The initial development of a jet caused by fluid, body and free surface interaction. part 3. an inclined accelerating plate

Abstract

The free surface and flow field structure generated by the uniform acceleration of a rigid plate, inclined at an angle a ∈(0, π/2) U (π/2, π) 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 a = π/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 a = π/2 is a bifurcation point in this local structure. For a ∈ (0, π/2), a weak jet rises up the plate when t = 0 + , with thickness 0(t 2 ) as t → 0 + , independent of a, with the free surface slope at the plate being O (t π α-2 ) as t → 0 + ; this slope is O(1/log(t)) as t → 0 + when a = π/2. However, when a ∈ (π/2, π), the jet becomes significantly stronger, with a highly nonlinear structure, and the thickness now depending on and increasing with a, being O (t γ ), where γ = (1-π/4α) -1 . In this case, moreover, a classical solution to the evolution problem is possible only when a ∈ (π/2, α c ], where α c ≈ 1.791 ≈ 102.6°. When a = α c , a 120° comer 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 « 1. We conjecture that no classical solution exists when t = 0 + for plate angles a ∈(α c , π). In practice, surface tension allows a solution to exist in some finite neighbourhood of t = 0, a result that will be presented in a later paper.