The effect of surface contamination on thermocapillary flow in a two-dimensional slot

1984 ◽  
Vol 139 ◽  
pp. 443-459 ◽  
Author(s):  
G. M. Homsy ◽  
E. Meiburg
Keyword(s):  
Perturbation Theory ◽  
Surfactant Concentration ◽  
Outer Core ◽  
Thermocapillary Flow ◽  
Interfacial Stress ◽  
General Effect ◽  
Two Dimensional ◽  
Elasticity Parameter ◽  
Two Parameters ◽  
Effect Of Surface

We consider the effect of insoluble surfactants on the steady thermocapillary flow in a differentially heated slot treated previously by Sen & Davis (1982). The equation of state for interfacial tension is taken to be linear in both temperature and surfactant concentration. We treat the problem in the limit of shallow slots and low thermal Marangoni numbers so that the effect of surfactants is described by only two parameters: a surface Péclet number Pe and an elasticity parameter denoted by E, the ratio of the compositional elasticity to the tension difference due to the imposed temperature difference. Using lubrication theory and matched asymptotic expansions, we reduce the problem to a single nonlinear integral–algebraic equation (for the outer core variables), which we solve both numerically and in various asymptotic limits by perturbation theory. It is shown that the general effect of surfactants is to retard the strength of the motion, but that this retardation is not necessarily uniform in space. Surprisingly, there are only extreme cases in which condensed surfactant layers will form, these being E [Lt ] 1, Pe [Gt ] 1. Sharp gradients in surfactant concentrations will not form in the general case of E = O(1). This behaviour is due to the strong coupling between the flow and the interfacial stress, and is contrasted with certain well-known forced-convection problems.

Comparison of Numerical and Perturbation Solutions of Two-Dimensional Nonlinear Water-Wave Problems

1976 ◽  
Vol 20 (03) ◽  
pp. 160-170
Author(s):  
Nils Salvesen ◽  
C. von Kerczek
Keyword(s):  
Perturbation Theory ◽  
Numerical Solutions ◽  
Order Perturbation ◽  
Order Perturbation Theory ◽  
Wave System ◽  
Two Dimensional ◽  
Third Order ◽  
Flow Past ◽  
Wave Problems ◽  
Good Agreement

Numerical solutions of the nonlinear problem of the steady two-dimensional potential flow past a submerged line vortex are obtained using the finite-difference iterative technique previously presented by the authors. These solutions are compared in detail with third-order perturbation theory solutions. It is found that very good agreement is obtained for cases of positive circulation of the vortex with strength large enough to produce downstream waves whose steepness is within 15 percent of the maximum possible steepness of irrotational free waves. These computed waves are as steep as the steepest waves obtained in a certain experiment involving the flow past a two-dimensional hydrofoil. For negative circulation, there is substantial difference between the numerical results and third-order perturbation theory. The failure of the perturbation theory is discussed. Details of the far-downstream wave system obtained by the numerical method are compared with other numerical solutions and very high-order perturbation theory solutions of the free-wave problem. Very good agreement is obtained in most cases.

Adsorption of surfactants at liquid/vapour and liquid/liquid interfaces

Surfactants ◽  
2019 ◽  
pp. 73-112
Author(s):  
Bob Aveyard
Keyword(s):  
Surfactant Concentration ◽  
Surface Concentration ◽  
Ionic Surfactants ◽  
Thermodynamic Quantities ◽  
Two Dimensional ◽  
Liquid Interfaces ◽  
Surfactant Mixtures ◽  
Surfactant Monolayers ◽  
Solvent Molecules ◽  
The Relationship

The variation of interfacial tension of a solution with surfactant concentration in bulk can be used, in conjunction with the Gibbs adsorption equation, to probe the behaviour of adsorbed surfactant monolayers. An adsorption isotherm equation expresses the relationship between bulk and surface concentrations of surfactant, and is used to determine thermodynamic quantities of surfactant adsorption. The variation of the surface pressure of a surfactant monolayer with the surface concentration is described by a surface equation of state, which reflects something of the nature of a monolayer. The way in which inorganic electrolytes modify the adsorption and monolayer behaviour of ionic surfactants is explained, and adsorption from surfactant mixtures is also introduced. In the Appendix, the discussion is extended to the treatment of adsorbed monolayers as two-dimensional solutions of surfactant with solvent molecules, rather than as two-dimensional gases.

scholarly journals Derivation of Ice Sliding Properties from The Numerical Modelling of Surging Ice Masses

1979 ◽  
Vol 23 (89) ◽  
pp. 420-421 ◽  
Author(s):  
W. F. Budd ◽  
B. J. McInnes ◽  
I. Smith
Keyword(s):  
Shear Stress ◽  
Numerical Modelling ◽  
Sliding Friction ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Flow Properties ◽  
Two Dimensional Model ◽  
Two Parameters ◽  
Basal Shear ◽  
Best Fit

Abstract It is difficult to deduce sliding properties from the numerical modelling of ordinary glaciers because the flow law of ice is still not known well enough to clearly differentiate sliding from internal deformation of the ice. For glaciers undergoing high-speed surges it appears that the majority of the total speed is due to sliding. Furthermore the average basal shear stress of the ice mass is lowered during the surge. This suggests that surging glaciers can be modelled by incorporating a sliding friction law which has the effective friction coefficient decreasing for high velocities. A relation of this type has been found for ice sliding on granite at −0.5°C by Barnes and others (1971) and has also been obtained for rough slabs with ice at the pressure-melting point by Budd and others (1979). A simple two-dimensional model was developed by Budd and McInnes (1974) and Budd (1975), which was found to exhibit the typical periodic surge-like characteristics of real ice masses. Since the sliding-stress relation for the low velocities and stresses was not known, and was not so important for the surges, it was decided to use the condition of gross equilibrium (i.e. that the ice mass as a whole does not accelerate) together with a single-parameter relation for the way in which the friction decreases with stress and velocity to prescribe the basal shear-stress distribution. The low-stress-velocity relation can thus be obtained as a result. This two-dimensional model has now been parameterized to take account of the three-dimensional aspects of real ice masses. A number of ice masses have since been closely matched by the model including three well-known surging ice masses: Lednik Medvezhiy, Variegated Glacier, and Bruarjökull. Since the flow properties of ice are so poorly known—especially for longitudinal stress and strain-rates—the model has been run with two unknown parameters: one a flow-law parameter (η) and the other a sliding parameter (ø). The model is run over a wide range of these two parameters to see if a good match can be made to the real ice masses and if so what the values of the parameters η and ø are for best fit. The matching of the three above ice masses gave very similar values for each of the two parameters η and ø, the value of η being within the range of values expected for the flow properties of temperate ice as determined by laboratory experiments. Using the same values of η and ø it is found that the ordinary glaciers modelled so far do not develop surging but that they could do if the value of ø were increased or if the mass-balance input were sufficiently increased. For Lednik Medvezhiy a detailed analysis of the friction coefficient with velocity was carried out and it was found that the values required for best fit showed a very close agreement to the sliding friction curve of Barnes and others (1971) at −0.5°C. It is concluded that this type of sliding relation can account for the major features of glacier surge phenomena. Finally it is apparent that the numerical modelling technique can be used very effectively to test any large-scale bulk sliding relation by the analysis of real surges of ice masses and in addition can provide further insight into the sliding relation in association with other stresses in the ice mass.

scholarly journals Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950111 ◽  
Author(s):  
Mohammed-Salah Abdelouahab ◽  
René Lozi ◽  
Guanrong Chen
Keyword(s):  
Dynamical System ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Search Algorithm ◽  
Two Dimensional ◽  
Mixed Mode Oscillations ◽  
The Stability ◽  
Two Parameters ◽  
Complex Phenomena ◽  
Autonomous Dynamical System

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an [Formula: see text]-asymptotically [Formula: see text]-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

scholarly journals DIFFICULTIES OF AN INFRARED EXTENSION OF DIFFERENTIAL RENORMALIZATION

1994 ◽  
Vol 09 (07) ◽  
pp. 1067-1096 ◽  
Author(s):  
L. V. AVDEEV ◽  
D. I. KAZAKOV ◽  
I. N. KONDRASHUK
Keyword(s):  
Perturbation Theory ◽  
Fourier Transformation ◽  
Two Dimensions ◽  
Test Scheme ◽  
Two Dimensional ◽  
Differential Identities ◽  
Feynman Integrals ◽  
Differential Renormalization ◽  
Self Consistent ◽  
Infrared Singularities

We investigate the possibility of generalizing the differential renormalization of D. Z. Freedman, K. Johnson and J. I. Latorre in an invariant fashion to theories with infrared divergencies via an infrared [Formula: see text] operation. Two-dimensional σ models and the four-dimensional ɸ4-theory diagrams with exceptional momenta are used as examples, while dimensional renormalization serves as a test scheme for comparison. We write the basic differential identities of the method simultaneously in co-ordinate and momentum space, introducing two scales which remove ultraviolet and infrared singularities. A consistent set of Fourier-transformation formulae is derived. However, the values for tadpole-type Feynman integrals in higher orders of perturbation theory prove to be ambiguous, depending on the order of evaluation of the subgraphs. In two dimensions, even earlier than this ambiguity manifests itself, renormalization-group calculations based on the infrared extension of differential renormalization lead to incorrect results. We conclude that the procedure of extended differential renormalization does not perform the infrared [Formula: see text] operation in a self-consistent way.

Different void fluctuation in ring-like events and jet-like events in 16O–AgBr interactions at 60 AGeV

2014 ◽  
Vol 23 (11) ◽  
pp. 1450067 ◽  
Author(s):  
M. Mondal ◽  
S. Biswas Ghosh ◽  
A. Mondal ◽  
D. Ghosh ◽  
A. Deb
Keyword(s):  
Phase Transition ◽  
Production Process ◽  
Dimensional Analysis ◽  
High Energy ◽  
Two Dimensional ◽  
Hadron Phase ◽  
High Energy Collisions ◽  
Two Parameters

This paper reports a study of fluctuations and possible signature of quark–hadron phase transition in high energy collisions. The study is based on the ring-like and jet-like events for pions produced in 16 O – AgBr interactions at 60 AGeV. We have performed two-dimensional analysis (η-φ space) of fluctuation of voids for two types of azimuthal substructures of produced pions (ring-like and jet-like) following connecting void approach given by R. C. Hwa and Q. H. Zhang. The values of two parameters "c" and "s" reveal different void pattern fluctuation in ring-like and jet-like events, which hints toward different mechanism in their production process.

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