Surfactant driven flows overlying a hydrophobic epithelium: film rupture in the presence of slip

Both for tear films and along the airways within the lung, one has an extremely thin fluid layer overlying a biological substrate; in both cases surfactants either of natural origin, or artificially introduced, are important in driving fluid flows. There is evidence that slip can occur when hydrophilic liquids, similar to mucus or aqueous tear films, overlie hydrophobic epithelium. Utilizing results from recent experimental findings we examine the possible influence of slip upon tear film rupture, important in so-called dry eye, and upon surfactant-induced flows within the lung, used in surfactant replacement therapy.     

Pattern formation in thin liquid films with charged surfactants

The dynamic evolution of positively charged surfactants at the interface of a thin film resting atop a negatively charged base is investigated. The lubrication approximation is used to develop coupled equations governing the dynamic evolution of such a system in the presence of charge effects coupled with van der Waals forces. The equations are investigated numerically and analytically, and, for certain parameter ranges, pattern formation is observed reminiscent of that accompanying thermocapillary-driven thin films. Spatial nonuniformities in the charge of the underlying substrate are also studied as a possible tool for film rupture wavelength selection.     

Synchronizing Moore and Spiegel

This paper presents a study of bifurcations and synchronization {in the sense of Pecora and Carroll [Phys. Rev. Lett. 64, 821-824 (1990)]} in the Moore-Spiegel oscillator equations. Complicated patterns of period-doubling, saddle-node, and homoclinic bifurcations are found and analyzed. Synchronization is demonstrated by numerical experiment, periodic orbit expansion, and by using coordinate transformations. Synchronization via the resetting of a coordinate after a fixed interval is also successful in some cases. The Moore-Spiegel system is one of a general class of dynamical systems and synchronization is considered in this more general context. (c) 1997 American Institute of Physics.     

Spherically symmetric inhomogeneous dust collapse in higher dimensional space-time and cosmic censorship hypothesis

We consider a collapsing spherically symmetric inhomogeneous dust cloud in higher dimensional space-time. We show that the central singularity of collapse can be a strong curvature or a weak curvature naked singularity depending on the initial density distribution.     

Analysis of tear film rupture: effect of non-Newtonian rheology

We investigate the rupture mechanism of a precorneal thin mucus coating sandwiched between the aqueous tear film and the corneal epithelial surface with a monolayer of surfactant overlying the aqueous layer. The Ostwald constitutive relation is employed to model mucus and a linear equation of state describing the relationship between surface tension and surfactant concentration is adopted. Three nonlinear coupled evolution equations governing the transport of surfactant, mucus, and total liquid layer thicknesses, based on lubrication theory and a perturbation expansion technique, have been derived. The resulting equations are solved numerically in order to explore the influence of the rheological properties of mucus, aqueous-mucus thickness ratio, aqueous-mucus interfacial tension, Marangoni number, and surfactant concentration on both the onset of instability and tear film evolution in the presence of van der Waals interactions, which could rupture the tear film. Our results reveal that the influence of rheological properties, aqueous-mucus thickness ratio, and interfacial tension on the time required for film rupture can be significant and varies considerably, depending on the magnitude of the Hamaker constants governing the strength of the van der Waals forces.     

On the dynamics of liquid lenses

The spreading of a lens of one liquid on the surface of another liquid is examined. Lubrication theory is used to derive a coupled system of equations for the air-liquid and liquid-liquid interfaces. In the case of highly viscous lenses, extensional stresses are promoted and an additional equation for the lens velocity is derived. The potential singularity at the three-phase line is relieved by a microscopic precursor layer of the spreading fluid assumed to be present ahead of the macroscopic lens. This layer is stabilised via the inclusion of disjoining pressure effects in the lens. The results of our full parametric study show that, for weak gravitational forces, the shape of the lens at equilibrium depends solely on the surface tension ratio for sufficiently deep substrate thicknesses. For thin substrates, the underlying liquid film deforms severely near the point of deposition exhibiting flattening and dimpling.     

Scattering by small defects in the neighbourhood of a fluid-solid interface

The scattering of incident plane elastic waves by a varietyof different defects that lie upon a fluid-solid interface isconsidered here using matched asymptotic expansions. The expansionscheme is developed in terms of a parameter , the ratio of typicaldefect length scale to a typical wavelength of the incidentfield, taken to be small. Three different canonical situations occur and these are illustratedvia three specific examples treated here: a rigid strut, anedge crack, and a rigid strip. In each case the leading-ordermatching is performed to identify the leading-order contributionof the defect to the acoustic field in the far field. In particular,each defect is identified with a source of dipole response ininterfacial stress of displacement. It is shown in the limit as s<<s1 that in the inner problemsthe fluid and solid pieces uncouple in a particularly convenientmanner allowing analytical solutions to be deduced. These arethen matched with appropriate outer solutions.     

Guided elastic waves and perfectly matched layers

Elastic waveguides support propagating modes that have two possible features, negative group velocities and long wavelengths that, for some frequencies, degrade the accuracy or otherwise poison existing numerical schemes that utilise perfectly matched layers (PMLs) to mimic infinite domains. We illustrate why negative group velocities and long waves are potentially an issue and describe how these problems are overcome. Detailed numerical simulations confirm the accuracy of the modified scheme and provide both theoretical and pragmatic estimates for the parameters within the PML model, in particular for the damping function. We also contrast and compare different implementations of the PML model using spectral and finite difference methods.     

Embedding formulae for diffraction by non-parallel slits

Embedding formulae for diffraction theory encode the diffractioncoefficients for some given wave incidence on a scatterer interms of the directivity from a single or reduced number ofscattering problems. If one deduces the relation between thesedirectivities, then the resulting formulae enable rapid computationsand allow one to concentrate computational resources accordingly.Unfortunately, the range of applicability of embedding formulaeis currently rather restricted. In this article, we demonstratehow embedding is applied to plane-wave scattering by non-parallelstrips or slits. Primarily, we concentrate upon the problemof a line crack, or strip, inclined to a flat infinite surfaceand we derive and implement the embedding formula. Various othergeneralizations are possible given these formulae and we outlinethem.     

Drop manipulation and surgery using electric fields

We study the dynamics of a slender drop sandwiched between two electrodes using lubrication theory. A coupled system of evolution equations for the film thickness and interfacial charge density is derived and simplified for the case of a highly conducting fluid. The contact line singularity is relieved by postulating the existence of a wetting precursor film, which is stabilised by intermolecular forces. We examine the motion of the drop as a function of system parameters: the electrode separation, beta, an electric capillary number, C, and a spatio-temporally varying bottom electrode potential. The possibility of drop manipulation and surgery, which include drop spreading, translation, splitting and recombination, is demonstrated using appropriate tuning of the properties of the bottom potential; these results could have potential implications for drop manipulation schemes in various microfluidic applications. For relatively small beta and/or large C values, the drop assumes cone-like structures as it approaches the top electrode; the latter stages of this approach are found to be self-similar and a power-law exponent has been extracted for this case.