## 3 Results  
The results concern mainly the temporal evolution of the velocity field, the pressure, and the wall displacement. Figure 4 shows the blood speed distribution inside the vessel at different times. To discuss this figure, one should keep in mind the profile of the inlet velocity depicted in Figure 2. At t = 0.1 s, the fluid was mainly directed to the first inlet. The section between the first and second outlets started to move. However, the remaining part of the aorta was almost at rest. At t = 0.2 s, the fluid also reached the second and the third exits. Progressively, the blood flowed into the vessel accelerating or decelerating. The un-steady flow remained laminar. Figure 4: Blood velocity field distribution inside the vessel at different times; (**a**) t = 0.1 s (**b**) t = 0.2 s (**c**) t = 0.4 s (**d**) t = 0.6 s (**e**) t = 0.7 s and (**f**) t = 0.8 s.  
From the results of Figure 4, one can also compute the flow rate in each output as well as the stroke volume. Figure 5 exhibits the temporal evolution of the maximal velocity. The maximum value reached varied from 0.1 to 0.32 m/s depending on time. The relative curve is coarsely similar to that of the input velocity. It is constituted by two peaks that have different amplitudes and different frequencies. A least-square fit allowed to obtain the amplitude of these peaks. We obtained 0.35 m/s for the first peak and 0.26 m/s for the second one. The discrepancy between the computation results and the theoretical results obtained by the least-square fit was in a tolerable limit. The mean uncertainty was about 10%. It was also interesting to analyze the pressure distribution versus time. Figure 6 exhibits the pressure distribution at t = 0.1 s and t = 0.7 s. It is clear that the pressure has a complex distribution. Furthermore, this distribution depends strongly on time, since the input conditions are unsteady. Figure 5: Temporal evolution of the maximal velocity. Figure 6: Pressure distribution inside the vessel at different times; (**a**) t = 0.1 s and (**b**) t = 0.7 s.  
The values of the wall shear stress applied to the blood vessel walls during a cardiac cycle vary strongly in time and position. This has an important effect on cardiovascular diseases. Figure 7 shows the temporal evolution of the maximal pressure. This figure confirms the large variation in pressure during a cardiac cycle. Evidently, this has an important effect on the wall shear stress and consequently on cardiovascular diseases. Like the maximal velocity, the temporal evolution of the maximal pressure consisted of two pulses. The amplitudes of the pulses were 12.65 kPa and 9.62 kPa. The mean uncertainty between the computational results and the data obtained from the least-square fit was lesser than 8%. Figure 8 depicts the displacement of the blood vessel wall at t = 0.1 s and t = 0.7 s. It is clear that a large deformation of the vessel occurred mainly in the vicinity of the embranchments. However, all the vessels may be subjected to large deformation. The temporal evolution of maximal displacement during a cardiac cycle was investigated. Figure 9 presents the related results. Similarly to the maximal velocity and the maximal pressure, the temporal  
Figure 8: Displacements in the blood vessel using a hyperelastic model; (**a**) t = 0.1 s and (**b**) t = 0.7 s.  
Figure 7: Temporal evolution of the maximal pressure. evolution of the maximal pressure consisted of two pulses. Using the least-square fit, we found that the amplitudes were 8.33 \(\upmu\)m and 6.26 \(\upmu\)m. The mean uncertainty due to the fit was about 3%. It appears clearly that the maximal displacement depended on time and could reach 8 \(\upmu\)m. These values are in accordance with the important shear stress on the vessel. This has an impact on cardiovascular diseases such as stenosis. Our results have the same order of magnitude as previous works [31; 35; 36; 37]. For instance, [PERSON] and [PERSON] obtained a radial deformation ranging from 1.5 to 2.2 \(\upmu\)m and a maximal velocity about 0.6 m/s.