Analysis of the boundary conditions for a Hele–Shaw bubble

Abstract

Effective boundary conditions are derived to be used with the classical Hele–Shaw equations in calculating the shape and motion of a Hele–Shaw bubble. The main assumptions of this analysis are that the displaced fluid wets the plates, and that the capillary number Ca and the ratio of gap width to characteristic bubble length ε are both small. In a small region at the sides of the bubble, it is found that the thin-film thickness scales with ε2/5 Ca4/5, rather than the Ca2/3 scaling that is valid over most of the thin film above and below the bubble.

Quasi-static liquid–air drainage in narrow channels with variations in the gap

“…There is moreover an additional term which is the above mentioned O( 2 ) correction that comes from the interface curvature in the (x, y) plane, expressed here in dimensional form. Further generalisation has been discussed by other authors such as [9][10][11] who considered a confined bubble in a Hele-Shaw type situation. When the (x, y) plane interface slope is not small, it has been shown that the additional term is indeed the in-plane interface curvature multiplied by π/4 [9][10][11].…”

“…Further generalisation has been discussed by other authors such as [9][10][11] who considered a confined bubble in a Hele-Shaw type situation. When the (x, y) plane interface slope is not small, it has been shown that the additional term is indeed the in-plane interface curvature multiplied by π/4 [9][10][11]. Moreover, in the case where the interface displacements are not locally perpendicular to the interface one should consider the normal velocity U n of the air-liquid interface so that a normal capillary number Ca n = µU n /γ must be used instead of Ca in (1) [9][10][11].…”

“…When the (x, y) plane interface slope is not small, it has been shown that the additional term is indeed the in-plane interface curvature multiplied by π/4 [9][10][11]. Moreover, in the case where the interface displacements are not locally perpendicular to the interface one should consider the normal velocity U n of the air-liquid interface so that a normal capillary number Ca n = µU n /γ must be used instead of Ca in (1) [9][10][11]. Reinelt also found an additional perturbative term to the above pressure drop relation associating the viscous capillary dissipation and the in-plane curvature [9,10].…”

“…In the following we shall concentrate on the limit of very small capillary number Ca 1, so that we shall not consider these higher-order corrections. In the frame of this approximation (x, y) variations of the pressure field are not considered for they are of O(Ca) (see, for example, [11]). Hence the pressure in the liquid and in the gas are spatially uniform when only O(Ca 2/3 ) corrections are included.…”

Mutual Adaptation of a Faraday Instability Pattern with its Flexible Boundaries in Floating Fluid Drops

“…The small thickness assumption allows a lubrication approximation transforming the 3D boundary value problem in a 2D one while the small wavelength approximation allows an average on larger length-scales. An asymptotic analysis adapted from [12] shows that the function yðxÞ describing the drop shape satisfies a two-dimensional Laplace law modified by the radiation pressure along the normal. It is a Riccati equation:…”