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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.

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# Shear Stress between Two Rotating Discs

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