Compound Drops and Bubbles

Sadhal, S. S.; Ayyaswamy, P. S.; Chung, J. N.

doi:10.1007/978-1-4612-4022-8_8

Cited by 28 publications

(35 citation statements)

References 88 publications

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“…A boundary elements method is used to solve the Stokes equations (1) and (2) with the boundary conditions (4)- (6) and a second order Runge-Kutta method is used to calculate the evolution of free surface f given by Eq. (3).…”

Section: Methodsmentioning

confidence: 99%

“…[1][2][3][4][5][6][7][8] Results of these studies are of interest in metallurgical and chemical industries, for example, where liquids of low viscosity, such as liquid metals and aqueous solutions, need to be handled. Bubbles in these liquids can be used to modify the concentrations of different substances and promote chemical reactions between them, to clean liquids from impurities captured by adhesion or diffusion processes, and for many other purposes.…”

Section: Introductionmentioning

confidence: 99%

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Bubble growth by injection of gas into viscous liquids in cylindrical and conical tubes

López-Villa¹,

Medina²,

Higuera³

2011

Physics of Fluids

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The effect of partial confinement on the shape and volume of bubbles generated by injection of a constant flow rate of gas into a very viscous liquid is studied numerically and experimentally. Numerical solutions of the Stokes equations for the liquid and the evolution equation for the surface of a bubble, and experiments with two different liquids, show that cylindrical and conical walls concentric with a gas injection orifice in the horizontal bottom of the liquid may strongly affect the shape and volume of the bubbles, and can be used to control the size of the generated bubbles without changing the flow rate of gas. A well-known scaling law for the volume of the bubbles generated by injection of a high flow rate of gas in a very viscous unconfined liquid is extended to take into account the presence of cylindrical or conical walls around the injection orifice.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bubble growth by injection of gas into viscous liquids in cylindrical and conical tubes

López-Villa¹,

Medina²,

Higuera³

2011

Physics of Fluids

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“…The motion of bubble swarms in the liquid phase occurs in several natural and industrial processes, for example, wastewater treatment, fermentation, chemical industry, and metallurgical processes [1]. In those processes, gas holdup is a crucial parameter because it exerts a major impact on mass and heat transfer between the gas and liquid phases [2].…”

Section: Introductionmentioning

confidence: 99%

Effect of Surfactants on Gas Holdup in Shear-Thinning Fluids

Huang

Fan

2017

International Journal of Chemical Engineering

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In this study, the gas holdup of bubble swarms in shear-thinning fluids was experimentally studied at superficial gas velocities ranging from 0.001 to 0.02 m⋅s −1 . Carboxylmethyl cellulose (CMC) solutions of 0.2 wt%, 0.6 wt%, and 1.0 wt% with sodium dodecyl sulfate (SDS) as the surfactant were used as the power-law (liquid phase), and nitrogen was used as the gas phase. Effects of SDS concentration, rheological behavior, and physical properties of the liquid phase and superficial gas velocity on gas holdup were investigated. Results indicated that gas holdup increases with increasing superficial gas velocity and decreasing CMC concentration. Moreover, the addition of SDS in CMC solutions increased gas holdup, and the degree increased with the surfactant concentration. An empirical correlation was proposed for evaluating gas holdup as a function of liquid surface tension, density, effective viscosity, rheological property, superficial gas velocity, and geometric characteristics of bubble columns using the experimental data obtained for the different superficial gas velocities and CMC solution concentrations with different surfactant solutions. These proposed correlations reasonably fitted the experimental data obtained for gas holdup in this system.

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“…The condition in (1.3) is the so-called Henry condition, cf. [14,20,19,4,3]. In this condition the coefficient β = β(x) is strictly positive and piecewise constant:…”

mentioning

confidence: 99%

Nitsche-XFEM with Streamline Diffusion Stabilization for a Two-Phase Mass Transport Problem

Lehrenfeld¹,

Reusken²

2012

SIAM J. Sci. Comput.

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Abstract. We consider an unsteady convection diffusion equation which models the transport of a dissolved species in two-phase incompressible flow problems. The so-called Henry interface condition leads to a jump condition for the concentration at the interface between the two phases. In [A. Hansbo, P. Hansbo, Comput. Methods Appl. Mech. Engrg. 191 (20002)], for the purely elliptic stationary case, extended finite elements (XFEM) are combined with a Nitsche-type of method, and optimal error bounds are derived. These results were extended to the unsteady case in [A. Reusken, T. Nguyen, J. Fourier Anal. Appl. 15 (2009)]. In the latter paper convection terms are also considered, but assumed to be small. In many two-phase flow applications, however, convection is the dominant transport mechanism. Hence there is a need for a stable numerical method for the case of a convection dominated transport equation. In this paper we address this topic and study the streamline diffusion stabilization for the Nitsche-XFEM method. The method is presented and results of numerical experiments are given that indicate that this kind of stabilization is satisfactory for this problem class. Furthermore, a theoretical error analysis of the stabilized Nitsche-XFEM method is presented that results in optimal a-priori discretization error bounds.AMS subject classification. 65N12, 65N30, be a convex polygonal domain that contains two different immiscible incompressible phases. The (in general time dependent) subdomains containing the two phases are denoted by Ω 1 , Ω 2 , withΩ =Ω 1 ∪Ω 2 . The interface Γ :=Ω 1 ∩Ω 2 is assumed to be sufficiently smooth. A model example is a (rising) droplet in a flow field. The fluid dynamics in such a flow problem is usually modeled by the incompressible Navier-Stokes equations combined with suitable conditions at the interface which describe the effect of surface tension. For this model we refer to the literature, e.g. [3,8,15,21,9]. By w we denote the velocity field resulting from these Navier-Stokes equations. We assume that div w = 0 holds. Furthermore, we assume that the transport of the interface is determined by this velocity field, in the sense that V Γ = w · n holds, where V Γ is the normal velocity of the interface and n denotes the unit normal at Γ pointing from Ω 1 into Ω 2 . In this paper we restrict ourselves to the case of a stationary interface, i.e., we assume w · n = 0. This case is (much) easier to handle than the case of an non-stationary interface Γ = Γ(t). We restrict to this simpler case because even for that the issue of stabilization of the Nitsche-XFEM method for convection-dominated transport problems has not been investigated, yet. The case of a non-stationary interface will be studied in a forthcoming paper. We comment on this further in Remark 6 at the end of the paper. We consider a model which describes the transport of a dissolved species in a two-phase

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Compound Drops and Bubbles

Cited by 28 publications

References 88 publications

Bubble growth by injection of gas into viscous liquids in cylindrical and conical tubes

Bubble growth by injection of gas into viscous liquids in cylindrical and conical tubes

Effect of Surfactants on Gas Holdup in Shear-Thinning Fluids

Nitsche-XFEM with Streamline Diffusion Stabilization for a Two-Phase Mass Transport Problem

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