Surface tension effects in a wedge

Corner solutions of the Laplace–Young equation

“…It is clear from figure (2) that the far field's behaviour on the corner is indeed negligible, and that asymmetry effects from the boundary conditions are also lost (through exponentially small terms) once an interior flat region is formed. The appearance of an interior flat region seems to occur at around 2 -3 capillary lengthscales from the origin, which is in agreement with the linearised solutions of Fowkes and Hood (13). The extent of the loss of asymmetry effects on the corner asymptotics is further demonstrated by noting that three significant figure agreement between the heights on the θ = ±α walls occurs for r < 0.5, R = 6, r < 2.2, R = 8 and r < 5.7 when R = 10.…”

“…It is again clear that the power series solution provides an accurate description of the correct corner behaviour, and these results also show that while logarithmic terms may be needed to cope with every possible corner asymptotic expansion as shown by Miersemann (10), they are not needed for usual practical applications that possess an interior flat region. Finally, for the linearised approximation to (2.1) and (2.2) given by |∇u| 2 << 1, Fowkes and Hood (13) give the following simple globally valid solution when α = π/4 u(r, θ) = cos γ e −r sin(π/4−θ) + e −r sin(π/4+θ) . It is straightforward to show that when α = π/4 and γ near π/2, our series solution (2.4), for the full nonlinear equations, reproduces (4.3).…”

“…These asymptotic results have considerable practical applications to industrial problems, such as the dipping of cuboid capacitors into a metallic paste and the subsequent determination of the wetted region on which any capacitor connection must occur, see King et al (14). An exact analytical solution to the linearised Laplace-Young equation for arbitrary γ and a range of wedge angles 2α, has been found and subsequently used to analyse this capacitor problem in Fowkes and Hood (13).…”

Exact solutions of the Laplace–Young equation

“…King et al (1999) demonstrated that in this latter case a full asymptotic analysis requires some knowledge of far-field behaviour. Other work has looked at establishing formal power series solutions near the corner of a wedge (Norbury et al 2005) and asymptotic expressions valid far from the corner have been derived by Fowkes & Hood (1998). Some recent numerical simulations using finite volume methods on an unstructured mesh have given further information on the detailed form of the free surface in a case of a medium-angled wedge, see Scott et al (2005).…”

Computation of capillary surfaces for the Laplace–Young equation

“…The accuracy of our numerical method is verified on several levels in both one and two dimensions. Checks are performed against (i) the exact one-dimensional plane wall solutions for both finite and semi-infinite domains (Landau and Lifshitz, 4), (ii) the linearized analytical solutions of Fowkes and Hood (1) applying in semi-infinite wedges, (iii) numerical solutions of the full non-linear ordinary differential equation for describing the radially symmetric surface occurring between two concentric cylinders and (iv) the regular power series expansion about the wedge corner at r = 0 to any order in r, of the full non-linear equation forũ(r, θ) developed by Norbury et al (5). We find that once γ < 80 deg (cos γ > 0.2), non-linearity is important in practice and that even a little rounding of corners has a dramatic effect on reducing the liquid rise in the corner.…”