垂直平板在粘性不可压缩流体中的突然启动

Nonlinear Wave Ahead an Accelerating Vertical Plate in Weakly Fluid

本文研究加速平板所产生的非线性水波。加速度对于小的时间t而言,表示为时间t的幂级数a(t)= ∑nant^(n-1),对冲击问题a_1≠0。 n=1 我们得到对于小的时间t,接触角cotθ=a=a_1/g,为常数有限值的结论,其中0<θ≤π/2,用渐近展开法,得到零阶至三阶压强P,流速u及波幅η。结果表明:它们是x的指数函数,u_1与P_0有关,η_2与v_1有关等。只有在高阶近似时P和η才与粘性有关。 采用接触角原理,现在的结果消除了原来势流解在平板冲击时波幅上的奇异性(x=0处,η→∞)这一方法对复杂形状可给出数值解。

Nonlinear wave ahead of an accelerating vertical plate are studied in this paper. The plate is assumed to be rigid and it moves horizontally with an acceleration a(t) towards the fluid of constant depth h. The fluid, which is initially at rest, is assumed to be weakly compressiable. For small time t, the acceleration a(t) in asymptotic expansion method is represented by a powerseries in t. i. e. a(t) = , for impules problem a1≠0.The contact angle θ between the free surface and the plate is founded to be constant for small t, that is cot θ = a1/g.The present analytical solution indicate that: we can obtained pressure p and velocity u from first order to third order and the free surface elevation η of the second order by asymptotic expansion solution. They are exponential function of x. The viscous effects is in the higher order of η, which is inversely proportional of Re = h2/vt.The present theory, with the contact angle condition, gives a finite surface elevation on the plate and the singularity is vanished.