The influence of the non-Newtonian properties of blood on the flow in large arteries: unsteady flow in a 90° curved tube

An orthogonal coordinate system along a generic spatial curve has been introduced, and the Navier-Stokes equations for a steady incompressible viscous flow have been explicitly written in this frame of reference. As an application the flow in a helical pipe has been studied, and, formdii 5f curvature and torsion small compared with the radius of the pipe, the flow has been considered as a perturbed Poiseuille flow. The result is that for curvatures and torsions of the same order and for low Reynolds number the curvature induces on the flow a first-order effect on the parameter e =κa, where κ is the curvature and a the radius of the pipe, while the effect of the torsion on the flow is of the second order in E. This last result disagrees with those of Wang (1981), who, adopting a non-orthogonal coordinate system, found a first-order effect of torsion on the flow.

On the effect of torsion on a helical pipe flow

Pulsatile flow in an in vitro model of the human carotid bifurcation was studied by flow visualization using hydrogen bubble techniques. A glass model was constructed after determining an average geometry from 57 biplanar angiograms of 22 subjects ranging from 34 to 77 years of age. The flow pulse used was a half-sine wave superimposed upon a mean flow. Maximum and minimum values of the instantaneous Reynolds number were 1200 and 400, respectively, based upon conditions in the common carotid model artery; the frequency parameter was 6.0. The division of flow into the internal external branches was 70:30. Visualization by hydrogen bubbles demonstrated significant deviations from steady flow behavior. Flow separated in the carotid sinus over the entire cycle, but the location and extent of separation varied strongly. The direction of flow near the walls of the model changed sharply during the cycle except for the region near the apex of the bifurcation where the orientation of streaklines was more nearly unidirectional at all times. Bubbles entering the separated flow region tended to remain entrapped there for several cycles. Rapid dispersion of bubbles occurred in the internal branch near the end of systole, suggesting the presence of flow disorder. The location of low wall shear stresses, directionally varying stresses, and longer residence times for fluid elements appears to coincide with the localization of early atheromatous plaques in human carotid specimens.

Pulsatile flow in a model carotid bifurcation.

RESULTS: In the volunteers, flow progressed as follows: an initial jet of blood is skewed towards the anterior right wall of the ascending aorta, blood is reflected laterally off the anterior right wall around the vessel towards the inner curvature creating opposing helices seen as clockwise and counterclockwise vortices, and subsequently retrograde flow develops between the two helices along the inner curvature. In the descending aorta, flow forms a clockwise helix. This may be caused by the geometry of the aortic arch, which curves inferiorly before descending along the spinal column. In patients, the most common pattern seen was where flow travels along the anterior right wall of the ascending aorta and then folds inferiorly along the inner curvature instead of entering the transverse aorta, creating a large region of retrograde flow (Figure 3). Streamline analysis of flow from the aortic valve showed a dominant right handed helical flow in nearly 80% of volunteers. Using graded vortices from extracted planes, we found that in fact volunteers had two opposing helices, while in patients only 21% of patients had two helices. Also, we found that in every volunteer where two vortices were present, one was clockwise and one counterclockwise. This was the case in only 75% of patients with two vortices; while in the remaining 25%, both vortices were clockwise. In every subject studied with opposing helices, the counterclockwise vortex was to the left of the clockwise one. Quantitative data confirmed the observed decrease in velocity at the transverse aorta at peak systole (Figure 1). In the volunteers, there was a 10% decrease in the mean velocity at peak systole across the aortic lumen (Vavg) from AsAo to TvAo (47.9 mm/s decrease with SD of 82.6, p = 0.023) and a marked 29% increase in Vavg from TvAo to DsAo (125 mm/s increase with SD of 75.6, p < 0.001). In the patients, there was a 57% increase in Vavg from AsAo to TvAo (145 mm/s decrease with SD of 112, p < 0.001) and a smaller statistically insignificant 12% increase in Vavg from TvAo to DsAo (47.3 mm/s increase with SD of 108, p = 0.16). Additionally, the ascending aortic aneurysm patients had larger proportion of the cardiac cycle with net retrograde flow (41% of cardiac cycle vs 17% of cardiac cycle, p < 0.001). CONCLUSIONS: We have both qualitatively and quantitatively described the different flow profiles in patients with ascending aortic aneurysms and healthy volunteers. The velocity and flow profiling such as that described in this study can be a valuable tool for the pre-operative work-up and post-surgical care of patients with aortic aneurysms, and potentially could help risk-stratify patients who are congenitally pre-disposed to this condition. Figure 2: Extracted planes from mid-systole orthonormal to the ascending aorta from a healthy 66-year-old female. (A) Magnitude data hidden allowing grading of vortical flow patterns. Two vortices can be seen (arrows). (B) With magnitude data present, retrograde flow (arrowheads) can be analyzed. Figure 3: Example flow pattern in a 66-year-old man with an ascending aortic aneurysm. (A) Particle traces in mid-systole from left anterior oblique view demonstrating retrograde flow developing along the inner curvature of the arch (arrows). This is the most common pattern seen in patients. (B) Streamlines redemonstrating the vortical flow pattern (arrowheads) along the inner curvature.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/jmri.21082

Comparison of flow patterns in ascending aortic aneurysms and volunteers using four-dimensional magnetic resonance velocity mapping

Using data obtained from 3D digital subtraction angiography acquisitions, computational fluid dynamics techniques were used first to assess hemodynamic factors in geometrically correct models of 3 paraclinoid aneurysms and then again for assessment after virtual removal of the aneurysms and reconstruction of the parent artery. Simulations revealed an area of relatively low and rotating wall shear stresses at the location at which each aneurysm had developed. This phenomenon, to our knowledge, has not been previously described.

http://www.ajnr.org/content/ajnr/27/5/1113.full.pdf

Hemodynamics in a cerebral artery before and after the formation of an aneurysm.

In the stream-line motion of fluid in a curved pipe the primary motion along the line of the pipe is accompanied by a secondary motion in the plane of the cross-section. The secondary motion decreases the rate of flow produced by a given pressure gradient and causes an outward movement of the region where the primary motion is greatest. It is difficult to deduce these consequences from the exact equations of motion, but it is easy to do so if it is assumed that the actual secondary motion is replaced by a uniform stream; conditions in the central part of the section mainly determines the motion and here the secondary motion is approximately a uniform stream. The appropriate velocity of the stream can be determined from the relation that has been found experimentally between the rate of flow in a curved pipe and the pressure gradient.

/pdf/note-on-the-motion-of-fluid-in-a-curved-pipe-vu7n20dalm.pdf

W. R. Dean

Papers

Note on the motion of fluid in a curved pipe