6 results
Dynamic simulation of a coated microbubble in an unbounded flow: response to a step change in pressure
- M. Vlachomitrou, N. Pelekasis
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- Journal:
- Journal of Fluid Mechanics / Volume 822 / 10 July 2017
- Published online by Cambridge University Press:
- 07 June 2017, pp. 717-761
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A numerical method is developed to study the dynamic behaviour of an encapsulated bubble when the viscous forces of the surrounding liquid are accounted for. The continuity and Navier–Stokes equations are solved for the liquid, whereas the coating is described as a viscoelastic shell with bending resistance. The Galerkin Finite Element Methodology is employed for the spatial discretization of the flow domain surrounding the bubble, with the standard staggered grid arrangement that uses biquadratic and bilinear Lagrangian basis functions for the velocity and pressure in the liquid, respectively, coupled with a superparametric scheme with $B$-cubic splines as basis functions pertaining to the location of the interface. The spine method and the elliptic mesh generation technique are used for updating the mesh points in the interior of the flow domain as the shape of the interface evolves with time, with the latter being distinctly superior in capturing severely distorted shapes. The stabilizing effect of the liquid viscosity is demonstrated, as it alters the amplitude of the disturbance for which a bubble deforms and/or collapses. For a step change in the far-field pressure the dynamic evolution of the microbubble is captured until a static equilibrium is achieved. Static shapes that are significantly compressed are captured in the post-buckling regime, leading to symmetric or asymmetric shapes, depending on the relative dilatation to bending stiffness ratio. As the external overpressure increases, shapes corresponding to all the solution families that were captured evolve to exhibit contact as the two poles approach each other. Shell viscosity prevents jet formation by relaxing compressive stresses and bending moments around the indentation generated at the poles due to shell buckling. This behaviour is conjectured to be the inception process leading to static shapes with contact regions.
Short- to long-wave resonance and soliton formation in boundary-layer interaction with a liquid film
- M. VLACHOMITROU, N. PELEKASIS
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- Journal:
- Journal of Fluid Mechanics / Volume 660 / 10 October 2010
- Published online by Cambridge University Press:
- 12 July 2010, pp. 162-196
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Dynamic interaction between a boundary layer of air and a liquid film is investigated in this paper. The low air-to-film-viscosity ratio is considered in which case the boundary layer is quasi-steady on the time scale within which interfacial waves develop. The base flow consists of a boundary layer that drags a film of constant shear. Linear analysis, in the context of triple-deck theory, predicts the formation of a wavepacket of capillary waves that advances and spreads with time. The Froude number of de-/anti-icing fluids or water interacting with air falls well within the supercritical regime, i.e. Fr > FrCr. Numerical simulations of such flow systems were performed in the context of triple-deck theory, and they do not exhibit wave saturation or formation of uniform wavetrains. The long-term interaction is mainly dependent on film inertia as this is characterized by parameter
= (μ/μf)2(ρf/ρ), which involves film and air viscosity and density ratios, and the dimensionless film thickness, H0, and shear, λ, provided by the base flow. Weakly nonlinear analysis taking into consideration mean drift, i.e. generation of long waves, due to self-interaction of the linear wave to O(ϵ2) in amplitude of the initial disturbance, reveals resonance between the wavepacket predicted by linear theory and long waves when the group velocity of the former happens to coincide with the phase velocity, H0λ, of long interfacial waves. Numerical simulations with anti-icing fluids and water verify this pattern. In both cases, long waves eventually dominate the dynamics and, as they are modulated with time, they lead to soliton-type structures. Anti-icing fluids eventually exhibit oscillatory spikes whose mean value never exceeds 2H0, roughly. Water films exhibit a single spike that keeps growing, thus generating a large separation bubble.
Nonlinear interaction between a boundary layer and a liquid film
- M. VLACHOMITROU, N. PELEKASIS
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- Journal:
- Journal of Fluid Mechanics / Volume 638 / 10 November 2009
- Published online by Cambridge University Press:
- 07 October 2009, pp. 199-242
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The nonlinear stability of a laminar boundary layer that flows at high Reynolds number (Re) above a plane surface covered by a liquid film is investigated. The basic flow is considered to be nearly parallel and the simulations are based on triple deck theory. The overall interaction problem is solved using the finite element methodology with the two-dimensional B-cubic splines as basis functions for the unknowns in the boundary layer and the film and the one-dimensional B-cubic splines as basis functions for the location of the interface. The case of flow above an oscillating solid obstacle is studied and conditions for the onset of Tollmien–Schlichting (TS) waves are recovered in agreement with the literature. The convective and absolute nature of TS and interfacial waves is captured for gas-film interaction, and the results of linear theory are recovered. The evolution of nonlinear disturbances is also examined and the appearance of solitons, spikes and eddy formation is monitored on the interface, depending on the relative magnitude of Froude and Weber numbers (Fr, We), and the gas to film density and viscosity ratios (ρ/ρw, μ/μw). For viscous films TS waves grow on a much faster time scale than interfacial waves and their effect is essentially decoupled. The influence of interfacial disturbances on short-wave growth in the bulk of the boundary layer bypassing classical TS wave development is captured. For highly viscous films for which inertia effects can be neglected, e.g. aircraft anti-icing fluids, soliton formation is obtained with their height remaining bounded below a certain height. When water films are considered interfacial waves exhibit unlimited local growth that is associated with intense eddy formation and the appearance of finite time singularities in the pressure gradient.
Nonlinear oscillations of liquid shells in zero gravity
- N. A. Pelekasis, J. A. Tsamopoulos, G. D. Manolis
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- Journal:
- Journal of Fluid Mechanics / Volume 230 / September 1991
- Published online by Cambridge University Press:
- 26 April 2006, pp. 541-582
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It has been shown experimentally (Lee et al. 1982) that water drops with injected air bubbles inside them may be forced dynamically to assume the spherosymmetric shape. Linear analysis is unable to predict a centring mechanism, but provides two distinct modes of oscillation. Weakly nonlinear theory (Tsamopoulos & Brown 1987) indicates that centring of the bubble inside the drop occurs when the two interfaces move out of phase. A hybrid boundary element-finite element schemes is used here to study the complete effect of nonlinearity on the dynamics of the motion. The gas inside the liquid shell may be considered either incompressible or compressible by using a polytropic relation. In both cases, the present calculations show that besides the fast oscillation of the shell due to an initial disturbance, a slow oscillatory motion of the centres of the bubble and the drop is induced around the concentric configuration. This occurs in both modes of oscillation and is a direct result of Bernoulli's law. Furthermore, when this slow oscillation is damped by viscous forces, it is anticipated that it will lead to a spherosymmetric shape.
Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling
- E. LAC, D. BARTHÈS-BIESEL, N. A. PELEKASIS, J. TSAMOPOULOS
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- Journal:
- Journal of Fluid Mechanics / Volume 516 / 10 October 2004
- Published online by Cambridge University Press:
- 24 September 2004, pp. 303-334
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The dynamic response of an initially spherical capsule subject to different externally imposed flows is examined. The neo-Hookean and Skalak et al. (Biophys. J., vol. 13 (1973), pp. 245–264) constitutive laws are used for the description of the membrane mechanics, assuming negligible bending resistance. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creeping-flow conditions are assumed to prevail. The capillary number $\varepsilon $ is the basic dimensionless number of the problem, which measures the relative importance of viscous and elastic forces. The boundary-element method is used with bi-cubic B-splines as basis functions in order to discretize the capsule surface by a structured mesh. This guarantees continuity of second derivatives with respect to the position of the Lagrangian particles used for tracking the location of the interface at each time step and improves the accuracy of the method. For simple shear flow and hyperbolic flow, an interval in $\varepsilon $ is identified within which stable equilibrium shapes are obtained. For smaller values of $\varepsilon $, steady shapes are briefly captured, but they soon become unstable owing to the development of compressive tensions in the membrane near the equator that cause the capsule to buckle. The post-buckling state of the capsule is conjectured to exhibit small folds around the equator similar to those reported by Walter et al. Colloid Polymer Sci. Vol. 278 (2001), pp. 123–132 for polysiloxane microcapsules. For large values of $\varepsilon $, beyond the interval of stability, the membrane has two tips along the direction of elongation where the deformation is most severe, and no equilibrium shapes could be identified. For both regions outside the interval of stability, the membrane model is not appropriate and bending resistance is essential to obtain realistic capsule shapes. This pattern persists for the two constitutive laws that were used, with the Skalak et al. law producing a wider stability interval than the neo-Hookean law owing to its strain hardening nature.
Boundary layer flow of air past solid surfaces in the presence of rainfall
- DIMITRIS N. SMYRNAIOS, NIKOLAOS A. PELEKASIS, JOHN A. TSAMOPOULOS
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- Journal:
- Journal of Fluid Mechanics / Volume 425 / 25 December 2000
- Published online by Cambridge University Press:
- 01 December 2000, pp. 79-110
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The steady two-dimensional laminar flow of an air stream, flowing past a solid surface at high Reynolds number, is examined in the presence of rainfall. As raindrops sediment on the surface they coalesce and form a continuous water film that flows due to shear, pressure drop and gravity, in general. In the limit as the boundary layer and film thickness remain smaller than the radius of curvature of the surface a simplified lubrication-type formulation describes the flow field in the film, whereas the usual boundary layer formulation is applied in the gas phase. In the case of a flat plate and close to the leading edge, x → 0, a piecewise-self-similar solution is obtained, according to which creeping flow conditions prevail in the film and its thickness grows like x3/4, whereas the Blasius solution is recovered in the air stream. Numerical solution of the governing equations in the two phases and for the entire range of distances from the leading edge, x = O(1), shows that the film thickness increases as the rainfall rate, r˙, increases or as the free-stream velocity, U∞, decreases and that the region of validity of the asymptotic result covers a wide range of the relevant problem parameters. In the case of flow past a NACA-0008 airfoil at zero angle of attack a Goldstein singularity may appear far downstream on the airfoil surface due to adverse pressure gradients, indicating flow reversal and eddy formation inside the liquid film, and, possibly, flow separation. However, when the effect of gravity becomes evident in the film flow, as the Froude number decreases, and provided gravity acts in such a way as to negate the effect of the adverse pressure gradient, the location of the singularity is displaced towards the trailing edge of the airfoil and the flow pattern resembles that for flow past a flat plate. The opposite happens when gravity is aligned with the adverse pressure gradient. In addition it was found that there exists a critical water film thickness beyond which the film has a lubricating effect delaying the appearance of the singularity. Below this threshold the presence of the liquid film actually enhances the formation of the singularity.