Hi,

I have developed a code which solves the (incompressible) N-S eqns in an elastic-deformable tube. It could be considered as a first step to examine blood flow in artery.

Now, I am looking for papers where people have examined this problem, in order to validate my code. I found a lot of work where there is a graft, a stenosis or even an atherosclerosis. But, I can not find a paper where the author(s) studies the flow in an elastic-deformable tube.

Could someone point me to such references?

Thank you in advance, Vasilis

Isn't blood flow non-newtonian?

Numerical investigation of blood flow. Part II: In capillaries Communications in Nonlinear Science and Numerical Simulation, Volume 14, Issue 4, April 2009, Pages 1396-1402 A. Jafari, P. Zamankhan, S.M. Mousavi, P. Kolari

Vasilis,

I'd be very interested in following up on your progress in this research. I'd be very interested to see just how stiff a wall can be used to allow the blood flow to occur.

I'm particularly interested in pulse motion within confined domains.

BTW, the Featflow team seem to have performed some studies for blood flow as well. You could try their website at www.featflow.de

Be nice to hear from you.

mw...

www.adthermtech.com/wordpress3

Yes the blood is non-Newtonian, but many researchers consider it to be a Newtonian one

Thank you for the paper

mw,

Could you elaborate on "how stiff a wall can be used to allow the blood flow to occur"? Regardless of how stiff the wall is, you will always have a flow. Nevertheless, if the wall is compliant the blood flow will deform the wall. Consequently, the deformed wall will affect the blood wall.

Most simulations which analyze the blood flow through arteries use a non Newtonian viscosity model if they account for the deformation of the walls. This has shown to produce pretty good agreement with experiments. For eg. check http://www.mate.tue.nl/mate/pdfs/215.pdf, its a combined numerical experimental simulation.

Hope this helps.

look at publication of Suncica Canic (http://www.math.uh.edu/~canic/) she does a good job in this regard, in particular shis work is very nice in terms of quality, e.g. see this one:

SIAM J. APPLIED DYNAMICAL SYSTEMS, 2003 Vol. 2, No. 3, pp. 431â€"463

or this one: Networks and Heterogeneous Media Vol. 2(3) 2007 397-423

good luck

mw wrote:

I'd be very interested to see just how stiff a wall can be used to allow the blood flow to occur.

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Perhaps a poor choice of wording on my behalf - let me re-phrase. What I meant to say was "If you were to stiffen the wall significantly, would there be any effect on the blood flow?"

The thought that I'm trying to work on is as follows: How does blood flow in a vein? Is it a combination of a mean flow + a conveyed pulse, or purely a series of pulses? What proportion is mean flow & what proportion is superimposed pulse?

Further to this, it is possible to have a pulse move a fairly significant distance down a tube, with mean flow almost zero - for a newtonian fluid, for instance. I'm not sure how this plays out for a non-newtonian fluid.

What role are the side walls playing in the blood flow motion?

mw...

www.adthermtech.com/wordpress3

Thank you for the paper, unfortunately the authors deal with the case where there is flow in a carotid bifurcation, and not blood flow in a tube.

Dear vadim,

Although Suncica Canic is looking at the problem from a mathematical perspective, her job could be a good benchmark against my result.

Thank you for pointing out to me her work.

mw,

Unfortunately, I can not answer all your questions, but I will answer your first question.

I have examined the stability of creeping flow over compliant surfaces, and I found that the wall compliance does affect the flow stability. I considered the fluid to be Newtonian. I also found, that the flow stability depends on the driving force, i.e. Poiseuile or Couette flow and the solid material. I did not examine the case of pulsate flow, and thus I can not (for the moment) answer all your questions.

Because, flow in an elastic-deformable tube is a fairly complicated problem, I can not tell you how exactly the wall compliance will affect the blood flow. Its effect on the blood flow will depend on the geometry, the driving force, the Reynolds number (especially if you scale the governing eqns so that its reciprocal multiplies the viscous terms), and the solid material.

Thanks vasilis.

Setting up an inlet pulse is not that difficult, in practice. E-mail me & we discuss further if you're interested. I expect that the pulse could travel onwards fairly nicely, without much effect on the sidewalls, if set up correctly.

mw...

www.adthermtech.com/wordpress3

For real-work problems the rheological modeling of arterial blood flow is crucial.

As I understand it (I am NOT a physician), flow of blood in a healthy artery or vein is laminar and there is no flow separation.

When the arterial wall is altered in stiffness ('hardening of the arteries') or contour (cholesterol deposits, partial blockage), the flow is disturbed. Local separation can occur. This separation actually generates noise that can be identified by a trained observer using a stethoscope. This abnormal flow pattern is thought to accelerate deposits on the artery wall. The deposits are unstable, subject to shedding particles that can lead to heart attack or stroke.

The apparent viscosity of blood is modified by excess cholesterol due to poor diet, lack of exercise, or high glucose concentration due to poorly-controlled diabetes. This is thought to promote hardening of the vessel walls and flow separation because the artery cannot conform smoothly to the changes in fluid property. In a feedback loop, additional deposits are promoted in the 'dead' zone associated with the separation. This suggests that correct modeling of the rheology of flowing blood is critical to understanding behavior of the circulatory system.

Medical research scientists tend to see this destructive synergism as a chemistry problem. The thoughtful fluids practitioner who can bridge the gap between bench chemistry and the circulatory system will make a very real contribution to understanding (and perhaps treating) a major cause of heart attack and stroke.

I totally agree with you. Nevertheless, this is a first attempt to model this problem. I am not trying to capture all the physics that govern this problem, but to get an idea of how the wall compliance affects the blood flow.

otd stated:

For real-work problems the rheological modeling of arterial blood flow is crucial.

Excellent overview 'otd'... So, this would seem to be a key factor in understanding blood flow. The wall-effect could be added once the fluid properties were well understood & modeled. It would be good for the OP to bear this in mind at the outset of his research.

mw...

www.adthermtech.com/wordpres3

1. Numerous works have been done in this area. Thus, it is a good idea to do some literature search before going further. Some journals that publish a lot in this field are: Journal of Applied Physiology, Annals of Biomedical Engineering, and occasionally Journal of Computational Physics. 2. Blood flow at high shear rates behaves as Newtonian. It is only at low shear that Non-Newtonian properties comes up. Thus , a good approximation for blood rheology is Bingham model that combines Newtonian stress with a yield stress. 3. Compliance of arteries and veins is directly related to the speed of propagation of pressure waves. In aorta, the speed of pressure waves is as low as 2.5-5 m/s. The more stiff is a wall the less is the propagation of pressure waves. However, in a glass tube filled with blood the propagation of pressure waves is 1560 m/s. Thus, to model blood flow in arteries and veins one need to examine carefully, if its approximation by incompressible Navier-Stokes equations is still valid.

Angen

Angen wrote:

3. Compliance of arteries and veins is directly related to the speed of propagation of pressure waves. In aorta, the speed of pressure waves is as low as 2.5-5 m/s. The more stiff is a wall the less is the propagation of pressure waves. However, in a glass tube filled with blood the propagation of pressure waves is 1560 m/s. Thus, to model blood flow in arteries and veins one need to examine carefully, if its approximation by incompressible Navier-Stokes equations is still valid.

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mw writes:

The logic seems to do an about turn here - perhaps I'm missing something? Would not a glass tube be considered to be a non-compliant wall? How then the extremely high velocity, which is presumably the 'speed of sound'?

Do the waves have to be pressure waves? What if they were instead a waveform which moves in almost incompressible fluids, with little change in pressure - a momentum wave, to be more precise? With this form of waveform, a stiff wall is fine - no compliance is necessary to allow a pulse form to travel down the tube.

mw...

www.adthermtech.com/wordpress3

Yes, the glass tube wall is very stiff. This is why the propagation of pressure waves is very high as well. Propagation of pressure waves depends on elastic properties of wall and fluid compressibility (the more elastic wall the more compliant is the tube). This point is explained very well in Lighthill's classic work "Waves in Fluids" (including wave propagation in arteries). The numbers for propagation speed I cited in my previous message are based on experimental works.

Angen