Boundary integral solution of potential problems arising in the modelling of electrified oil films

Abstract

We consider a class of potential problems on a periodic half-space for the modelling of electrified oil films, which are used in the development of novel switchable liquid optical devices (diffraction gratings). A boundary integral formulation which reduces the problem to the study of the oil-air interface alone is derived and solved in a highly efficient manner using the Nyström method. The oil films encountered experimentally are typically very thin and thus an interface-only integral representation is important for avoiding the near-singularity problems associated with boundary integral methods for long slender domains. The super-algebraic convergence of the proposed methods is discussed and demonstrated via appropriate numerical experiments.

Numerical-asymptotic models for the manipulation of viscous films via dielectrophoresis

“…In the examples considered later, a closed form expression will be available for φ H ; however, in general it may be necessary to approximate φ H by, for example, a truncated Fourier series (if solving the half-plane problem via separation of variables) or quadrature (if computing φ H directly from the boundary integral formula (3.3)). It is shown in Chappell (2015) that the integral equation (3.17) has a unique bounded solution for any 1 , 2 > 0. In order to combine the above-described fluid and electric potential models, we note that the fluid equations depend on the normal and tangential derivatives of φ on Γ I (see (3.16)), rather than on φ itself.…”

“…In Chappell (2015) it is shown that (3.18) has a unique bounded solution in the Sobolev space H −1/2 (Γ I ) -see for example Kress (1989, § 8.2) for an introduction to Sobolev spaces. The solution procedure is thus to first find the potential φ correponding to the initially flat interface y = h 0 by solving (3.17), and to compute the required normal derivative via (3.18).…”

“…In this section we recast the problem (2.2)–(2.14) described in the previous section as a set of boundary integral equations on a single periodic section of the interface . We highlight that the boundary integral model for the electrostatic problem is taken from Chappell (2015) (note that therein, only electrostatics are considered) and that for the Stokes flow is adapted from those summarised in Pozrikidis (1992); in the present work these two models are coupled together via a (temporally discretised) kinematic boundary condition.…”

“…The tangential derivative may be computed from φ relatively simply using interpolation of Γ I by trigonometric polynomials as discussed later. The normal derivative will be computed from the result for φ by making use of the Dirichlet-to-Neumann operator (Chappell 2015), which results from solving the following first-kind integral equation for ∂φ α /∂ν 0 , α = 1, 2:…”

“…, then V is infinitely differentiable with respect to both x and x 0 (Chappell 2015). We then decompose the integral on the left-hand side of (3.18) as…”