In my lecture notes, we went through a problem with a cylinder approaching a stationary wall with speed $v$. Since we are interested in the thin layer approximation, we utilised lubrication theory. Approximating the distance between the cylinder and wall as
$h(x) = d(1+x^2/2ad)$
where $d$ is the shortest distance between the cylinder and wall.
To find the flow velocity $u$, we used the parallel flow equations to get:
$u=1/2\mu (dp/dx)y(h-y)$
But what i don't understand is why we have imlemented the boundary conditions that the flow velocity is 0 and both $y=0$ and $y=h$, surely, the velocity should be $-v%$ at $y=h$?
The cylinder is moving so it is also a function of time. The velocity is a vector usually denoted $\vec{v}$ or $\vec{u} = (u,v,w)$ where $u = \frac{\partial x}{\partial t}$ $v = \frac{\partial y}{\partial t}$. The boundary conditions would require, \begin{cases} u=0 & \text{ at } y=0, y=h(x,t) \\ v=0 & \text{ at } y=0\\ v=\frac{dh}{dt} & \text{ at } y=h(x,t)\\ \end{cases}