A test was also performed on the time step sensitivity

A test was also performed on the time-step sensitivity of the obtained results. The solutions compared were obtained with the steps of 0.005, 0.010 and 0.020s. As a result, it was found that the solution varied only slightly with decreasing time steps in the specified range; because of this, the main computations were carried out only with a step of 0.01s.
Computational results The computations revealed that Dean vortex pairs in which the fluid rotated in opposite directions formed in the thoracic segment of the common carotid artery. The Dean vortices in Fig. 4 are visualized by Q-criterion isosurfaces; they have the form of two structures of similar shape, elongated along the outer wall of the vessel model. Upon entering the cervical segment of the common carotid, the Dean vortices were transformed into a single vortex generating a swirling flow. The most intense swirling was generated in the junction between the thoracic and cervical segments. The swirl attenuated downstream, and a second vortex emerged approximately in the middle of the cervical segment, gradually increasing in size along the length of the vessel. Two parameters were used to characterize the intensity of the swirling flow: the integral swirl parameter S, which is defined as the dimensionless angular momentum flux, and the swirl parameter β widely used in practice: which is defined as the ratio between the maximum peripheral velocity to the maximum axial velocity. In these formulae r is the radial coordinate; R is the vessel radius; and are the circumferential and the axial velocities. Fig. 5 shows the variation of the two swirl parameters during the s1p receptor for both of the examined models. It can be seen that the most intense swirl occured during systole (from about 0.17s to 0.40s). During this time interval S≈0.08 and β≈0.2 for the statistically average model, while the swirl intensity was higher by about 30–50% for the model with maximum physiological tortuosity. Notably, the β/S ratio was approximately constant for both models for most of the cycle and amounted to 2.0–2.5. Fig. 6 shows the variation of the swirl parameters along the length of the cervical segment of the artery model. It is characterized by a smooth decrease that is almost linear in some regions. Despite a pronounced difference between the swirl intensities obtained for the two models in the beginning of the cervical segment (S≈0.1, β≈0.2 for Model 1; S≈0.2, β≈0.45 for Model 2), the intensities of the swirling flow leveled by the end of the cervical segment and were characterized by the values S≈0.05, β≈0.1. These values are recommended for use as inlet conditions in setting a numerical problem on computing the blood flow in the carotid bifurcation.
Conclusion Swirling flow in the tortuous carotid artery is formed under the influence of the spatial curvature of the artery and the pulsating nature of the flow. The most intense swirl is formed during the flow rate decrease phase at the junction of the thoracic and cervical segments, where the Dean vortices, typical for flow in curvilinear tubes, are transformed into a single vortex generating the swirling flow. As the swirl attenuates, a second vortex re-emerges downstream. The mean swirling intensity per systole (determined by the ratio of the maximum circumferential velocity to the maximum axial velocity) amounts to 0.20 for the model of the statistically average common carotid artery and to 0.25 for the model with maximum physiological tortuosity, which agrees with the clinical results [5].
Acknowledgment The study was carried out with the financial support of the Russian Foundation for Basic Research (Grant no. 15-01-07923).
Introduction Numerous papers containing theoretical [1–8] as well as numerical results [9,10] have been dedicated to shell simulation. A complete system of equations for shells is often derived from equations of three-dimensional elasticity theory by introducing certain simplifications. However, it is preferable to involve a direct approach to simulating shells as deformable surfaces, as well as various analytical techniques.