Lagrange equations for a system of bubbles est a liquid of low viscosity

Golovin, A. M.

doi:10.1007/bf00914452

J Appl Mech Tech Phys

1972

DOI: 10.1007/bf00914452

|View full text |Cite

Lagrange equations for a system of bubbles est a liquid of low viscosity

A. M. Golovin¹

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

1972

2009

Publication Types

Select...

Article3

Relationship

Self Cite0

Independent3

Authors

Journals

Cited by 3 publications

(1 citation statement)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To describe the perturbation dynamics, we use an energetic approach similar to that used in [1] to describe the dynamics of a system of bubbles in a low-viscosity liquid. The generalized forces in the Lagrange equations can be expressed in terms of the rate of energy dissipation E in the potential velocity field:…”

mentioning

confidence: 99%

Effect of viscosity on dynamics of bubble perturbations in a liquid

Voinov

2009

J Appl Mech Tech Phy

View full text Add to dashboard Cite

The dynamics of a bubble in a low-viscosity liquid is considered. The equation of dynamics of small perturbations with allowance for viscosity effects is determined.Let us consider a bubble with a variable radius in a low-viscosity liquid. Let the shape of the bubble surface S be close to spherical. The goal of the present paper is to derive the equation of dynamics of surface perturbations.Let us introduce a spherical coordinate system (r, θ, ϕ) with the origin in the sphere center. The expression for the bubble-surface radius r 0 includes a small perturbation proportional to the spherical harmonic Y n :Here, R is the radius of a sphere with an equivalent volume and a is the perturbation amplitude; the integral of Y 2 n over a unit sphere is assumed to be equal to unity. In what follows, the argument t at the functions R, r 1 , and a is omitted. It follows from the expression for the bubble volume V = (4/3)πR 3 that the value of the variable isThe unperturbed velocity field is potential. Let the perturbation of the velocity field around the bubble be also close to potential. This is possible if the product of the kinematic viscosity ν and the characteristic time τ is small: ντ l 2 (l = πR/n is the characteristic distance). The velocity potential v = ∇Φ is found by solving the Neumann problemin an approximate form:Here, the coefficient at 1/r is valid with accuracy to smallṘa 3 /R 2 andȧa 2 /R. To describe the perturbation dynamics, we use an energetic approach similar to that used in [1] to describe the dynamics of a system of bubbles in a low-viscosity liquid. The generalized forces in the Lagrange equations can be expressed in terms of the rate of energy dissipation E in the potential velocity field:The generalized coordinates are q 1 = a and q 2 = R. The Lagrange function is L = T f − σS (T f is the kinetic energy of the liquid and σ is the surface tension). The kinetic energy and the dissipation rate are determined by the formulasTyumen'

show abstract

mentioning

confidence: 99%

Effect of viscosity on dynamics of bubble perturbations in a liquid

Voinov

2009

J Appl Mech Tech Phy

View full text Add to dashboard Cite

show abstract

Motion of a pair of bubbles in a liquid of low viscosity

Golovin¹,

Петров²

1972

J Appl Mech Tech Phys

View full text Add to dashboard Cite

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

Voinov¹,

Golovin²

1973

Fluid Dyn

View full text Add to dashboard Cite

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Lagrange equations for a system of bubbles est a liquid of low viscosity

Cited by 3 publications

References 1 publication

Effect of viscosity on dynamics of bubble perturbations in a liquid

Effect of viscosity on dynamics of bubble perturbations in a liquid

Motion of a pair of bubbles in a liquid of low viscosity

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

Contact Info

Product

Resources

About