When we have two parallel plates separated by distance $h$, the plate at the top moves with velocity $V$, while the one at the bottom remains stationary, and we consider that the fluid flow between two parallel plates has a slip of, $$ \delta u \approx \ell \frac{du}{dy}$$

This is my initial approach, which could very well be incorrect:

I considered, $du/dy = V/h$ so $\delta u \approx \ell V/h $ and for the shear stress $\tau = \mu \frac{\partial u}{\partial y}$

For $$\frac{\partial u}{\partial y} = \frac{U_{top}-U_{bottom}}{h}$$

Where I considered the fluid's velocity at the top plate to be, $U_{top} = V - \delta u$ , and at the bottom, $U_{bot} = 0 + \delta u$.

In order to improve my understanding, I searched for a diagram that illustrated the phenomena, which shows (apologies for the low quality) :

Diagram for slip

This $b$ distance is the $\ell$ of my equation at the top. So I am wondering if it should be included as part of the $dy$ term, as $dy = h - \ell$ or should it be part of $\partial y = h - \ell $. Or perhaps my original analysis was correct? Let me know where I am going wrong. I am not unsure how to describe this system mathematically.

Thanks a lot, your time and help are very appreciated.

  • 1
    $\begingroup$ It's far from clear what your actual question is? $\endgroup$
    – Gert
    Jan 20, 2021 at 2:48
  • $\begingroup$ The question is in the last paragraph. "Should I include b, as part of my dy term, such that $ dy = h - b $, or as $\partial y = h - b$ or is the way I analyzed initially correct (where $dy = \partial y$ as can be seen above)? " It all depends on what is the proper way to analyze this system, but I did not want to ask for the solution. $\endgroup$
    – RMS
    Jan 20, 2021 at 12:45