TY - JOUR

T1 - The Initial Development of A Jet Caused By Fluid, Body and Free Surface Interaction. Part 3. An Inclined Accelerating Plate

AU - Needham, David

AU - Chamberlain, PG

AU - Billingham, J

PY - 2008/11/1

Y1 - 2008/11/1

N2 - The free surface and flow field structure generated by the uniform acceleration of a rigid plate, inclined at an angle alpha is an element of (0, pi/2) boolean OR (pi/2, pi) to the exterior horizontal, as it advances into an initially stationary and horizontal strip of inviscid, incompressible fluid, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating vertical plate, when alpha = pi/2, as studied in King and Needham (J. Fluid. Mech. 268 (1994)). Particular attention is devoted to the inner region in the vicinity of the intersection point between the plate and the free surface. It emerges that the angle alpha = pi/2 is a bifurcation point in this local structure. For alpha is an element of (0, pi/2), a weak jet rises up the plate when t = 0(+), with thickness O(t(2)) as t -> 0(+), independent of alpha, with the free surface slope at the plate being O(t pi/alpha -2) as t -> 0(+); this slope is O(1/log(t)) as t -> 0(+) when alpha = pi/2. However, when alpha is an element of (pi/2, pi), the jet becomes significantly stronger, with a highly nonlinear structure, and the thickness now depending on and increasing with alpha, being O(t(gamma)), where gamma = (1 - pi/4 alpha)(-1). In this case, moreover, a classical solution to the evolution problem is possible only when alpha is an element of (pi/2, alpha(c)], where alpha(c) approximate to 1.791 approximate to 102.6 degrees. When alpha = alpha(c), a 120 degrees corner forms on the free surface when t = 0(+) at the initial intersection point of the plate and free surface, and convects self-similarly into the inner region for 0 <t

AB - The free surface and flow field structure generated by the uniform acceleration of a rigid plate, inclined at an angle alpha is an element of (0, pi/2) boolean OR (pi/2, pi) to the exterior horizontal, as it advances into an initially stationary and horizontal strip of inviscid, incompressible fluid, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating vertical plate, when alpha = pi/2, as studied in King and Needham (J. Fluid. Mech. 268 (1994)). Particular attention is devoted to the inner region in the vicinity of the intersection point between the plate and the free surface. It emerges that the angle alpha = pi/2 is a bifurcation point in this local structure. For alpha is an element of (0, pi/2), a weak jet rises up the plate when t = 0(+), with thickness O(t(2)) as t -> 0(+), independent of alpha, with the free surface slope at the plate being O(t pi/alpha -2) as t -> 0(+); this slope is O(1/log(t)) as t -> 0(+) when alpha = pi/2. However, when alpha is an element of (pi/2, pi), the jet becomes significantly stronger, with a highly nonlinear structure, and the thickness now depending on and increasing with alpha, being O(t(gamma)), where gamma = (1 - pi/4 alpha)(-1). In this case, moreover, a classical solution to the evolution problem is possible only when alpha is an element of (pi/2, alpha(c)], where alpha(c) approximate to 1.791 approximate to 102.6 degrees. When alpha = alpha(c), a 120 degrees corner forms on the free surface when t = 0(+) at the initial intersection point of the plate and free surface, and convects self-similarly into the inner region for 0 <t

U2 - 10.1093/qjmam/hbn019

DO - 10.1093/qjmam/hbn019

M3 - Article

VL - 61

SP - 581

EP - 614

JO - Quarterly Journal of Mechanics and Applied Mathematics

JF - Quarterly Journal of Mechanics and Applied Mathematics

SN - 0033-5614

ER -