Stability of inclined flow of a liquid film with soluble surfactants and variable mass density

If a molecule or atom of material has molecular weight A, the number density of such units is given by n=Nρ/A, where N is Avogadro's number and ρ is the mass density of the material. The amount of scattering from each unit can be written by assigning an imaginary cross-sectional area σ to each unit. If the current I0 is incident on a thin slice of material of thickness z and the current I remains unscattered, then the scattering cross-section σ is defined by I=IOnσz. For a specimen that is not thin, the definition must be applied to each imaginary thin slice and the result I/I0 =exp(-nσz) is obtained by integrating over the whole thickness. It is useful to separate the variable mass-thickness w=ρz from the other factors to yield I/I0 =exp(-sw), where s=Nσ/A is the scattering cross-section per unit mass.

Download Full-text

Anti-plane shear waves in periodic elastic composites: band structure and anomalous wave refraction

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2015.0152 ◽

2015 ◽

Vol 471 (2180) ◽

pp. 20150152 ◽

Cited By ~ 13

Author(s):

Sia Nemat-Nasser

Keyword(s):

Band Structure ◽

Phase Velocity ◽

Negative Refraction ◽

Mass Density ◽

Shear Waves ◽

Variable Mass ◽

Three Dimensional ◽

Positive Energy ◽

Plane Shear ◽

Elastic Composites

For anti-plane shear waves in periodic elastic composites, it is shown that negative energy refraction can be accompanied by positive phase-velocity refraction and positive energy refraction can be accompanied by negative phase-velocity refraction , and that this can happen over a broad range of frequencies. Hence, in general, negative refraction does not necessarily require antiparallel group and phase-velocity vectors. Details are given for layered composites and the results are extended to, and illustrated for, two-dimensional periodic composites, revealing a wealth of information about the refractive characteristics of this class of composites. The composite's unit cell may consist of any number of constituents of any variable mass density and elastic modulus, admitting large discontinuities . A powerful variational-based solution method is used that applies to one-, two- and three-dimensional composites, irrespective of their constituents being homogeneous or heterogeneous. The calculations are direct, accurate and efficient, yielding the band structure, group-velocity, energy-flux and phase-velocity vectors as functions of the frequency and wavevector components, over an entire frequency band.

Download Full-text

Eigenvalues of strings and cylinders with variable mass density

Communications in Analysis and Geometry ◽

10.4310/cag.2000.v8.n2.a6 ◽

2000 ◽

Vol 8 (2) ◽

pp. 393-443

Author(s):

Richard Snyder Laugesen

Keyword(s):

Mass Density ◽

Variable Mass

Download Full-text

Effect of variable mass density on the kinematics of scalar gradient

Physics of Fluids ◽

10.1063/1.3609281 ◽

2011 ◽

Vol 23 (7) ◽

pp. 075107 ◽

Cited By ~ 6

Author(s):

M. Gonzalez ◽

P. Paranthoën

Keyword(s):

Mass Density ◽

Variable Mass

Download Full-text

Stability of symmetric inclined flow of a liquid film in the presence of phase transition

Journal of Soviet Mathematics ◽

10.1007/bf01849528 ◽

1993 ◽

Vol 63 (4) ◽

pp. 446-448

Author(s):

V. I. Panchuk ◽

G. A. Chechko

Keyword(s):

Phase Transition ◽

Liquid Film ◽

Inclined Flow

Download Full-text

Effects of Surfactant on the Motion of a Large Bubble in a Capillary Tube

ASME 2011 9th International Conference on Nanochannels, Microchannels, and Minichannels, Volume 2 ◽

10.1115/icnmm2011-58067 ◽

2011 ◽

Author(s):

Gokalp Gursel ◽

Ufuk Olgac ◽

Metin Muradoglu

Keyword(s):

Film Thickness ◽

Liquid Film ◽

Evolution Equations ◽

Stokes Equations ◽

Capillary Tube ◽

Navier Stokes ◽

Large Bubble ◽

Front Tracking Method ◽

Wide Range ◽

Soluble Surfactants

The finite-difference/front-tracking method is used to study the motion and deformation of a large bubble moving through a capillary tube in the presence of both insoluble and soluble surfactants. Emphasis is placed on the effects of surfactant on the liquid film thickness between the bubble and the tube wall. The numerical method is designed to solve the evolution equations of the interfacial and bulk surfactant concentrations coupled with the incompressible Navier-Stokes equations. A non-linear equation of state is used to relate interfacial surface tension to surfactant concentration at the interface. The film thickness is first computed for the clean bubble case and the results are compared with the lubrication theory in the limit of small capillary numbers, i.e., Ca ≪ 1, and found to be in good agreement with the predictions of Bretherton [1]. Thereafter, the method is used to investigate the effects of insoluble and soluble surfactants on the film thickness for a wide range of governing non-dimensional numbers. It is found that both the insoluble and soluble surfactants have a thickening effect on the liquid film, which compares well with both the experimental results of Krechetnikov and Homsy [2] and analytical predictions of Daripa and Pasa [3].

Download Full-text