Laplace equation in rectengualar channel (Fluid mechanics)

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Homework Statement

Estimate the speed a potential flow in gravity field would have in direction $y$ in rectangle channel with depth [itex] h [/iteh] and length $l$. The fluid is incompressible and on the surface $z = 0$ we have boundary condition $\dfrac{\partial^2 \phi}{t^2} + g\dfrac{\partial \phi}{\partial z} = 0$

Homework Equations

$\nabla^2 \phi = 0, \vec v = \nabla \phi(x,y,z,t),$
The free surface is described by $\zeta(x,y,t) = \dfrac{1}{g} \dfrac{\partial \phi}{\partial t}$ with the
ansatz $\phi = Z(z) e^{-\omega t}e^{i(k_1x + k_2y)}$ which is for channel with depth $h$ and infinity length.

The Attempt at a Solution

We put the ansatz in Laplace and obtain $Z^{\prime \prime} - \underbrace{(k_1^2 + k_2^2)}_{k}Z = 0$

The solution is with boundary condition $\nabla \phi (-h) = 0 \Rightarrow \dfrac{d }{d z } Z(-h) = 0$

We obtain a solution $\phi( \vec r, t ) = \dfrac{ig}{\omega}A \dfrac{ch(k(z+h))}{ch kh} e^{i(k_1x + k_2y - \omega t)}$

My problem is how to express the boundary limits on the walls at $(0,-h) \cup (l,-h)$ expanding in the y direction.

 

[Asim Riaz]

The solution for the vertical velocity is[itex] \dfrac{\partial \phi}{\partial t} = - \omega \phi = - \dfrac{gAchkh}{ch(k(z+h))}e^{i(k_1x + k_2y - \omega t)} But I am not sure how to express the boundary limits on the walls at [itex] (0,-h) \cup (l,-h) expanding in the y direction.