Working Out Shaft Torque in a Stirred Tank with a Cylindrical Cavern Formation (using a non Newtonian fluid)

Here is some background to the problem (in a stirred tank):

“With yield stress non-Newtonian (viscoplastic) fluids, it is possible to generate an agitated volume around the impeller, defined as a cavern, surrounded by a stagnant region where the shear stress is insufficient to overcome the apparent yield stress of the fluid.”

Sometimes you can get a cylindrical cavern around the impeller, see the below image.

“By performing a force balance between the applied torque, Γ and the shear stress acting on the surface of a cylinder, we can define the boundary by setting the shear stress equal to the yield stress τ = τY. The total torque is given by:
$$Gamma = frac{pi}{2} tau_{y}H_{C}D_{C}^2+frac{pi}{6}tau_{y}D_{C}^3$$

I just can’t get the second term. The first term I can get by doing:
$$Gamma_{1}=tau_y cdot Area_{Curved} cdot frac{D}{2} = pi cdot frac{D^2}{2} cdot H_{c} cdot tau_{y}$$

This gets me the first term…but the second term I just can’t get, this is what I’m doing:

$$Gamma_{2}=tau_{y} cdot Area_{Faces} cdot frac{D}{2} =tau_{y} cdot 2 pi cdot frac{D^2}{4} cdot frac{D}{2} = tau_{y} cdot pi cdot frac{D^3}{4} $$

Argh, so I’m getting D^2/4 instead of D^2/6 for the second term and I just can’t work it out, if anyone can help I’d appreciate it. Thanks.

MIH.