Lagrange equations for a system of bubbles of varying radii in a liquid of small viscosity

Voinov, O. V.; Golovin, A. M.

doi:10.1007/bf01019283

Cited by 16 publications

(13 citation statements)

References 4 publications

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“…The Lagrange equation of motion of a bubble immersed in a liquid of constant density and viscosity in an irrotational flow can be written as follows:

\frac{d}{d t} true(\frac{\partial T}{\partial \dot{q}} true) - \frac{\partial T}{\partial q} + \frac{\partial V}{\partial q} + \frac{1}{2} \frac{\partial D}{\partial \dot{q}} = 0

where T is the total kinetic energy of the fluid and air contained in the bubble, V is the total potential energy, and D is the total rate of viscous energy dissipation in the system. For the whole volume of the liquid column (ϑ) presented in Figure (volume of the computational domain), the total kinetic energy is given by the following formula:

T = \frac{1}{2} \int_{ϑ} 2 π r ρ u^{2} d x d r

The total potential energy can be calculated as follows:

V = V_{g} + V_{s}

where Vg is the gravitational potential energy:

V_{g} = \int_{ϑ} 2 π r \cdot ρ g d x d r

and Vs is a surface energy component given by:

V_{s} = A σ

where A is the bubble surface area.…”

Section: Methodsmentioning

confidence: 99%

Energy balance in viscous liquid containing a bubble: Rise due to buoyancy

Zawała

2016

Can J Chem Eng

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The total energy balance of a system consisting of a bubble (radius 0.74 mm) freely rising in viscous liquid under the action of buoyancy was considered. Values of bubble acceleration, local and terminal velocities, and shape deformations calculated by numerically solving the Navier-Stokes equation are compared with experimentally determined values. Additionally, the energies of the system associated with bubble motion (kinetic, potential, and rate of viscous energy dissipation), obtained from simulations, are given. On the basis of energy balances obtained for the system, two independent relations describing the motion of a bubble (Lagrange and Joseph equations) were used to show that, for steady-state conditions, the constant (terminal) velocity of the bubble is a result of balance between buoyancy force and the drag originating from energy dissipation due to fluid viscosity. According to the calculations performed, the discrepancy between the expected and actual results of the Lagrange and Joseph equations for the system considered was $1 %. The estimated added mass coefficient was in close agreement with expected theoretical value for deformed spheres, and increased with decreasing tube radius.

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“…The Lagrange equation of motion of a bubble immersed in a liquid of constant density and viscosity in an irrotational flow can be written as follows:

\frac{d}{d t} true(\frac{\partial T}{\partial \dot{q}} true) - \frac{\partial T}{\partial q} + \frac{\partial V}{\partial q} + \frac{1}{2} \frac{\partial D}{\partial \dot{q}} = 0

T = \frac{1}{2} \int_{ϑ} 2 π r ρ u^{2} d x d r

The total potential energy can be calculated as follows:

V = V_{g} + V_{s}

where Vg is the gravitational potential energy:

V_{g} = \int_{ϑ} 2 π r \cdot ρ g d x d r

and Vs is a surface energy component given by:

V_{s} = A σ

where A is the bubble surface area.…”

Section: Methodsmentioning

confidence: 99%

Energy balance in viscous liquid containing a bubble: Rise due to buoyancy

Zawała

2016

Can J Chem Eng

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“…It is well known that the problem of motion of a fluid with bubbles is a Lagrangian problem in the case where the flow of the fluid is completely determined by bubble motion [9]. Exactly this situation is considered in the present paper.…”

Section: A~ = O X E ~ ~3\ U Sj; O-~nmentioning

confidence: 96%

“…A formula for the kinetic energy that takes account of terms of order not higher than f12 was given in [9,10]. In the calculation of the total kinetic energy of the system "fluid-gas bubbles," we do not take account of the kinetic energy of the gas, because the mass of the gas is small compared to the mass of the fluid.…”

Section: A~ = O X E ~ ~3\ U Sj; O-~nmentioning

confidence: 99%

Kinetic model of bubbly flow

Teshukov¹

2000

J Appl Mech Tech Phys

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A kinetic approach based on the approximate calculation of the fluid flow potential and formulation of Hamilton's equations for generalized coordinates and momenta of bubbles is employed to describe processes of collective interaction of gas bubbles moving in an inviscid incompressible fluid. Kinetic equations governing the evolution of the distribution function of bubbles are derived. These equations are similar to Vlasov equations.Kinetic approaches for the description of fluid flows with gas bubbles have been developed in a number of recent papers [1][2][3][4]. Some of the systems of equations obtained are similar in structure to Vlasov equations which are used to describe plasma flows. In the derivation of these equations, Hamilton's ordinary differential equations that describe the motion of individual particles are employed. If gas bubbles moving in an inviscid incompressible fluid are treated as particles, then for the derivation of the above-mentioned ordinary differential equations, one needs to "know the fluid flow potential in the region between the particles. For the simplified situation where the bubbles are considered incompressible, an approximate Hamiltonian describing the motion of the bubbles for a rarefied bubbly medium was obtained by Russo and Smereka [4] and used to derive a system of kinetic equations governing the evolution of the one-particle distribution function. The assumption on the incompressibility of bubbles can be used to describe real flows with bubbles of sufficiently small size where the surface tension, which maintains the shape of the bubbles, is considerably greater than the variations of the hydrodynamic pressure. This model can be used for description of concentration waves for small pressure differences.The motion of a system of compressible bubbles in a fluid is often modeled using averaged equations, supplemented with the Rayleigh-Lamb equation for a single bubble. Various models of this type differing in additional terms of equations describing real effects are discussed in [5,6]. Certain hydrodynamic effects related to motion of bubbles in a fluid were considered in a monograph by Lavrent'ev and Shabat [7]. A kinetic approach for modeling bubbly flows in which the evolution of the bubble distribution function is governed by equations similar to Boltzmann equations or Vlasov equations, allows one to describe the motion more thoroughly and to derive average equations using a regular procedure. In particular, this approach can, in principle, yield certain basic relations that are postulated in a hydrodynamic description.In the present paper, we derive a system of kinetic equations for bubbly flow that describes the motion of compressible gas bubbles in an inviscid incompressible fluid. We first formulate the system of Hamilton's equations for generalized coordinates of spherical bubbles (spatial coordinates of the centers and radii) and the corresponding momenta. This system is easily written if the potential of the irrotational fluid flow in the region between the bubb...

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“…Voinov and Golovin [20] showed that if an infinite Newtonian viscous fluid contains N spherical bubbles whose radii are a j , j = 1, . .…”

Section: Lagrangian Theorymentioning

confidence: 99%

“…Chincholle [4] did, but not to the order needed in the present work. Voinov & Golovin [20] pioneered the use of Rayleigh's dissipation function with Lagrange's equations in irrotational bubble theory, and extended the proof to the case of bubbles of varying radius.…”

Section: Introductionmentioning

confidence: 99%

Growing bubbles rising in line

Harper

2001

Journal of Applied Mathematics and Decision Sciences

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Over many years the author and others have given theories for bubbles rising in line in a liquid. Theory has usually suggested that the bubbles will tend towards a stable distance apart, but experiments have often showed them pairing off and sometimes coalescing. However, existing theory seems not to deal adequately with the case of bubbles growing as they rise, which they do if the liquid is boiling, or is a supersaturated solution of a gas, or simply because the pressure decreases with height. That omission is now addressed, for spherical bubbles rising at high Reynolds numbers. As the flow is then nearly irrotational, Lagrange's equations can be used with Rayleigh's dissipation function. The theory also works for bubbles shrinking as they rise because they dissolve.

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Lagrange equations for a system of bubbles of varying radii in a liquid of small viscosity

Cited by 16 publications

References 4 publications

Energy balance in viscous liquid containing a bubble: Rise due to buoyancy

Energy balance in viscous liquid containing a bubble: Rise due to buoyancy

Kinetic model of bubbly flow

Growing bubbles rising in line

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