A mathematical model is presented for surfactant-driven thin weakly viscoelastic film flows on a flat, impermeable plane. The Oldroyd-B constitutive relation is used to model the viscoelastic fluid. Lubrication theory and a perturbation expansion in powers of the Weissenberg number (We) are employed, which give rise to non-linear coupled evolution equations governing the transport of insoluble surfactant and thin liquid film thickness. Spreading on a Newtonian film is recovered to leading order and corrections to viscoelasticity are obtained at order We. These equations are solved numerically over a wide range of viscosity ratio (ratio of solvent viscosity to the sum of solvent and polymeric viscosities), pre-existing surfactant level and Peclet number (Pe). The effect of viscoelasticity on surfactant transport and fluid flow is investigated and the mechanisms underlying this effect are explored. Shear stress, streamwise normal stress and the temporal rate of change of extra shear stress generated from gradients in surfactant concentration dominate thin viscoelastic film flows whereas only shear stresses play a role in Newtonian thin film flows. Our results also reveal that, for weak viscoelasticity, the influence of viscosity ratio on the evolution of surfactant concentration and film thickness can be significant and varies considerably, depending on the concentration of pre-existing surfactant and surfactant surface diffusivity.

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.

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.

The results of an experimental study on both pulsating and steady Newtonian fluid flow in an initially stretched rubber tube subjected to external vibration are reported. A circulating loop system was designed to maintain constant hydrostatic pressure throughout the tests so that the influence of external excitation on the fluid flow could be properly distinguished. The effects of fluid flow velocity and initial stretch rates on the dynamicrespon se and damping of the tube conveying fluid were examined, and it was observed that damping ratios increase with increasing flow velocities, and generally decrease with increasing initial stretch rates for the tube conveying fluid. It was also noted that dynamic responses increase with increasing initial stretch rates, and decrease with increasing flow velocities. The effect of external vibration on fluid flow rates is small in a tube with a thickness-to-radius ratio

A mathematical model is developed for lung injury treatments involving the delivery of therapeutic chemicals, such as drugs and gene vectors, into the lung using simultaneous tracheal instillation of exogenous pulmonary surfactant. The influence of exogenous surfactant dose, flow rate, bulk liquid viscosity, pulmonary absorption rate and chemical molecular diffusivity on the chemical delivery to the lung is investigated. Our results reveal that different pulmonary absorption rates lead to significantly different distribution patterns and change the time taken for the total amount of chemical to be absorbed along the airways. The various factors can also influence where the majority of the chemical is placed within the lung and this is relevant to the targeting of drugs to particular lung generations.

A general approach to modelling the vibration of prestressed thin cylindrical shells conveying fluid is presented. The steady flow of fluid is described by the classical potential flow theory, and the motion of the shell is represented by Sanders'theory of thin shells. A strain-displacement relationship is deployed to derive the geometric stiffness matrix due to the initial stresses caused by hydrostatic pressure. Hydrodynamic pressure acting on the shell is developed through dynamic interfacial coupling conditions. The resulting equations governing the motion of the shell and fluid are solved by a finite element method. This model is subsequently used to investigate the small-vibration dynamic behaviour of prestressed thin cylindrical shells conveying fluid. It is validated by comparing the computed natural frequencies, within the linear region, with existing reported experimental results. The influence of initial tension, internal pressure, fluid flow velocity and the various geometric properties is also examined.

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 rheology and a linear equation of state describing the relationship between surface tension and surfactant concentration is adopted. Five coupled equations governing the transport of surfactant, mucus and total liquid layer thicknesses, based on lubrication theory, have been derived. The resulting equations are solved numerically in order to explore the effect of system parameters such as mucus rheology, aqueous-mucus thickness ratio, aqueous-mucus interfacial tension and Marangoni number on tear film evolution in the presence of van derWaals forces, which could induce film rupture.

The non-linear equations of motion of a flexible pipe conveying unsteadily flowing fluid are derived from the continuity and momentum equations of unsteady flow. These partial di!erential equations are fully coupled through equilibrium of contact forces, the normal compatibility of velocity at the fluid} pipe interfaces, and the conservation of mass and momentum of the transient fluid. Poisson coupling between the pipe wall and fluid is also incorporated in the model. A combination of the "nite di!erence method and the method of characteristics is employed to extract displacements, hydrodynamic pressure and flow velocities from the equations. A numerical example of a pipeline conveying fluid with a pulsating flow is given and discussed.

This paper presents a method for the dynamic analysis of initially tensioned orthotropic thin-walled cylindrical tubes conveying steady #uid #ow, based on Sanders' non-linear theory of thin shells and the classical potential #ow theory. The method is relatively straightforward, using a hydrodynamic pressure formulation derived from the velocity potential, a dynamic coupling condition at the #uid}structure interface and two-noded frustum elements to assess the dynamic behaviour of these tube/#uid systems accurately. A non-linear strain}displacement relationship is also deployed to derive the geometric sti!ness matrix due to the initial stresses and hydrostatic pressures. The equations of motion for the tube and #uid are solved by a "nite element method, and this is validated by comparing the natural frequencies obtained with other published results. The in#uence of material properties, fluid flow velocities and initial axial tensions on the natural frequencies is then illustrated and discussed.