TY - JOUR

T1 - On the entry of a wedge into water: The thin wedge and an all-purpose boundary-layer equation

AU - Fraenkel, L.E.

AU - Keady, Grant

PY - 2004

Y1 - 2004

N2 - In 1932, H. Wagner formulated the problem of the entry into water of an infinite wedge moving vertically downwards with constant speed. Among much else, Wagner noted that, in the absence of gravity, viscosity and surface tension, a similarity transformation removes time from the problem. Many other authors have considered the problem since 1932. The present paper settles a question, left open in earlier work, concerning the contact angle pibeta; this angle is shown, together with the wedge angle (or vertex angle) 2pialpha, in Figure 1(b). The question is whether the supremum pi(β) over bar of pibeta, over the whole set of solutions having 0 < 2πα < pi, is equal to pi/4 or to a smaller value. The answer is that (β) over bar < 1/4 (and the proof suggests that 1/4 - <(beta)over bar> is not small relative to the range of); this has long been indicated by numerical work, but ( as far as we know) has not been proved rigorously until now. The paper also introduces an integral equation of boundary-layer type that allows numerical calculation without extrapolation of the limiting solution as alpha --> 0 and of the value beta(0) corresponding to alpha = 0. ( Such calculation is not possible with the full integral equation governing the problem.) It turns out that this same integral equation of boundary-layer type also governs the other two critical cases of the problem: beta --> 0 and beta --> 1/4; therefore it may be called an all-purpose boundary-layer equation. Numerical calculation with this equation indicates that beta(0) = (β) over bar and that beta(0) = 0.100 +/- 0.002, which is essentially in agreement with earlier, extrapolated values.

AB - In 1932, H. Wagner formulated the problem of the entry into water of an infinite wedge moving vertically downwards with constant speed. Among much else, Wagner noted that, in the absence of gravity, viscosity and surface tension, a similarity transformation removes time from the problem. Many other authors have considered the problem since 1932. The present paper settles a question, left open in earlier work, concerning the contact angle pibeta; this angle is shown, together with the wedge angle (or vertex angle) 2pialpha, in Figure 1(b). The question is whether the supremum pi(β) over bar of pibeta, over the whole set of solutions having 0 < 2πα < pi, is equal to pi/4 or to a smaller value. The answer is that (β) over bar < 1/4 (and the proof suggests that 1/4 - <(beta)over bar> is not small relative to the range of); this has long been indicated by numerical work, but ( as far as we know) has not been proved rigorously until now. The paper also introduces an integral equation of boundary-layer type that allows numerical calculation without extrapolation of the limiting solution as alpha --> 0 and of the value beta(0) corresponding to alpha = 0. ( Such calculation is not possible with the full integral equation governing the problem.) It turns out that this same integral equation of boundary-layer type also governs the other two critical cases of the problem: beta --> 0 and beta --> 1/4; therefore it may be called an all-purpose boundary-layer equation. Numerical calculation with this equation indicates that beta(0) = (β) over bar and that beta(0) = 0.100 +/- 0.002, which is essentially in agreement with earlier, extrapolated values.

U2 - 10.1023/B:engi.0000018157.35269.a2

DO - 10.1023/B:engi.0000018157.35269.a2

M3 - Article

VL - 48

SP - 219

EP - 252

JO - Journal of Engineering Mathematics

JF - Journal of Engineering Mathematics

SN - 0022-0833

IS - 3-4

ER -