I'm looking for references e.g., on theory and practical mathematical equations, CFD work, etc.., to a classic chemical engineering/flow problem - calculation/estimation of average pressure drops, shear stresses, and perhaps instantaneous radial flow profiles from Doppler-measured flow/fluid velocity in small circular tubes. Where things get a bit complicated is the flow is pulsatile/nonsteady and the fluid is nonnewtonian (human blood). More complication is added by the fact that the situation I'm interested in involves the flow created in a test model consisting of 4 - 6 mm ID tubing that is folded back onto itself creating a closed loop of circumference in range of 30 - 50 cm. Here, the flow is set up in blood within the loop via microstepper motor control of a repeating motion profile e.g., 1 Hz/60 'beats' per minute, subjected to the loops when a small ball-and-cage check valve is present in the loop. The repeating motion profile, valve action, and fluid momentum create the pulsatile flow. All I've been able to dig up so far is that I need to consult work by Womersley, and perhaps be concerned about a Dean effect due to the curved nature of the tubing.

Do any of you CFD experts have any thoughts here or references to CFD publications/work that may be pertinent here? Any comments and feedback would be welcome and appreciated.

You haven't mentioned the flow rate, but I assume it should be small. Under such a circumstance, since the ratio of the loop to the ID is almost 100, you can assume the flow to be "locally" 1-D, or equivalently "locally" a channel flow with pulsatile inlet BC. An "exact" formulation for such a flow can be found in Schlichting (or other advanced fluid dynamics books). It should not be too difficult to modify this to fit your particular exit BC.

This should give you a fairly accurate (for the time and money you'd spend) estimate of the time-dependent pressure drop and velocity profiles, etc.

If you need to include other complications, then you'll have to solve the problem using CFD: unless someone else has solved exactly the same problem you have, references to other CFD results will be just as "valuable" as the exact solution I mentioned above!

Adrin Gharakhani

The problem should be described as following: 1). pulsatile flow, the Womersley Number is around 5 by your experimental condition, I think. 2). non-newtonian flow 3). curve tube, where the secondary flow and the blood cell motion in radial direction are intersting.

as for 1), I recommend you the classical book: McDonald's Blood Flow in Arteries, by WW Nichols, et al and the web site: //www4.ncbi.nlm.nih.gov/PubMed, where you can search a lot of papers on CFD of blood flow in artety.

as for 2), it is a little bit difficult, and the reference is limited. maybe "Pulsatile flow of non-newtonian fluid through arterial stenoses, J. biomechanics, V29, N7, PP899, 1996" and the references therein are helpful.

the 3), you also can search the above web site, there are a number of papers on the numerical analysis of curve tube blood flow.

I think it depends on your interests, for example: the impedance of the tube, wall stress and the stress on the blood cell, non-newtonian effects, or secondary flow and the blood cell motion. some non-interest aspects need to be escaped. as I know no one have taken into acount all of the above aspects in one work of CFD, it is very difficult, even if not impossible.

I think you may get some informations about transitional flows in curved channels where the authors obtained results at Dean Numbers from 50 to 1100 which include laminar channel flow, laminar flow with Dean vortex pairs, flow with secondary instabilities, channel flow with local regions of augmented turbulence, and turbulent channel flows. 1. Flow Visualization of Dean Vortices in a Curved Channel with 40 to 1 aspect ratio [P. M. Ligrani( Prefessor, Mechanical Engineering Department, University of Utah), and R. D. Niver.], Physics of Fluids, Vol. 31, No. 12, PP.3605-3617, December 1988.

Although, the authors have applied the state-of-the art technology mainly in gas turbine blade cooling passages, the technology can also be applied in passages in biological systems such as what you are looking for. This is a very complex area you are working at. Please let me know if I can be of any help in future. And good luck.