# Shear Stress between Two Rotating Discs

1. Aug 1, 2013

Hey everyone,

As part of my PhD project I'm trying to design a small bioreactor in which I will be growing animal cells. This bioreactor will basically consist of a hollow cylindrical vessel that has an impeller in it which will keep the liquid medium in the vessel well mixed. The impeller I am using consists of two flat parallel circular discs which will have the same axis of rotation (i.e. they will both be mounted on the end of a single narrow shaft and submerged in the medium). The discs will be spinning at the same angular velocity (ω=10.47 rad/s) and will have the same radius (0.0268 m). There will be a gap of 1cm between the discs (0.01 m). The fluid has a viscosity of 8.9 x 10^-4 Pas.

My problem is that I need to be able to know to a good degree of accuracy what magnitude of shear stress the cells will be subjected to when they pass through this 1cm gap between the discs. So I need to have a way of calculating the theoretical shear stress acting on the fluid between the two discs as a result of their rotation. I know that the shear stress should increase as you move from the centre of the discs out to the edge of the discs because the velocity (v) at any point on the discs is given by v=ω*radius (so increasing radius = increasing velocity = increasing shear stress).

Does anybody know of a formula or method I could use to calculate this shear stress? I've been struggling with this for a couple of weeks so any help would be hugely appreciated! If you want any additional details or images to help describe the problem feel free to ask. Thanks.

2. Aug 1, 2013

### Staff: Mentor

If the whole liquid rotates together with the disks, I would expect zero shear stress. If the whole liquid does not rotate (significantly), I would expect significant shear stress close to the edges of the disks.
Based on those extreme examples, I think you have to take the geometry of the setup into account.

As a lower bound, the velocity change between the edge of the disk and the outer cylinder is the velocity of the disk - together with the viscosity, you can convert this to some sort of average shear stress - which serves as a lower bound for the maximal value.

3. Aug 1, 2013

Hi mfb,

Thanks for the reply, that makes sense. So just to give you more details, I have created a CFD simulation of the problem and looked at the velocity of the flow between the discs. The velocity is at a max at the disc surfaces (around 0.3 m/s) but then drops very dramatically to practically zero when you move about a mm or so away from the discs, so I'm guessing this would fall into the latter extreme that you mentioned.

For my bioreactor I need to be able to vary the shear stress being applied to the cells and see how this stress affects their growth. For this purpose, it would be very convenient if I could take an average value for the shear so that for every rotational speed setting I use there would be one set value for the shear that this setting produces on the cells between the discs. I wonder if I changed the design so that the discs were very close together and the boundary layers at the two discs were partially merging, would this ensure that any cells passing between the gap would definitely experience shear? I'm guessing that if the boundary layers were fully merged there would just be no velocity gradient between the discs at all and so there would be no shear, but if I just partially merged them maybe it would ensure that any cell passing through the gap would definitely experience a decent level of shear, and hopefully because the boundary layers are merged, the difference between the level of shear experienced in the centre of the gap wouldn't be quite so dramatically different from the shear experienced at the disk surface as it has been before now?

4. Aug 1, 2013

### Staff: Mentor

I would try to find a system with a more homogeneous shear stress. Some tiny regions with high values and a large volume with nearly no shear stress does not look like a good setup for that.

5. Aug 1, 2013

Let's just say for argument's sake that I can't change the design. Given this, do you know any formula I could use to calculate the shear stress between the disks? Or even a mathematical function I could use to calculate the shear stress at any given point between the discs? (That way I could at least plot the shear stress profile)

Thanks again.

6. Aug 1, 2013

### Staff: Mentor

Your simulation should be able to give that.