# Shear rate of a fluid in a container with a sliding PTFE top plate. Will the liquid move with the plate?

I am trying to work out how a liquid inside a container will be affected by the back and forth sliding of a Teflon coated plate such as the one in the picture.

I know how to work out the shear rate of a fluid between two plates moving at different velocities using the v/h equation, but it doesn't take into account things like the walls on my container, or the fact that the plates may be hydrophobic.

My goal is to move the plate (which is in contact with the liquid) back and forth without the liquid moving. So my question revolves around firstly whether it is even possible to move the screen back and forth without significantly displacing a certain liquid it is in contact with underneath, and secondly how I would go about calculating to what extent the liquid would be disturbed.

• Your question looks very much like Stokes second problem. Are you familiar with that? If yes, what are the differences? Feb 8 '16 at 9:25
• Regarding hydrophobia: I'm not very familiar with those, but to my knowledge this is being modeled by surface tension or more specific, the contact angle. From my understanding the water will still stick to the plate (btw. plate = screen?) even if the latter is hydrophobic. So for the sake of a force being transmitted from the plate to fluid, hydrophobia of the plate should not matter. Maybe I'm mistaken here. Feb 8 '16 at 9:42
• The phrase "without significantly displacing a certain liquid" is very subjective; can you give a more precise constraint, e.g. a limit on fluid velocity at a certain point in the container? Any motion of the screen/plate will induce flow in the container, it depends on the frequency and magnitude of the plate movement, the viscosity of the fluid, the dimensions of the container, etc. Calculating the velocity profile in the container for this case would likely require a CFD package. Feb 11 '16 at 13:39
• Yes I've realised from reading back over my question that it is fairly subjective. I've basically come to the conclusion though that the screen won't be able to touch the resin due to the factors you pointed out. Back to the drawing board. Feb 14 '16 at 14:52

if you want to limit the effect of the plate oscillations on the motion of the liquid as much as possible you need to make the time scale for viscous diffusion much larger than that of the plate oscillations.

If the plate oscillates with an angular frequency $\omega$, the liquid has a kinematic viscosity $\nu$ and the length scale is the height of the box $H$ then you want the ratio: $$\epsilon=\frac{\nu}{\omega H^2}\ll1$$ here $\frac{H^2}{\nu}$ is the viscous time scale and $\frac{1}{\omega}$ is the oscillation time scale.

As mentioned by @ChrisJohns, this requires very low viscosities or large angular frequencies. However, there will alway be some pertubation of the liquid by the plate but $\epsilon\ll1$ should limit it.

Take note, that this assumes that the height of the box is the characteristic length scale; this assumes the absence of any interference by walls. I would redesign the box to have a height much smaller than any of the other two dimensions if possible but still such that $\epsilon\ll1$.

In order for the fluid to not be affected at all by the movement of the plate it would require either zero friction between the plate and the liquid or for the fluid to have zero viscosity.

There is also the fact that the mechanics of moving the plate will inevitably involve some flexing, however small, of the tank and plate and transfer of vibrations into the tank by whatever actuator mechanism you use.

There are also external factors like convection in the fluid and environmental vibrations.

So zero perturbation is not going to be possible and you need to decide what would be an acceptable level. As well as the friction properties of the surface and the fluid in question the rate of movement of the plate and its scale will be crucial factors.