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

Abstract

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.

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

“…In our case, as it is illustrated in Figure 2, the physical coordinates (r, z, t) are transformed to the computational coordinates (η, ξ, t), with 0 ≤ η ≤ 1, 0 ≤ ξ ≤ 1, and J = r η z ξ − z η r ξ denoting the Jacobian of the mapping ( Figure 2). The mapping between the two domains is implemented by using the elliptic mesh generation technique since its superiority over other techniques when severe interfacial distortions are expected has been demonstrated in many studies [27][28][29][30]. denoting the Jacobian of the mapping ( Figure 2).…”

“…denoting the Jacobian of the mapping ( Figure 2). The mapping between the two domains is implemented by using the elliptic mesh generation technique since its superiority over other techniques when severe interfacial distortions are expected has been demonstrated in many studies [27][28][29][30]. The elliptic mesh generation technique was developed and implemented by Christodoulou and Scriven [27] and Tsiveriotis and Brown [28] and is based on solving a set of partial differential equations for the coordinates of each mesh point.…”

“…The actual form of the elliptic equations in combination with the imposition of the appropriate boundary conditions can produce high quality meshes in terms of the smoothness, orthogonality, and density of the grid. In this study, we employed the quasi-elliptic transformation and the boundary conditions that Dimakopoulos and Tsamopoulos [29] proposed since the implementation of the latter technique has been reported [29,30] to generate meshes that can successfully follow large interfacial distortions:…”

“…The first equation (Equation 26 ε is an empirical parameter that controls the extent of mesh smoothness versus its orthogonality and it ranges between 0 and 1. Its value is defined by trial and error, and in our case was set equal to 0.1, a value proposed in [30]. The ξ-curves, which are nearly parallel to the interface and are prescribed so that they follow its deformation, are generated by Equation (27).…”

“…This type of splitting when solving for the unknown variables demands very small time steps in order to facilitate convergence of the NR procedure and convergence of the temporal refinement to a certain solution. This is in marked difference with the treatment in [30], where time steps on the order of 10 −2 were used. In the latter study, the interface is viscoelastic and its shape, which is determined together with the other unknowns, is used as an essential condition in the elliptic mesh generation methodology.…”