The initial development of a jet caused by fluid, body and free surface interaction. part 3. an inclined accelerating plate
The initial development of a jet caused by fluid, body and free surface interaction. part 3. an inclined accelerating plate
复制标题
由流体、物体和自由表面相互作用引起的射流的最初发展。
DOI：
--
复制
发表时间：
2008
期刊：
影响因子：
--
通讯作者：
J. Billingham
中科院分区：
文献类型：
--
作者：
D. Needham;P. G. Chamberlain;J. Billingham
本文利用匹配渐近展开方法，在小时间极限条件下，研究了刚性板以a∈(0,π/2) U （π/2, π）向外水平方向倾斜时，当其进入初始静止且水平的无粘不可压缩流体带时，均匀加速度所产生的自由表面和流场结构。这项工作推广了当a = π/2时均匀加速的垂直板的情况，如King和Needham （J. Fluid）所研究的。268(1994))。特别注意的是在板和自由表面之间的交点附近的内部区域。结果表明，角a = π/2是该局部结构的一个分岔点。对于a∈(0,π/2)，当t = 0 +时，一股弱射流从板上升起，厚度为0(t2), t→0 +，与a无关，板处自由面斜率为0(t π α-2), t→0 +；当t→0 + a = π/2时斜率为0 （1/log(t)）然而，当a∈(π/2, π)时，射流变得明显更强，具有高度非线性的结构，厚度现在取决于a并随a增加，为O (t γ)，其中γ = (1-π/4α) -1。在这种情况下，只有当a∈(π/2, α c]，其中α c≈1.791≈102.6°时，进化问题才有经典解。当a = α c时，当t = 0 +时，自由表面上在板与自由表面的初始交点处形成一个120°角，并自相似地对流到0 < t«1的内部区域。我们推测当t = 0 +时，板角a∈(α c, π)不存在经典解。实际上，表面张力允许解存在于t = 0的有限邻域中，这个结果将在后面的论文中提出。
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.