# Slip-Condition for Fluid between Parallel Plates [closed]

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) :

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.

• It's far from clear what your actual question is?
– Gert
Jan 20, 2021 at 2:48
• 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.
– RMS
Jan 20, 2021 at 12:45