1. Introduction. The **numerical** **simulation** of **Fluid**-**Structure** **Interaction** (FSI) problems is an important topic in wide areas of engineering and medical research. Concerning the latter, of great importance is the modelling of **blood** **flow** in the large **arteries** of the human cardiovascular system, where pulsatile flows combined with a high degree of deformability of the arterial walls together cause large displacement effects that cannot be neglected when attempting to accurately model the **flow** dy- namics of the system. High fidelity computational **fluid** dynamics and structural mechanics solvers based on, for example, the Finite Element Method (FEM) need to be combined in a framework that is challenging both from a mathematical as well as implementation viewpoint. For an overview of cardiovascular modelling techniques we refer to [42, 44] and the book [14]. The complexity and nonlinearity of FSI problems has until recently limited the scope of physically meaningful simulations to just small and isolated sections of **arteries**. When attempting to consider the entire cardiovas- cular system as a complex network of different time and spatial scales, Model Order Reduction (MOR) techniques can accurately and reliably reduce the nonlinear FSI models to computationally more cost-efficient ones.

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The scope of this study is on the **numerical** modelling and **simulation** in one- dimensional FSI **blood** **flow** cases. One-dimensional, incompressible, Newtonian **flow** is considered in this study. Continuity equation, momentum equation and pressure-area constitutive relation are coupled and solved numerically with the employment of compatibility conditions at the boundary nodes. Besides, finite element method with SUPG stabilization formulation is employed as space discretization and first-order forward difference is employed as time discretization. For straight vessel, two types of pressure-area constitutive relations are coupled together with continuity equation and momentum equation, that are, nonlinear elastic model and collapsible model, which are termed as Model 1 and Model 2 respectively. Pressure differences for p-A Model 1 range from 400 Pa to 2500 Pa while pressure differences for p-A Model 2 range from 10 mmHg to 45 mmHg.

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flapping wings, helping us to understand the three-dimensional and time-dependent formation mechanism. The wakes consist of interlocked elliptical vortex rings on each wing. For high reduced frequencies, an upstroke and a downstroke vortex **structure** were produced which contain starting, stopping, root and tip vortices. It was shown that, for decreasing reduced frequency, the upstroke vortex **structure** diffuses and the tip and root vortices of the downstroke dominate the wake. The vortex configurations occurring are not consistent with those of two idealized wake configurations postulated for birds (Rayner, 1979a; Rayner, 1979b) and differ from those recorded for birds (Spedding et al., 2003; Rosén et al., 2004; Hedenström, 2006). The analysed vortex structures are related to the wake of birds with an inflexible wing motion and may occur during acceleration in the forward direction at medium and fast flight speeds. Furthermore, the results show similarities to those of bats (Muijres et al., 2008; Hedenström et al., 2009) and an insect model (Van den Berg and Ellington, 1997a; Van den Berg and Ellington, 1997b). Although the model does not represent real bird geometry and wingbeat kinematics, **flow** characteristics that may appear in the aerodynamic performance of birds are described. The non- appearance of the root vortex during bird flight has been discussed, but the question has not yet been answered. Further **numerical** and experimental studies will be necessary using avian models without a gap and with real bird geometry and structural wing behaviour. In this case, **numerical** representation of a real bird wing for FSI simulations will be desirable to predict the contribution of the FSI to lift and drag production.

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FSIs are increasingly being implemented in the study of vascular diseases and AAAs [7]-[11]. Thubrikar et al. [12] investigated wall stress distributions in three-dimensional models of human abdominal aortic aneurysms based on CT-scans. Chong and How [13] measured the **flow** patterns in an endovascular stent-graft for abdo- minal aortic aneurysm repair. They observed that low velocity regions in the main trunk as well as **flow** separa- tion in the stump region and the curved segment of the iliac limbs were associated with thrombosis in the clini- cal situation. Volodos et al. [14] investigated forces exerted on the rigid, bifurcated EVG. They found that the iliac bifurcation angle, EVG size and **blood** pressure impact the migration forces significantly. In summary, most of the research focused on AAA-wall stress or EVG-lumen **flow** separately, i.e. without consideration of the coupled **fluid**-**structure** interactions between the lumen **blood**, EVG wall, sac stagnant **blood**, and AAA wall. FSI occurs whenever the problem involves the **flow** of **fluid** causing the deformation of a solid **structure**. This deformation, in turn, changes the boundary conditions of the **fluid** field. Because **blood** **flow** and AAA/EVG are coupled in a complicated way, the pulsatile **flow** will affect the movement of the AAA/EVG walls and wall mo- tion in turn influences the **flow** fields, i.e. the lumen and sac **blood**. The dynamic **interaction** between the **flow** and wall may influence the predicted wall stress. Di Martino et al. was the first to consider the dynamics of AAA and installed graft separately and report patient specific wall stress results of coupled **fluid**-solid interac- tion (FSI) **simulation** and suggested that FSI is a useful tool to investigate the physical sight of the **blood** **flow** and aneurysm wall rupture [15].

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Ducoin et al. [27, 32] conducted both a **numerical** and experimental investigation into the effect of flexibility on a stationary foil in uniform **flow**. They compared coupled and uncoupled simulations to assess the effect of FSI with ANSYS CFD and Complete Structural Design. Uncoupled simulations were computed after the CFD simulations were solved with the foil assumed rigid and then the resulting forces were used to compute the structural deflections. Results from experimental visualisation of the foil displacement compared well to the coupled **numerical** simu- lations. The maximum vertical displacement in the coupled **simulation** was 10.7% higher in the coupled case while the change in lift was 10.6% higher. The change in lift was correlated to the change in pressure distribution at the foil tip due to the deflection. This corresponded to a large variation in the minimum pressure at the leading edge of the tip suction side. Ducoin et al. [31] used a RANSE k − ω SST model coupled with a transition model. They found that the structural response is strongly linked to hydrodynamic phenomena such as boundary layer transition and leading edge vortex shedding. An investigation of the effect of pitch veloc- ity revealed that the increased speed influenced the boundary layer transition and hydrofoil loading which resulted in a higher incidence angle before stall [29, 31]. Hysteresis was evident after stall on the downstroke. Fluctuating displacements have been observed when leading edge vortex shedding occurs during stall, for both **numerical** and experimental approaches.

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In deep water engineering, current force is one of the dominate load. However, the ef- fect of **structure** on current force is mainly reflected in the projected area of the vertical direction of **flow**, shape of piles, pile spacing and so on, but the effect of **structure** dis- placement and deformation on the **flow** is ignored. Although this calculation method is easy to use in engineering, it’s not suitable for deep waters engineering. As the **structure** has a great influence on the **flow** of water in deep water because of the large deforma- tion of the **structure**. It is significant to study the **flow** force of the **structure** in deep wa- ter, from the viewpoint of **fluid** **structure** **interaction**.

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hydrodynamic force of the mesh. However, the influence of current on trawl net was usually ignored in the above researches for simplicity. It is necessary to analyze the **interaction** between trawl net and underwater currents. There are several approaches are adopted to investigate the **interaction** the net and current including the velocity reduction method and the porous media method. Comparatively speaking, the velocity attenuation method (Løland, 1993; Berstad et al., 2005; Zhan et al., 2006) is too simple to calculate the **flow** field. Porous media method has been more and more widely used with the improvement of computer ability. Helsley et al. used porous media method to analyze the **flow** field downstream of the cage and the distribution of nutrients (Helsley et al., 2005). Patursson et al. used the porous medium method to analyze the distribution of the **flow** field around the mesh at different angles of attack (Patursson et al., 2006), and gave the method to determine the drag coefficient in the porous medium model (Patursson et al., 2010). Zhao et al. used porous media method to study mesh and cage, and pointed out that the thickness of porous media model has little effect on the **simulation** results (Zhao et al., 2011, 2013a, 2013b, 2015; Bi et al., 2014a, 2014b). However, the porous media method has its own shortcomings, such as the mesh needs to be re-meshed under large deformation. In order to resolve the current net-**fluid** **interaction** problems above, Yao et al. (2016b) proposed a novel hybrid volume method to analyze the **interaction** between the net mesh and the surrounding **flow** field, which was used to calculate the **interaction** between the **fluid** and net cage under large deformation condition.

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Two dimensional **blood** **flow** through tapered **arteries** under stenotic conditions was analysed by Chakravarty and Mandal ( 2000 ). Long et al. (2001) investigated pulsatile **flow** through arterial stenosis numerically. **Numerical** **simulation** of pulsatile **blood** **flow** in straight tube stenosis model was performed to investigate the post stenotic **flow** phenomena. **Flow** features such as velocity profiles, **flow** separation zone and wall shear stress distribution in the post stenotic region are described. Results shows that the formation and development of **flow** separation zone in the post stenotic region are very complex. **Flow** n a catheterised curved artery with stenos was studied by Dash et al. (1999). Here the investigation is done through mathematical analysis. **Blood** is modelled as an incompressible Newtonian **fluid** and the **flow** is assumed to be steady and laminar. The effect of Catheterization on various physiologically important **flow** characteristics is studied for different values of catheter size and Reynolds number of the **flow**.

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Recent advances in CFD and the availability of supercomputers make a realistic computational model of a ureter and peristaltic **flow** possible and it can provide a tool to develop and explore further research. Extensive literature [2, 3, 4] has been devoted to finding mechanical properties of the ureter in both animals and humans and the majority of the studies indicated a viscoelastic behaviour for the ureter. The first attempts to create a model were based on the theoretical analysis and **numerical** solutions for viscosity dominated **flow** through a uniform tube with peristaltic motion [6]. In their study the hydrodynamics of **flow** through a distensible tube of a finite length with a peristaltic motion is discussed. Later, different rates of the frequency of peristaltic motion have been applied to the same model. The dynamics of the upper urinary tract have been investigated extensively and the pressure/**flow** relation in different conditions has been studied. [5, 6, 7, 8, 9, 10]. However, the **fluid** **structure** **interaction** has been not stated in any of the study mentioned above.

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Johnston et al. [15] used several non-Newtonian **fluid** models including Carreau **fluid** model in their investigate on the wall shear stress distribution in **blood** **flow** through stenostic **arteries** and pointed out that Carreau **fluid** model is the best among all the other **fluid** models, since it showed very good agreement with their experimental data. Chang et al [16] numerically analyzed the **fluid**-solid **structure** **interaction** between **blood** and arterial wall and found that the wall shear stress is highest at the throat of the stenosis. Akbar [17] studied the effects of heat and mass transfer in **blood** **flow** through a tapered stenotic artery, modeling **blood** as Carreau **fluid** model. Akbar and Nadeem [18] investigated the steady **flow** of **blood** in a tapered artery with mild stenosis, treating **blood** as Carreau **fluid** model and used perturbation method to obtain the asymptotic solutions.Sankar [19] mathematically analyzed the pulsatile **flow** of **blood** in a tapered artery with overlapping stenosis, treating **blood** as Carreau **fluid** model.

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The aortic arch model used here only partially simulated the real situation because it neglected the effect of the branches of the arch [3,4] and the aortic arch's non- planarity [35]. Because this was an initial attempt; the model was purposely kept simple in order to give an insight into the stress distribution on the aortic arch with **interaction** between a pulsatile **flow** and the wall. Further- more, more than 80% of tears and dissections are at the aorta, not at branches of the arch [26,36]. In the future investigations, the branches will be added to the model. The mechanical properties of **arteries** have been reported extensively in the literature. The type of nonlinearity is a consequence of the curvature of the strain-stress function, which shows that an artery becomes stiffer as the distend- ing pressure increases. Horsten et al. [37] showed that elasticity dominates the nonlinear mechanical properties of arterial tissues. In **numerical** **simulation** of the biome- chanics in **arteries** [6-8,10], the wall was assumed to be an elastic constitutive model. Of course, a non-linear consti- tutive model could provide more biological aspects of the biomechanics. In future work we plan to implement the nonlinear properties of aortic wall within the code by means of user-subroutines.

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The **numerical** solutions obtained from CFD **simulation** are presented and compared in this section followed by results from the static structural analysis. Results for both CFD simulations and static structural analysis are based on non-dimensionalized distances based on a 1:18 scale. Fig. 4 presents the velocity contours for (a) gentle breeze and (b) storm conditions. It is observed that the airflow tends to go around the tree rather than through it. The magnitudes of the **flow** field and ensuing recirculation region are much stronger for the storm condition in comparison to the gentle breeze.

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Different time steps had little effect in the process of dropping, but the leukocyte bounce time and height dif- fers. Leukocyte dropped from the initial position to the bottom of **flow** chamber under the influence of gravity where the distance from the initial position to the **blood** vessel wall is 5.3e−3m. Z-direction movement was more regular than x-direction and y-direction. Leukocyte bounced at position 4.36e−4m instead of landing on the **blood** vessel wall, with jumping height of 1.69e−3m (5e− 4s). After that, leukocyte began to decline to the near vascular wall and then bounce again periodically. The results of other time step were similarity. In 1.25 second of movement, leukocyte bounced six times for time step 5e−4s, five times for time step 6e−4s and te−4s. The time and distance from bottom while leukocyte jumping were shown in Table 2. The average value and variance of leukocyte bounce cycle was shown in Table 3. Clearly, the Jumping cycle time of time step 6e−4s was the big- gest and most stable. However, the height in jumping of time step 7e−4s was becoming bigger and bigger with the increase in number of hitting. Table 4 showed the leu- kocyte jump height in each cycle.

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Mazumdar [7] (Figure-1). From the CFD **simulation**, the vibration, von-Mises stresses on the BMHV components, including the connection pins, valve housing, and leaflet deformation were all obtained. To investigate the effect of vibration on the **blood** **flow** profile, the heartbeat was varied in the CFD model at values of 80, 90, 100, 110, and 120 BPM to obtain the **blood** velocity vector in the **arteries**. As mentioned, the CFD modelled was coupled with the Static **Structure** model to study the effect of varying the heart beats on the valve **structure**. Considering

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According to Clarke [11] there is some debate over whether these vortices should be reflected back into the exterior **flow** or removed from the model completely. Indeed, most of the discussion focused on the vorticity destruction mechanism. Morton [43] denied that the vorticity decayed because of wall diffusion. However, Fage and Jo- hansen [16] noted that in the case of the flat plate, the vortices – physical ones, not to be confused with blob vortices – which approach the rear of the plate will ordinar- ily be deleted by the action of viscosity. Therefore, this latter approach was used by removing the vortices whenever they come within the plate. Another problem rises in the application of this procedure, that is whether or not to remove the vortices during the intermediate substep which are required to have an average over the strength of the nascent vortices. In the current scheme it is of lesser importance than in methods where vortices are shed from all the wall since the number of created vortices is limited to two per full time-step. Thus, even for long-term simulations (e.g. 1000 iterations), it has been found that a maximum of two vortices (e.g. the two nascent vortices) were deleted at each time step. However, there is no limit to the maximum number of vortices to be deleted at each time step, and the deletion scheme is usually used for method with vortices shed continuously along the wall.

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To predict the **flow** characteristics of **blood** through the use of suitable **fluid** model, Cho and Kensey (1989) analyzed several non-Newtonian **fluid** models including Power law and Carreau **fluid** models and compared their results with the experimental results of others. Johnson et al. (2004) propounded that the Carreau **fluid** model is a generalized Newtonian **fluid** model which is most appropriate for modeling **blood** when it flows through narrow **arteries** at low shear rates. They also reported that the Carreau **fluid** model does not over predict the **fluid** behavior near the arterial wall when the **blood** viscosity is high with a significant non-Newtonian influence on the **flow**. Chang et al (2007) observed that Carreau **fluid** model has the high flexibility that it could be used to model **blood** when it flows through both larger diameter and smaller diameter **arteries**. Since, Carreau **fluid** model’s constitutive equation has four physical parameters, one can obtain more detailed information on the **flow** characteristics of **blood** in a wide range of shear rates (Akbar, 2014). Carreau **fluid** model’s constitutive equation can be reduced to represent Newtonian **fluid** behavior when the power law index is unity or the time constant is zero or both. Hence, in view of the aforementioned arguments, it is more appropriate to model the **blood** as Carreau **fluid** when it flows through narrow diameter **arteries**.

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To study the performance of the combination of the combined compact IRBF and the fully coupled approaches in simulating viscous **flow**, we con- sider a transient **flow** problem, namely Taylor-Green vortex [15]. This prob- lem is governed by the N-S equations (40)-(42) and has the analytical solu- tions

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Different reasons can lead to displacement of the critical point upwards or downwards. For example, a slow down of the working-**fluid** stream through the evaporator pipes leads to a lowering of the critical point towards the boiler flame chamber bottom. An extreme situation occurrs in the case of a complete and immediate interuption of the working-**fluid** stream in the pipe when the total liquid working **fluid** evaporates very quickly. This is an anticipated situation in the **numerical** **simulation**.

The pulsatile **flow** of **blood** through stenosed **arteries** is analyzed by assuming the **blood** as a two- **fluid** model with the suspension of all the erythrocytes in the core region as a non-Newtonian **fluid** and the plasma in the peripheral layer as a Newtonian **fluid**. The non-Newtonian **fluid** in the core region of the artery is assumed as a i Herschel-Bulkley **fluid** and ii Casson **fluid**. Perturbation method is used to solve the resulting system of non-linear partial diﬀerential equations. Expressions for various **flow** quantities are obtained for the two-**fluid** Casson model. Expressions of the **flow** quantities obtained by Sankar and Lee 2006 for the two-**fluid** Herschel- Bulkley model are used to get the data for comparison. It is found that the plug **flow** velocity and velocity distribution of the two-**fluid** Casson model are considerably higher than those of the two- **fluid** Herschel-Bulkley model. It is also observed that the pressure drop, plug core radius, wall shear stress and the resistance to **flow** are significantly very low for the two-**fluid** Casson model than those of the two-**fluid** Herschel-Bulkley model. Hence, the two-**fluid** Casson model would be more useful than the two-**fluid** Herschel-Bulkley model to analyze the **blood** **flow** through stenosed **arteries**.

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heat transfer of the thermophoretic **fluid** **flow** past an exponentially stretched surface inserted in porous media in the presence of internal heat generation/absorption, infusion, and viscous dissemination. Afify [8] examined the MHD free convective heat and **fluid** **flow** passing over the stretched surface with chemical reaction. A nu- merical analysis of insecure MHD boundary layer **flow** of a nanofluid past a stretched surface in a porous media was carried out by Anwar et al. [9]. Nadeem and Haq [10] studied the magnetohydrodynamic boundary layer **flow** with the effect of thermal radiation over a stretching surface with the convective boundary conditions.

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