification of the chelating structure. The two solvents influence the interaction of HPCO, of HOCO and of HPPO with the metals: $$\hbox {Fe}^{2+}$$ , $$\hbox {Mg}^{2+}$$ , $$\hbox {Zn}^{2+}$$ and $$\hbox {Cu}^{2+}$$ . The transform certain monovalent bonds into divalent bonds and vice versa, or promote or prevent the metal from integrating the structure of the ligand. And NBO calculations show that the donor atoms yield their charge by half or entirely equal to the metal as a function of bond number: monovalent or divalent bond.

Instability of a Diffusive Boundary Layer beneath a Capillary Transition Zone

Natural convection induced by carbon dioxide (CO2) dissolution from a gas cap into the resident formation brine of a deep saline aquifer in the presence of a capillary transition zone is an important phenomenon that can accelerate the dissolution process, reducing the risk of CO2 leakage to the shallower formations. Majority of past investigations on the instability of the diffusive boundary layer assumed a sharp CO2–brine interface with constant CO2 concentration at the top of the aquifer, i.e., single-phase system. However, this assumption may lead to erroneous estimates of the onset of natural convection. The present study demonstrates the significant effect of the capillary transition zone on the onset of natural convection in a two-phase system in which a buoyant CO2 plume overlaid a water-saturated porous layer. Using the quasi-steady-state approximation (QSSA), we performed a linear stability analysis to assess critical times, critical wavenumbers, and neutral stability curves as a function of Bond number. We show that the capillary transition zone could potentially accelerate the evolution of the natural convection by sixfold. Furthermore, we characterized the instability problem for capillary-dominant, in-transition, and buoyancy-dominant systems. In the capillary-dominant systems, capillary transition zone has a strong role in destabilizing the diffusive boundary layer. In contrast, in the buoyancy-dominant systems, the buoyancy force is the sole cause of the instability, and the effect of the capillary transition zone can be ignored. Our findings provide further insight into the understanding of the natural convection in the two-phase CO2–brine system and the long-term fate of the injected CO2 in deep saline aquifers.

Effect of gravity on the stability of viscoelastic thermocapillary liquid layers

Thermocapillary convection appears in many polymer processing operations. In order to understand the effect of gravity for its stability, three-dimensional linear stability analysis is performed for a viscoelastic fluid. The critical Marangoni number is derived as a function of Bond number, Prandtl number and elasticity. When the Prandtl number is large, the increasing of gravity effect often makes the flow more stable, and the coupling of gravity to elasticity and thermocapillary force excites many different kinds of preferred modes. For small Prandtl number, the flow is always destabilized by gravity effect. The work done by gravity becomes a new energy source for perturbation. However, it can be either positive or negative, which is not directly related to the variation of critical Marangoni number with gravity. The effect of gravity on the instability mechanism and the properties of preferred modes are demonstrated for different elasticity.

On the onset of instabilities in a Bénard-Marangoni problem in an annular domain with temperature gradient

This manuscript addresses the linear stability analysis of a thermoconvective problem in an annular domain. The flow is heated from below, with a linear decreasing horizontal temperature profile from the inner to the outer wall. The top surface of the domain is open to the atmosphere and the two lateral walls are adiabatic. The effects of several parameters in the flow are evaluated. Three different values for the ratio of the momentum diffusivity and thermal diffusivity are considered: relatively low Prandtl number (Pr = 1), intermediate Prandtl number (Pr = 5) and high Prandtl number (ideally Pr ? ? , namely Pr = 50). The thermal boundary condition on the top surface is changed by imposing different values of the Biot number, Bi. The influence of the aspect ratio (?) is assessed for through by studying several aspect ratios, ?. The study has been performed for two values of the Bond number (namely Bo = 5 and 50), estimating the perturbation given by thermocapillarity effects on buoyancy effects. Different kinds of competing solutions appear on localized zones of the ?-Bi plane. The boundaries of these zones are made up of co-dimension two points. Co-dimension two points are found to be function of Bond number, Marangoni number and boundary conditions but to be independent on the Prandtl number.

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

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.

Using Parabolic Flights to Examine Quantitatively the Stability of Liquid Bridges under Varying Total Body Force

Liquid bridges were flown aboard a Boeing 727-200 aircraft in a series of parabolic arcs that produced multiple periods of microgravity. During the microgravity portion of each arc, geff, the effective total body acceleration due to external forces became negligibly small so that cylindrical liquid bridges could be suspended across two coaxial support posts. Near the bottom of each arc, geff slowly increased to a maximum of 1.84g, causing the liquid bridges to deform and in some cases collapse. Although the physics of liquid bridges subject to varying total body force is well-established and has been analyzed extensively both theoretically and experimentally, specific hardware was designed to vary geff in a precise way that overcomes the gravity-related limitations and high g-jitter associated with parabolic flights. Bridge-stability was examined for axial and lateral orientations with respect to geff by measuring the slenderness ratio as a function of Bond number at the instant of bridge collapse. Results exhibit remarkable agreement with theory as well as with the experimental results obtained in a magnetic levitation-based experiment. The parabolic flight method offers technical originality and provides experimental insights for researchers in the microgravity field. Here we present hardware development, experimental considerations, and results, and demonstrate that parabolic flight is a viable alternative to extant techniques for quantitative experiments on fluids.

Film drainage of viscous liquid on top of bare bubble: Influence of the Bond number

We present experimental results of film drainage on top of gas bubbles pushed by gravity towards the free surface of highly viscous Newtonian liquid with a uniform interface tension. The temporal evolution of the thickness of the film between a single bubble and the air/liquid interface is investigated via interference method. Experiments under various physical conditions (range of viscosities and surface tension of the liquid, and bubble sizes) evidence the influence of the deformation of the thin film on the thinning rate and confirm the slow down of film drainage with Bond number as previously reported by numerical work of Pigeonneau and Sellier [Phys. Fluids 23, 092102 (2011)]10.1063/1.3629815. Considering the liquid flow in the cap squeezed by buoyancy force of the bubble, we provide an approximation of thinning rate as a function of Bond number that agrees with experimental and numerical data. Qualitatively, the smaller the area of the thin film compare to the surface of the bubble, the faster the drainage.

Dynamics and stability of two-potential flows in the porous media

The experimental and numerical results of the capillary-force-driven climb of wetting liquid in porous media, which is opposed by the gravity force, are analyzed with respect to the emergence of a multiphase flow front and flow stability of the climbing liquid. Two dynamic characteristics are used: (i) the multiphase flow front thickness as a function of time, and (ii) the capillary number as a function of Bond number, where both numbers are calculated from the harmonic average of pores radii. Throughout the climb, the influence of capillary, gravity, and viscous force variations on the flow behavior is investigated for different porous media. For a specific porous medium, a unique flow front power law function of time is observed for the capillary flow climbs with or without gravity force. Distinct dynamic flow front power law functions are found for different porous media. However, for capillary climb in different porous media, one is able to predict a unique behavior for the wetting height (the interface between wetted and dry regions of porous medium) using the capillary and Bond number. It is found that these two numbers correlate as a unique exponential function, even for porous media whose permeabilities vary for two orders of magnitude. For climbs without the gravity force (capillary spreads), the initial climb dynamics follows this exponential law, but for later flow times and when a significant flow front is developed, one observes a constant value of the capillary number. Using this approach to describe the capillary climb, only the capillary versus Bond number correlation is needed, which is completely measureable from the experiments.

Predicting discharge dynamics of wet cohesive particles from a rectangular hopper using the discrete element method (DEM)

Accurate prediction of the discharge rate from hoppers is important in many industrial processes involving the handling of granular materials. The present work investigates the parameters affecting the discharge rate of a wet cohesive system from a quasi-3-D, rectangular hopper using the discrete element method (DEM). The cohesion between the particles is described by a pendular liquid bridge force model and the strength of the cohesive bond is characterized by a Bond number. The Beverloo correlation is applied to cohesive systems by modifying the Beverloo constant as a function of Bond number. The predictions obtained from this modified correlation fit the simulation data reasonably well. In addition, the effect of hopper angle in cohesive systems is shown to follow a trend similar to cohesionless systems, where the discharge rate is insensitive to changes in hopper angle except below a critical angle (with respect to the vertical) where the discharge rate increases rapidly. This critical angle of flow decreases with increasing cohesion.

Dynamics of drop coalescence at fluid interfaces

Drop coalescence was studied using numerical simulations. Liquid drops were made to coalesce with a body of the same liquid, either a reservoir or a drop of different size, each with negligible impact velocity. We considered either gas or liquid as a surrounding fluid, and experimental results are discussed for the gas–liquid set-up. Under certain conditions, a drop will not fully coalesce with the liquid reservoir, leaving behind a daughter drop. Partial coalescence is observed for systems of low viscosity, characterized by a small Ohnesorge number, where capillary waves remain sufficiently vigourous to distort the drop significantly. For drops coalescing with a flat interface, we determine the critical Ohnesorge number as a function of Bond number, as well as density and viscosity ratios of the fluids. Studying the coalescence of two drops of different sizes reveals that partial coalescence may occur in low-viscosity systems provided the size ratio of the drops exceeds a certain threshold. We also determine the extent to which the process of partial coalescence is self-similar and find that the viscosity of the drop has a large effect on the droplet's vertical velocity after pinch off. Finally, we report on the formation of satellite droplets generated after a first pinch off and on the ejection of a jet of tiny droplets during coalescence of a parent drop significantly deformed by gravity.

Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves

We derive the Camassa–Holm equation (CH) as a shallow water wave equation with surface tension in an asymptotic expansion that extends one order beyond the Korteweg–de Vries equation (KdV). We show that CH is asymptotically equivalent to KdV5 (the fifth-order integrable equation in the KdV hierarchy) by using the non-linear/non-local transformations introduced in Kodama (Phys. Lett. A 107 (1985a) 245; Phys. Lett. A 112 (1985b) 193; Phys. Lett. A 123 (1987) 276). We also classify its travelling wave solutions as a function of Bond number by using phase plane analysis. Finally, we discuss the experimental observability of the CH solutions.

Brownian dynamics simulation of linear polymers under elongational flow: Bead–rod model with hydrodynamic interactions

Brownian dynamics (BD) simulations of a linear freely jointed bead–rod polymer chain with excluded volume (EV) interaction have been performed under elongational flow with and without the use of fluctuating hydrodynamic interactions (HI). The dependence of the chain size, shape and intrinsic elongational viscosity on the elongational rate ε̇ are reported. A sharp coil–stretch transition is observed when ε̇ exceeds a critical value, ε̇c. The inclusion of the HI leads to a shift in the coil–stretch transition to higher flow values. Chain deformation due to elongational flow is observed to first consist of the alignment of the chain with the direction of flow without significant chain extension followed by additional alignment of the bond vectors with the flow direction and chain extension as flow rate is increased further. The distribution function for the chain’s radius of gyration becomes significantly broader within the transition region which implies an increase in fluctuations in the chain size in this region. The structure factors parallel and perpendicular to the flow direction illustrate different elongational rate dependencies. At high rates, the structure factor in the direction of the flow exhibits an oscillating dependence which corresponds to the theoretically predicted shape for a rigid-rod model. The mean squared orientation of each bond within the chain with respect to the flow direction as function of bond number is nearly parabolic in shape with the highest degree of orientation found within the chain’s interior. The dependence of the critical elongational rate, ε̇c, on the chain length, N, is observed to be ε̇c∼N−1.96 when hydrodynamic interactions are not employed and ε̇c∼N−1.55 when they are invoked. These scaling exponents agree well with those obtained in previous BD simulations of bead-FENE (i.e., finitely extensible nonlinear elastic) spring chains as well as with the theoretical predictions of ε̇c∼N−2 and ε̇c∼N−1.5 without and with hydrodynamic interactions based on the Rouse and Zimm models, respectively.

Stability of the Shape of a Viscous Drop under Buoyancy-Driven Translation in a Hele-Shaw Cell

It has been shown (N. R. Gupta, A. Nadim, H. Haj-Hariri, and A. Borhan, J. Colloid Interface Sci.218, 338 1999) that a circular drop translating in a Hele-Shaw cell under the action of gravity is linearly stable for nonzero interfacial tension. In this paper, we use the boundary integral method to examine the nonlinear evolution of the shape of initially noncircular drops translating in a Hele-Shaw cell. For prolate initial deformations, it is found that the drop reverts to a circular shape for all finite Bond numbers considered. Initially oblate drops, on the other hand, are found to become unstable and break up if the initial shape perturbation is of sufficiently large magnitude. The critical conditions for the onset of drop breakup are examined in terms of the magnitude of the initial deformation as a function of Bond number. Two branches of marginal stability are identified and the effects of viscosity ratio and asymmetric initial perturbations on the stability diagram are discussed.

Stability of Magnetically Levitated Liquid Bridges of Arbitrary Volume Subjected to Axial and Lateral Gravity

The paramagnetic fluid composed of manganese chloride tetrahydrate dissolved in water was used to form bridges held between two parallel collinear rods. Magnetic levitation was used to control the effective gravity felt by the bridges. Their stability was studied as a function of Bond number and volume of the supported liquid. Data are compared with theoretical predictions for liquid bridges as well as with experimental results, where available.

Gravity segregation at the pore scale in cores under miscible and low interfacial tension conditions including in-situ tomography

Gravity effects at the pore/core scale have been examined in two sets of experiments. The first used uniform-sized beadpacks and visual observations and the second core material and CT scanning. The uniform beadpacks were flooded with a fluid mixture which could be conditioned to form 2 phases in situ. The interfacial tension and density difference were controlled by varying the temperature. Fluid segregation due to density difference, pore size and interfacial tension and thus a function of Bond Number, was studied. Experiments with the wetting and the non-wetting phase in the majority were included. In the CT scanning experiments, a uniform alumina core had a horizontal miscible displacement performed in it and the displacement was followed by regular CT scanning along the core. Then two static phases (using a fluid system similar to the one used for the beadpacks) were formed in situ within the core, and the positions and saturation of wetting and non-wetting phases observed over a period of time (~ 24 h). The effects of gravity cannot be fully correlated by the current definition of Bond Number; saturation influence has to be added. If gravity effects are neglected during the interpretation of core test data errors can be created when using the results for reservoir engineering purposes: for example in dispersion calculations, relative permeability estimations and scale-up predictions.

Experimental and Theoretical Studies of Liquid Sloshing at Simulated Low Gravity

Analyses and experimental comparisons are given for liquid sloshing in a rigid cylindrical tank under conditions of moderately small axial accelerations; in particular, the theory is valid for Bond numbers larger than 10. The analytical results are put in the form of an equivalent mechanical model, and it is shown that the sloshing mass and the natural frequency of the first mode, for a liquid having a 0 deg contact angle at the tank walls, are smaller than for high-g conditions. The experimental data, obtained by using several small-diameter tanks and three different liquids, are compared to the predictions of the mechanical model; good correlation is found in most cases for the sloshing forces and natural frequency as a function of Bond number.