Three-dimensional stability of free convection vortices in the presence of a magnetic field

Three-dimensional (3D) stability of 2D vortical flow of a liquid metal in a cavity of square cross section is examined. Vortices are produced as a result of free convection and internal heating in the cavity in the presence of a magnetic field. Low-magnetic-Reynolds-number equations are used for the base flow and stability formulation. Finite element methodology is used to discretize the problem. Efficient calculation of the dominant eigenvalues is afforded by the Arnoldi method, while neutral stability diagrams are constructed using continuation techniques. The number of vortices exhibited by the base flow switches from one to two as the internal heating crosses a threshold value. The dominant instability mechanism is the Gortler instability in the case of a single vortex and elliptical instability in the case of two vortices. In elliptic instability, axial vorticity is symmetric, it is characterized by two lobed structures aligned with one of the two principal directions of strain and the dominant eigenmode assumes the form of a traveling wave. The magnetic field opposes buoyancy, alters the direction of maximal strain by accentuating wall shear layers in comparison with the vortex pair in the core and leads to smaller frequencies at criticality

Linear stability analysis and dynamic simulations of free convection in a differentially heated cavity in the presence of a horizontal magnetic field and a uniform heat source

The steady state and stability of two-dimensional free convection flow in a square cavity is examined, in the presence of a uniform internal heat source and a uniform magnetic field that is perpendicular to gravity and parallel to an imposed temperature gradient. The finite element method is used for calculating the steady and dynamic state of the system in the parameter space defined by the dimensionless numbers, Gr, Ha, Pr, and S. The trapezoidal rule is used for time integration. Linear stability analysis is performed by solving a generalized eigenvalue problem. The Arnoldi method is used for the calculation of eigenvalues with significant savings in storage and CPU time requirements. The base solution normally exhibits two recirculation regions when the heat production term is large enough. Stability analysis predicts a Hopf bifurcation to a periodic branch. A neutral stability diagram is constructed for a range of values of Ha, Gr, and S for liquid lithium, Pr=0.0321. Internal heat generation, i.e,. increasing S, enhances instability by decreasing the critical value of Grashof, Gr(cr), determining the onset of the Hopf branch, whereas intensifying the magnetic field, i.e., increasing Ha, stabilizes the flow by increasing Gr(cr). Dynamic simulations confirm the above structure, identify the oscillatory solution branch as a supercritical Hopf bifurcation for the entire parameter range that was examined, and recover the time constants predicted by stability analysis. As Gr increases or as Ha decreases symmetric arrangement of the two rolls is eliminated and the steady flow configuration loses stability when Gr>Gr(cr). Subsequently, time periodicity sets in leading to more or less efficient heat removal in terms of lowering or increasing the average cavity temperature, in comparison with the steady flow configuration for the same Gr, depending on whether Ha lies below or above a critical value, respectively, for fixed S and Pr. (C) 2006 American Institute of Physics

Numerical study of the interaction between a pulsating coated microbubble and a rigid wall. II. Trapped pulsation

The dynamic response of an encapsulated bubble subject to an acoustic disturbance in a wall restricted flow is investigated, when the viscous forces of the surrounding liquid are accounted for. The Galerkin finite element methodology is employed and the elliptic mesh generation technique is used for updating the mesh. As the bubble accelerates towards the wall, the dominant force balance is between Bjerknes forces and the viscous drag that develops. In this process a prolate shape is acquired by the bubble, due to excessive compression at the equator region. When the bubble reaches the wall lubrication pressure develops in the near wall region that resists further approach. As long as the sound amplitude remains below a threshold value determined by the onset of parametric shape mode excitation saturated, or "trapped,"pulsations are performed around a certain small distance from the wall. The balance between Bjerknes attraction and the lubrication pressure that arises due to shell bending along the flattened shell portion that faces the wall generates an oblate shape. Elongation is now observed along the equatorial plane where a local liquid overpressure is established generating large tensile strain. The time-averaged deflection of the microbubble at the pole that lies close to the wall is determined by the bending and stretching resistances of the shell in the manner described by Reissner's linear law for a static compressive load on an elastic shell, corrected for the effect of surface tension. The oscillatory part of the bubble motion in that same region, the contact region, follows the forcing frequency and consists of a pressure driven and a shear flow in the form of a Stokes layer where a significant amount of instantaneous wall shear is generated. The thickness of the film that occupies the Stokes layer is on the order of a few tenths of nm and is determined by the balance between liquid and shell tangential viscous stresses. The steady streaming effect on the wall shear is absent owing to the negligible phase difference between the volume and center of mass pulsations. © 2021 American Physical Society

Static response of coated microbubbles compressed between rigid plates: Simulations and asymptotic analysis including elastic and adhesive forces

The static response of coated microbubbles is investigated with a novel approach employed for modeling contact between a microbubble and the cantilever of an atomic force microscope. Elastic tensions and moments are described via appropriate constitutive laws. The encapsulated gas is assumed to undergo isothermal variations. Due to the hydrophilic nature of the cantilever, an ultrathin aqueous film is formed, which transfers the force onto the shell. An interaction potential describes the local pressure applied on the shell. The problem is solved in axisymmetric form with the finite element method. The response is governed by the dimensionless bending, k^b=kb/χR02, pressure, P^A=PAR0/χ, and interaction potential, W^=w0/χ. Hard polymeric shells have negligible resistance to gas compression, while for the softer lipid shells gas compressibility is comparable with shell elasticity. As the external force increases, numerical simulations reveal that the force versus deformation (f vs d) curve of polymeric shells exhibits a transition from the linear O(d) (Reissner) regime, marked by flattened shapes around the contact region, to a non-linear O(d1/2) (Pogorelov) regime dominated by shapes exhibiting crater formation due to buckling. When lipid shells are tested, buckling is bypassed as the external force increases and flattened shapes prevail in an initially linear f vs d curve. Transition to a curved upwards regime is observed as the force increases, where gas compression and area dilatation form the dominant balance providing a nonlinear regime with an O(d3) dependence. Asymptotic analysis recovers the above patterns and facilitates estimation of the shell mechanical properties. © 2018 Author(s)

Nonlinear interaction between a boundary layer and a liquid film

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 (rho/rho(w), mu/mu(w)). 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

Static arrangement of a capillary porous system (CPS): Modelling

The static arrangement is studied of a thin CPS wafer which is filled from below with a liquid metal. The CPS is modelled as a thin cylindrical disk that is resting on a flat wall. It is in contact with a reservoir that provides liquid lithium. Isothermal conditions are considered and a liquid metal layer is assumed to have been established on top of the CPS and reached an axisymmetric static arrangement. A numerical solution is obtained via the finite element methodology that solves the Young-Laplace equation which incorporates surface tension, gravitational, pressure and electrostatic forces. The layer thickness is predicted at static equilibrium as a function of the imposed pressure drop across the wafer, i.e. between the reservoir and the surrounding medium, and the wetting and dielectric properties of the liquid metal. It is seen that at large reservoir overpressure surface tension balances pressure forces and the liquid metal assumes the form of an almost hemispherical drop of small radius. Gravity is not important in this limit. As the pressure drop decreases the drop assumes an oblate shape and a thin film is gradually formed that entirely covers the CPS and extends onto the wetted rigid substrate. In this range, gravity balances pressure drop and surface tension and the film thickness is on the millimeter range, which is relatively large and has negative implications on the stability of the liquid metal layer as the electric field strength increases. Below a certain pressure drop the film in conjectured to become very thin, on the order of μm, and the disjoining pressure is expected to balance the imposed pressure drop across it. Such static arrangements have been reported in the literature and are favored in terms of stability of the CPS against j→×B→ effects. © 2016 EURATO

Dynamic simulation of a coated microbubble in an unbounded flow: Response to a step change in pressure

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 -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. © 2017 Cambridge University Press

Numerical Study of a Liquid Metal Oscillating inside a Pore in the Presence of Lorentz and Capillary Forces

In order to ensure stable power exhaust and to protect the walls of fusion reactors, liquid metals that are fed to the wall surface through a capillary porous system (CPS) are considered as alternative plasma‐facing components (PFCs). However, operational issues like drop ejection and plasma contamination may arise. In this study, the unsteady flow of a liquid metal inside a single pore of the CPS in the presence of Lorentz forces is investigated. A numerical solution is performed via the finite element methodology coupled with elliptic mesh generation. A critical magnetic number is found (Bondm = 4.5) below which the flow after a few oscillations reaches a steady state with mild rotational patterns. Above this threshold, the interface exhibits saturated oscillations. As the Lorentz force is further increased, Bondm > 5.8, a Rayleigh–Taylor instability develops as the interface is accelerated under the influence of the increased magnetic pressure and a finite time singularity is captured. It is conjectured that eventually, drop ejection will take place that will disrupt cohesion of the interface and contaminate the surrounding medium. Finally, the dynamic response of different operating fluids is investigated, e.g., gallium, and the stabilizing effect of increased electrical conductivity and surface tension is demonstrated. © 2020 by the author

Numerical study of the interaction between a pulsating coated microbubble and a rigid wall. I. Translational motion

The dynamic response of an encapsulated bubble to an acoustic disturbance in a wall restricted flow is investigated in the context of axial symmetry, when the viscous forces of the surrounding liquid are accounted for. The Galerkin finite element methodology is employed and the elliptic mesh generation technique is used for updating the mesh. The bubble is accelerated towards the wall as a result of the secondary Bjerknes forces and consequently the translational velocity gradually increases in a nearly quadratic fashion as the bubble approaches the wall. Proximity to the wall affects the resonance frequency that is seen to be reduced as the initial distance between the bubble and the wall decreases, as long as the sound amplitude remains below a threshold value that is determined by the onset of parametric shape mode excitation. While the microbubble remains far from the wall an overpressure develops in the upstream region that causes flattening and bending of the shell. However, shell elasticity coupled with viscous shell stresses prevents jet formation. Thus the bubble remains spherical during the expansion phase of the pulsation and deforms mainly in the compressive phase, during which most of the translation takes place due to the reduced added mass effect. As it approaches the wall the maximum overpressure is moved to the downstream pole region and this generates an excess of viscous shell stresses during compression that balance compressive elastic stresses. As a result the latter are attenuated in the downstream region of the shell, in comparison with the bulk of the shell where they are balanced solely by the cross membrane pressure drop, leading to a gradually more pronounced prolate bubble shape. Viscous drag due to the surrounding liquid develops mainly in the bulk of the shell where it is balanced by viscous shell stresses in the tangential stress balance. Over a period of the pulsation it counteracts the Bjerknes force that accelerates the bubble, via a force balance that is almost instantaneously established due to the relatively large shell viscosity. This is in marked difference with the case of rising gas bubbles that acquire oblate shapes as a result of the balance between buoyancy and pressure drag. In the case of coated microbubbles the drag coefficient is seen to obey a law previously obtained for no-slip interfaces for large radial and relatively small translational Reynolds numbers. © 2021 American Physical Society