According to the World Health Organization (WHO), cardiovascular diseases (CVDs) remain the leading cause of mortality globally even today timmis2022european,gaidai2023global. Among various CVDs, valvular heart diseases, particularly aortic stenosis, mitral regurgitation, atresia, and rheumatic fever-derived heart valve disorders, account for a significant and growing burden, especially in ageing populations and in low-to middle-income countries where rheumatic fever exists coffey2021global,supino2006epidemiology,carapetis2016acute. These conditions impair the unidirectional flow of blood through the heart, leading to reduced cardiac efficiency and heart failure. If these conditions prevail for a long time and are untreated, they can finally lead to death. While pharmacological interventions may provide symptomatic relief for some time, they are often insufficient in stopping disease progression and restoring the natural functions of the heart valve. In such cases, surgical or transcatheter replacement of the naive valve with an artificial heart valve becomes necessary. The increasing demand for artificial valves is further driven by the rise in life expectancy, the success of early diagnostic methods, and the growing acceptance of minimally invasive techniques such as Transcatheter Aortic Valve Replacement (TAVR). Therefore, the development of biocompatible, durable, and hemodynamically efficient artificial valves is a critical need in modern cardiovascular healthcare, aiming not only to restore valve function but also to minimise complications such as thrombosis, hemolysis, and structural deterioration \citep{kumar2023scientific}.

Artificial heart valves are broadly categorised into two, namely, artificial mechanical heart valves (MHVs) and bioprosthetic or tissue artificial heart valves \citep{vongpatanasin1996prosthetic}. Artificial MHVs, made from biocompatible materials like metals (e.g., titanium, carbon) and polymers (e.g., ultra-high molecular weight polyethylene (UHMWPE), polytetrafluoroethylene (PTFE)), offer long-term durability and can perform hemodynamic function effectively for several years \citep{gott2003mechanical}. Despite these advantages, MHVs come with a significant drawback, which is the need for lifelong anticoagulation therapy to prevent thromboembolic complications due to their thrombogenic nature. On the other hand, bioprosthetic artificial heart valves, typically made from animal tissues such as bovine or porcine pericardium, offer better biocompatibility and usually do not require long-term anticoagulation \citep{manji2014bioprosthetic}. However, these valves are susceptible to structural valve degeneration (SVD) over time, often necessitating reoperation within 10-15 years. This degradation occurs even more rapidly in younger patients due to faster rates of calcification and tissue remodelling. Consequently, despite the burden of using life-long anticoagulation, MHVs remain the preferred choice for younger patients, as they offer superior long-term durability and significantly reduce the likelihood of multiple valve replacement surgeries over a lifetime \citep{tillquist2011cardiac}.

The artificial MHVs are further broadly classified into three categories, mainly depending on their structural design: ball and cage, tilting disc, and bileaflet. The ball and cage valve was probably the first kind of MHV introduced for clinical use, making it historically significant. However, unfortunately, this valve causes significant turbulent flow and high pressure drops because of its unique structural design, comprising a spherical ball enclosed in a cage, leading to an increased risk of hemolysis and thrombosis de2020diamond,wang2025cavitation. Subsequently, the tilting disc valve was introduced, which improved upon the drawbacks associated with the ball and cage valve by allowing more central flow. Despite the improvement in the structural design, this valve also has drawbacks, such as producing asymmetric flow patterns and high shear stresses manning2008detailed,govindarajan2010two,cenedese2005laboratory. Therefore, further investigations have led to the development of a new class of MHV, the bileaflet mechanical heart valve (BMHV), the most widely used advanced modern design MHV today dewall2000evolution,gott2003mechanical. This heart valve offers almost near-physiological flow characteristics, resulting in lower flow resistance and pressure gradients and a larger effective orifice area compared to older structural designs of MHVs, such as ball and cage and tilting disc. Still, this valve also has several limitations, which have been reported both in lab-scale studies and in clinical use chambers2013clinical,saito2016bileaflet,johnson2019thirty. For instance, despite having better hemodynamics than older MHVs, bileaflet valves still cause blood flow disturbances resulting from non-physiological flow patterns around the valve. These in turn generate high shear regions and leakage jets, which can ultimately lead to blood cell damage or promote the hemolysis process yoganathan2004fluid,cheng2004three,dasi2009fluid.

The presence of these inherent limitations in BMHVs has prompted further recent research efforts, with a primary focus on surface modifications, aiming at improving their hemocompatibility and biocompatibility as well as reducing associated complications \citep{yousefi2024surface}. These modifications, comprised either surface coatings or micro-nano texturing, ultimately led to the development of superhydrophobic surfaces, focusing on reducing the risks of blood clots by minimising blood cell adhesion to the valve surface. Several studies have demonstrated various surface modification techniques for MHVs and examined their interactions with blood components, such as platelets and red blood cells, to show improved biocompatibility \citep{yousefi2024surface}. However, despite the critical role that hemodynamics plays in determining the functional performance of artificial MHVs, only a limited number of investigations have focused on how these superhydrophobic surface modifications influence the associated flow dynamics. Among a few, Bark et al. \citep{bark2017hemodynamic} were the ones who applied a commercial superhydrophobic coating (named Ultra-Ever Dry) on a St Jude Medical BMHV and compared the hemodynamic performance as well as biocompatibility in the absence of coating. They found a lower tendency in the adhesion of blood cells (such as platelets and leukocytes) on the surface of the coated valve compared to the uncoated one. Furthermore, in their study, the hemodynamic parameters of clinical importance, such as pressure drop and vorticity intensity, were also found to be lower in the case of the coated valve, although the influence was minimal. In contrast, Hatoum et al. \citep{hatoum2020impact} reported that the application of a superhydrophobic surface coating on a 3D-printed BMHV led to an increase in both Reynolds shear stresses (RSS) and viscous shear stresses compared to the uncoated valve. Therefore, they concluded that the superhydrophobic coating does not necessarily improve the hemodynamic performance of a BMHV. They came to this conclusion after performing both experiments in a pulse duplicator and computational fluid dynamics (CFD) simulations under realistic pulsatile physiological conditions seen in a naive heart valve.

Therefore, the aforementioned discussion reveals a noticeable gap in the investigation of hemodynamics associated with superhydrophobic MHVs, particularly the bileaflet type. Moreover, the limited existing studies have reported contradictory outcomes, making it difficult to draw definitive conclusions about the hemodynamic benefits of superhydrophobic surfaces. Therefore, the present study aims to address this knowledge gap by systematically evaluating whether superhydrophobicity can indeed increase the hemodynamic performance of BMHVs and thereby reduce turbulent flow intensity and blood cell damage. To achieve this aim, we perform extensive large-scale direct numerical simulations (DNS) of a BMHV under physiologically realistic pulsatile flow conditions, incorporating slip boundary conditions to model the effects of superhydrophobic surfaces. The results are then directly compared and discussed to those of conventional no-slip boundary conditions at different stages of a cardiac cycle. Furthermore, this study considers both fully functional and varying degrees of dysfunctional valve conditions, allowing us to assess the role of surface slip across a range of clinical scenarios of an MHV. The hemodynamic insights gained from this investigation are expected to significantly guide the design and development of next-generation surface-modified MHVs for achieving better biocompatibility and performance.

In the present study, as discussed in the preceding section, we investigate the hemodynamics past a BMHV, which is widely used in clinical heart valve replacements emery2005st,baudet1995long. Specifically, we focus on a St. Jude Medical (SJM) BMHV of 23 diameter, incorporating superhydrophobic leaflet surfaces to induce slip effects under both functional and dysfunctional conditions, as shown in Fig. 1. This particular valve model has been extensively employed in prior experimental and numerical studies of BMHV hemodynamics Ge2005, Yun2014. A detailed description of the computational setup has been reported in our recent works chauhan2024hemodynamics,chauhan2024influence; therefore, only key aspects relevant to the current problem formulation are outlined here. The complete computational domain is illustrated in Fig. 2(a), while zoomed-in views of the fully functional (0% defective), 50% defective, and 100% defective valve configurations are shown in Figs. 2(b)–(d), respectively. The two leaflet surfaces are distinctly labelled to highlight regions of imposed slip, which are referenced later in the results and discussion (e.g., see Fig. 2(b)).

Furthermore, the non-Newtonian nature of blood is accounted for in this study by employing the power-law rheological model, which effectively captures its shear-thinning behaviour. The model parameters are obtained by fitting experimental shear viscosity data of real and whole blood, as illustrated in Fig. 3. The imposed volumetric flow rate waveform represents one cardiac cycle of approximately 860 , corresponding to a heart rate of 70 beats per minute, and is shown in Fig. 4 Yun2014. All simulations are conducted over three cardiac cycles to ensure periodicity, and the results discussed herein pertain to the last (third) cycle. Additionally, the angular displacement of the valve leaflets throughout the cardiac cycle is plotted on the secondary axis of the same Fig. 4. The current analysis focuses specifically on the time interval when the valve is fully open, during which both leaflets of the fully functional SJM valve are held stationary at a fixed angle of with respect to the xz-plane, as indicated in Fig. 4.

This study performs DNS to investigate the hemodynamics past the BMHV. In doing so, we solve the following set of governing equations by assuming the blood flow to be incompressible in nature:\\

Continuity equation:

Momentum equation:

In the above equations, is the velocity vector, is the position vector, is the time, is the pressure, is the extra-stress tensor, and is the density of blood (taken as ). The extra-stress tensor, , is calculated using the following constitutive relation:

Where is the strain-rate tensor and is the apparent shear viscosity of blood. In the present study, we have adopted the power-law rheological model to calculate the apparent viscosity of the blood, as follows:

Where , , , and refer to the blood consistency coefficient, flow behaviour index, zero-shear viscosity, and infinite (or high) shear viscosity, respectively. Figure 3 displays the corresponding fitting curve of the power-law model alongside the experimental shear rheological data of blood. From Eq. 4, it is evident that the power-law model predicts an infinite and a zero value of the apparent shear viscosity in the limits of low and high values of , respectively. Such predictions are not physically realistic and represent a fundamental drawback of this non-Newtonian rheological model. To address this limitation, a cut-off strategy is employed in the present study. Specifically, for low values of , the apparent viscosity is bounded as Pa·s, and for high values of , it is limited as Pa·s, as schematically shown in Fig. 3. This technique of enforcing bounded apparent viscosity is often used in numerical simulations that employ power-law-type rheological models.

The finite volume method (FVM) based open-source computational fluid dynamics (CFD) software OpenFOAM (version 7) \citep{openfoam} has been employed to solve the governing equations, namely, the continuity equation (Eq. 1), the momentum equation (Eq. 2), and the power-law constitutive model (Eq. 4), as outlined in the preceding section. Specifically, the transient solver pisoFoam has been used, which incorporates pressure–velocity coupling via the Pressure Implicit with Splitting of Operators (PISO) algorithm PISO2001. This algorithm is particularly suited for unsteady flow simulations and is known to be more efficient than the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm in such contexts. A detailed account of the discretisation strategies used for various terms in the governing equations is available in our recent works chauhan2024influence,chauhan2024hemodynamics, and hence is not repeated here. All numerical simulations were carried out over three cardiac cycles, i.e., up to , using 400 processors in parallel. Each simulation required approximately 72–84 hours of wall-clock time, amounting to about 28,000-34,000 total computational hours, on a supercomputing cluster equipped with Intel Xeon Platinum 8268 (2.9 GHz) processors and 960 GB RAM per node.

Furthermore, in the present study, we employ the following set of boundary conditions:

At the inlet boundary: For velocity, the pulsating flow rate waveform as prescribed in Fig. 4 was specified. A zero-gradient value was assigned for the pressure.

At the outlet boundary: Outflow boundary condition was used for the velocity. For pressure, we provide an average value of systolic () and diastolic () pressure, i.e., .

At the surface of leaflets: On these surfaces, a slip boundary condition was imposed for the velocity field. This type of boundary condition simulates stress-free sliding of blood along the leaflet surfaces, allowing tangential motion without resistance, unlike the no-slip condition, where the fluid adheres to the surface and the velocity vanishes. Mathematically, it can be expressed as and , where denotes the local normal direction to the surface. Additionally, a zero-gradient boundary condition was applied for the pressure at these surfaces, implying no pressure variation normal to the boundary.

At the artery walls: For velocity, we impose the standard no-slip condition and the zero-gradient condition for pressure.

Furthermore, thorough and systematic grid and time-step size convergence studies were carried out to determine the optimum grid density and time-step size for the present simulations. These studies are essential for balancing the accuracy of the results with the computational cost, a critical consideration in any CFD-based investigation. The complete details of the grid generation, time-step selection, and their respective convergence analyses are available in our previous study \citep{chauhan2024influence}, and are therefore not repeated here. However, some salient features, particularly about the grid structure and its generation, are briefly highlighted here. The initial grid structure for the computational domain was generated using the blockMeshDict utility in OpenFOAM. Subsequently, the snappyHexMeshDict utility was employed to create the complex leaflet geometries in the valve region and to locally refine the mesh in regions with steep gradients in velocity, pressure, and stress. The final mesh used in this study consists of polyhedral cells, which are known to enhance mesh quality and improve numerical stability due to their reduced sensitivity to stretching near the artery walls and leaflet surfaces. Additionally, the numerical solver and associated parameters—including grid resolution, time-step size, and solver tolerances—were selected systematically to ensure robustness and reliability without compromising accuracy. A comprehensive validation of the present numerical framework against existing in vitro experimental data and in silico numerical results of Yun et al. Yun2014 for flow past an SJM MHV has already been reported in our prior study \citep{chauhan2024hemodynamics}, and therefore, this exercise is also not repeated in this study.

The solution of the governing equations as outlined in Sect. 2 together with the boundary conditions (described in Sect. 3) is obtained in terms of the primitive flow variables, namely the velocity and pressure fields, throughout the computational domain. These fields are subsequently post-processed to compute several clinically relevant hemodynamic parameters. These include the wall shear stress () acting on the solid boundaries, the non-dimensional drag coefficient () on the surfaces of the two valve leaflets, the von Mises stress (), and the blood damage index (). All these parameters will be used to provide a quantitative comparison between the slip and no-slip conditions imposed on the valve leaflet surfaces. The following expressions are used in the present study to evaluate , , \citep{han2022models}, and \citep{garon2004fast}:

In the above equations, is the shear-rate (tangential component) on the surface of the solid artery wall, is the average velocity ( 0.148 ), is the area of the leaflet projected in the direction of flow, is the magnitude of the drag forces acting on the leaflet, is the Kronecker delta function, is the outward unit normal vector drawn on the surface of the leaflet, is the unit vector in the direction of flow, is the surface area of the leaflet, and is the average linear damage evaluated using the following formula proposed by Garon and Farinas \citep{garon2004fast}:

Where is the rate of hemolysis production per unit time, and is the volume of the entire computational domain.

As mentioned earlier, both no-slip and slip boundary conditions are imposed on the leaflet surfaces to enable a qualitative and quantitative comparison between the two cases. The pulsating flow rate waveform (see Fig. 4) is applied at the entry (inlet) plane of the region A, i.e., ventricular side chamber, which mimics the physiological variation of a single cardiac cycle lasting 860 , corresponding to a heart rate of 70 beats per minute Yun2014. All simulations are conducted over three cardiac cycles, with results primarily analysed and presented for the final periodic cycle at three key time instances: mid-acceleration phase (T1), peak systolic phase (T2), and mid-deceleration phase (T3), also indicated in Fig. 4. The post-processing and analysis primarily focus on the following hemodynamic and mechanical metrics: velocity magnitude contours, the Kymograph of axial velocity component, von Mises stress distributions, pressure drop between the inlet and outlet planes, WSS, , and , among others. These results are thoroughly presented and discussed to gain insight into the impact of slip effects arising from superhydrophobic modifications on the valve leaflets. It is important to note that the influence of blood's non-Newtonian characteristics on flow past a BMHV has been examined in detail in our previous work \citep{chauhan2024influence}. Therefore, in the present study, we primarily report results obtained using the power-law model, as blood predominantly exhibits shear-thinning behaviour.

We begin the discussion of the results with the velocity magnitude contours obtained at different surfaces of the heart valve leaflets (shown in sub-Fig. 2(b)) when the leaflets are treated with the slip boundary condition. Figure 5 presents the contours at three representative time instances of the cardiac cycle and for three different extents of defect conditions in the heart valve, namely, 0% (fully functional), 50%, and 100%. During the mid-acceleration phase of the cardiac cycle (i.e., = T1), for a fully functional valve, surface-1 (the one facing the valve housing) exhibits finite velocity magnitude, indicating the presence of slip except near the long axis of the leaflets, where low-velocity regions appear. In contrast, surface-2 shows a broader region of low-velocity magnitude on both leaflets compared to surface-1. At the peak systolic stage (i.e., = T2), the velocity magnitude on surface-1 increases compared to T1. However, unlike the smooth contours seen during T1, distinct high-velocity magnitude patches emerge, suggesting the onset of more complex flow behaviour. On surface-2, a visibly chaotic pattern of the velocity magnitude is observed on both leaflets, pointing toward the development of a turbulent-like state at this time of the cardiac cycle. This behaviour is attributed to the increased strength of blood flow at peak systole, which increases both velocity magnitude and its gradients, thereby intensifying slip effects and promoting turbulent-like flow dynamics, which will be further analysed in the subsequent discussion. During the mid-deceleration phase (i.e., = T3), the velocity magnitude decreases due to the adverse pressure gradient acting opposite to the initial flow direction. Surface-1 of both leaflets exhibits more extensive low-velocity regions than in the earlier phases (T1 and T2). However, surface-2 continues to display a chaotic pattern similar to that observed at T2, albeit with reduced intensity in the high-velocity zones. Notably, at any given time instance, T1, T2, or T3, the velocity magnitude distributions on the two leaflet surfaces are asymmetric, reflecting a transition from laminar to turbulent-like flow under these inflow conditions.

As one of the two leaflets (leaflet-2) becomes dysfunctional with a defect, the non-uniformity in the velocity magnitude distribution becomes noticeably more pronounced on surface-1 of leaflet-1, even during the T1 stage of the cardiac cycle. Due to the obstruction in the blood flow area caused by the partial dysfunction of leaflet-2, the majority of the blood is diverted through the region between leaflet-1 and the valve housing. This results in a reduced velocity magnitude in the vicinity of leaflet-2. Consequently, both surfaces of leaflet-2 exhibit relatively low velocity magnitudes, indicating diminished slip effects on these surfaces across all time instances. In contrast, the highest velocity magnitude is observed on surface-2 of leaflet-1, as shown in the third row of Fig. 5. Under the fully dysfunctional condition ( defect), the velocity magnitude distribution becomes even more non-uniform. Both surfaces of leaflet-2 display very low velocity magnitudes due to the complete blockage of blood flow in its vicinity, as evident from the second and fourth rows of the defect case in Fig. 5. On the other hand, the slip effect becomes more significant on the surfaces of leaflet-1, with surface-2 experiencing a particularly increased effect compared to the fully functional and defective scenarios, again supported by the third row of Fig. 5. Overall, with the introduction of slip conditions, these velocity distribution changes on the leaflet surfaces suggest alterations in the shear and normal forces acting near the heart valve. These changes are expected to have a significant impact on clinically important parameters, such as WSS, BDI, etc., which will be discussed in detail in the following section.

Figure 6 presents the surface distribution of velocity magnitude across eight representative planes (P1–P8) within the cardiovascular domain at (mid-acceleration stage of the cardiac cycle), considering both no-slip and slip boundary conditions on the heart valve leaflets under varying levels of dysfunction. Among these planes, P1-P6 are aligned normally to the -direction, where P1 lies upstream of the valve, P2 passes through the valve region, and P3-P6 are located downstream. Planes P7 and P8 are oriented normally to the - and -directions, respectively. For the fully functional valve with the no-slip condition (sub-Fig. 6(a)), a relatively uniform velocity field is observed in the central regions of the ventricular and aortic chambers (regions A and D in Fig. 2(a)), particularly evident in planes P1 and P4-P6. In contrast, velocity non-uniformity is notable in plane P3, which intersects the sinus region (region C), primarily due to flow separation near the trailing edges of the leaflets. This separation leads to the formation of low-velocity vortical structures. Within the valve region (plane P7), three distinct jets are visible: a dominant central jet (between the two leaflets) and two weaker lateral jets (between each leaflet and the valve housing), with the central jet exhibiting the highest velocity magnitude. Plane P8 reveals a concave high-velocity zone near the valve region. Under the slip condition for the same fully functional case (sub-Fig. 6(b)), the velocity distribution remains qualitatively similar to the no-slip case, with minor differences. Notably, the low-velocity zone in the sinus region appears slightly diminished, as seen in plane P8, indicating a subtle enhancement of flow due to slip. More pronounced differences emerge in the presence of leaflet dysfunction. For the 50% defect scenario, the high-velocity zone in plane P8 is more intense in the no-slip case (sub-Fig. 6(c)) compared to the slip case (sub-Fig. 6(d)). Additionally, a high-velocity region develops near the tip of leaflet-2 and extends into the sinus region, with a larger spatial extent under the slip condition, as evident in plane P7. In this configuration, most of the blood flows centrally between the leaflets, forming a prominent central jet. With a 100% defect, this central jet bends towards the leaflet-2 under the no-slip condition (sub-Fig. 6(e)), whereas such bending is not observed in the slip case (sub-Fig. 6(f)), as illustrated in plane P7. These observations highlight how leaflet dysfunction and boundary conditions collectively influence the velocity distribution and jet dynamics within the valve and downstream regions. At (systolic peak of the cardiac cycle), significant differences in velocity magnitude distribution are observed between the no-slip and slip conditions, even in the case of a fully functional heart valve, as shown in sub-Figs. 7(a) and (b), respectively. Notably, under the no-slip condition, the central jet formed between the two leaflets extends well into the sinus region, whereas it remains largely confined within the valve region under the slip condition. This distinction is clearly visible in both the P7 and P8 planes. However, the central jet appears more disrupted near the leading edge in the slip case compared to the no-slip one, suggesting increased flow instability due to the reduced wall friction. As leaflet-2 becomes dysfunctional, the flow is redirected predominantly through the region between leaflet-1 and the valve housing, resulting in the formation of a prominent lateral jet. Interestingly, this jet is shorter in the no-slip case than in the slip case. For example, this difference is clearly seen in the P4 plane of the sub-Figs. 7(e) and (f) corresponding to the 100% defect condition. A similar trend continues during the mid-deceleration phase of the cardiac cycle (), as illustrated in Fig. 8. At this time instance, jet structures become highly irregular and disrupted regardless of whether slip or no-slip boundary conditions are imposed on the leaflet surfaces. The high-velocity regions extend further downstream at T3 than at T1 or T2, especially when leaflet-2 is fully dysfunctional, as seen in sub-Figs. 8(e) and (f). Additionally, the flow field exhibits a much more chaotic and irregular velocity distribution due to the onset of flow reversal and backflow phenomena characteristic of this decelerating phase.

To further investigate the temporal evolution of the flow field, particularly in the region proximal to the heart valve, we analyze the kymographs of axial velocity along a probe line for both no-slip (first column) and slip (second column) boundary conditions under the same three functional conditions discussed previously, Fig. 9. For the fully functional heart valve, the axial flow field exhibits greater temporal disruption under the slip boundary condition (sub-Fig. 9(b)) when compared to the no-slip case (sub-Fig. 9(a)). This observation is consistent with the spatial distribution of velocity magnitude presented across various cross-sectional planes in the cardiovascular model, especially during the peak systolic phase (Fig. 7). Furthermore, in the no-slip case, the axial velocity field maintains relatively higher magnitudes beyond the flow deceleration phase, particularly after 2150 , than in the slip case. This behaviour persists across the dysfunctional valve scenarios as well. However, for the dysfunctional cases, the overall intensity of temporal fluctuations appears more pronounced under the no-slip condition. Overall, these findings suggest that momentum retention is better preserved under no-slip conditions, irrespective of the functional conditions of the heart valve. The slip boundary promotes increased unsteadiness and temporal disruption in the axial flow profile for the fully functional case, whereas the no-slip boundary does the same for the dysfunctional case, at least in the proximity of the heart valve.

Figure 10 depicts the isosurface of the Q-criterion within the present flow field, particularly in the surroundings of the heart valve. This is a commonly used visualisation technique in fluid dynamics for identifying and visualising vortices in a flow field, especially in turbulent or transitional flows. The Q-criterion is computed as: , where is the vorticity tensor, represent the strain-rate tensor, and signifies to their corresponding magnitudes. Physically, the Q-criterion characterises the local balance between rotational (vortical) and extensional (strain) components of the velocity gradient. Regions with positive Q-values correspond to zones where the local rate of rotation dominates over the strain rate, indicative of vortex cores where fluid elements tend to swirl around a common axis. Conversely, negative Q-values denote areas where strain is dominant, leading to elongational deformation rather than coherent rotation. These contrasting flow features are visualised in the first two and last two columns of Fig. 10, respectively. In the current analysis, we show Q-criterion isosurfaces only for the fully functional heart valve, captured at three characteristic stages of the cardiac cycle: T1 (mid-acceleration, first row), T2 (peak systole, second row), and T3 (early deceleration or backflow phase, third row). Results are compared between the no-slip and slip boundary conditions to assess the influence of wall shear constraints on the formation and evolution of vortical structures. During the T1 stage, small-scale vortices begin to emerge near the valve leaflets. These structures are more densely packed and intense in the no-slip condition (sub-Fig. 10(a)) compared to the slip condition (sub-Fig. 10(b)), highlighting the suppressive effect of slip boundaries on near-wall vorticity generation. As the flow accelerates and reaches peak systole (T2 stage), these vortices grow in size and extend into the sinus region, driven by the increased flow rate. Notably, the slip case exhibits a greater number of vortices between the two valve leaflets (sub-Fig. 10(f)) than the no-slip counterpart (sub-Fig. 10(e)), suggesting increased swirling motion in this region due to reduced near-wall viscous dissipation. At the T3 stage, flow reversal initiates retrograde motion, resulting in a pronounced increase in vortex formation throughout the sinus. Both slip and no-slip cases display a large population of vortical structures; however, the no-slip condition produces more fragmented and spatially irregular vortices (sub-Fig. 10(i)) compared to the more coherent structures observed under slip conditions (sub-Fig. 10(j)). This fragmentation may be attributed to higher shear gradients and flow deceleration effects near the walls. In addition to rotational structures, regions of elongational or extensional flow, characterised by negative Q-values, are also observed. At the T1 stage, elongational strain is more prominent in the no-slip case (sub-Fig. 10(c)) than in the slip condition (sub-Fig. 10(d)), particularly within the interleaflet gap. As the cardiac cycle progresses, the extent of these elongational zones increases, spreading into the sinus region. However, at the later stages (T2 and T3), no substantial differences in the distribution of strain-dominated regions are observed between the two boundary conditions, suggesting that the influence of wall slip becomes less significant during periods of deceleration and flow reversal.

Figure 11 illustrates the pressure distribution along the centerline of the valve and sinus regions at three distinct stages of the cardiac cycle (T1, T2, and T3) under various valve functional conditions, considering both no-slip and slip boundary conditions. At the T1 stage, the upstream pressure is consistently higher across all conditions due to the deceleration of blood flow near the valve leaflet. This pressure increases further with increasing valve defect, attributed to greater obstruction of central flow between the leaflets. As the flow moves toward the sinus region, pressure decreases due to vortex formation, followed by a partial recovery farther downstream. The overall pressure drop becomes more pronounced with increasing valve defect. A similar trend is observed during the T2 stage, although the upstream pressure is lower than in T1. Nevertheless, the total pressure drop across the valve increases further, regardless of valve function. In contrast, the T3 stage exhibits a reversed pressure gradient: downstream pressure exceeds upstream pressure due to a rapid deceleration of flow, resulting in backflow. However, the pressure variation during T3 is less significant than in T1 and T2. Notably, the influence of boundary conditions (no-slip and slip) on the centerline pressure distribution is minimal. Both cases follow nearly identical trends, with negligible differences, particularly on the upstream side of the valve.

The distribution of von Mises stress on the valve leaflet surface (calculated as per the formula presented in Eq. 7), as shown in Fig. 12, is analysed at various time instances of the cardiac cycle and under different defect levels, considering both no-slip and slip boundary conditions. This stress distribution is critical for multiple aspects of heart valve design. From a material development and selection perspective, von Mises stress provides insight into regions susceptible to mechanical fatigue, which is vital for ensuring the long-term durability of the valve. Additionally, regions of higher stress are linked to high shear and tensile forces, which can damage red blood cells and activate platelets, potentially triggering thrombus formation. Surface areas with increased high von Mises stress are also prone to micro-damage or cracking, facilitating protein adhesion and platelet aggregation, key factors in thrombogenicity. Significant variations in von Mises stress are observed across different phases of the cardiac cycle, valve dysfunctions, and boundary conditions. For a fully functional valve, the application of slip boundary conditions consistently results in lower stress magnitudes on the leaflet surface across all cardiac stages. During , (mid-acceleration phase), the stress is predominantly localised around the leaflet edges and hinge region under no-slip conditions (sub-Fig. 12(a)). In contrast, under slip conditions, the stress in these areas is markedly reduced (sub-Fig. 12(b)), and the top and bottom flat surfaces of the leaflet also exhibit lower stress. As the cardiac cycle progresses to (peak systolic phase), the von Mises stress increases considerably on the flat surfaces of the leaflet under no-slip conditions (sub-Fig. 12(c)), corresponding to increased blood velocity and flow-induced forces. However, this increase is significantly mitigated under slip conditions, particularly on the downstream side of the leaflet (sub-Fig. 12(d)). In the mid-deceleration phase of the cardiac cycle, the overall stress magnitude decreases, yet its spatial distribution becomes more non-uniform and irregular, especially under slip conditions (sub-Fig. 12(f)). When leaflet defects are introduced, the trends in stress variation due to boundary conditions and cardiac cycle stages generally are seen to be the same. However, the stress distribution becomes notably asymmetric between the two leaflets. In defective valves, the upper leaflet consistently exhibits higher von Mises stress compared to the lower leaflet at all stages of the cardiac cycle. Moreover, as the defect severity increases (e.g., from 0% to 100%), the peak stress on the upper leaflet increases substantially, regardless of the boundary condition or cardiac stage. This is clearly illustrated in sub-Figs. 12(c) and (o), which compare the no-slip cases at the T2 stage for 0% and 100% defect scenarios, respectively.

Figure 13 depicts the spatial distribution of WSS on the surfaces within the valve housing and sinus regions across different stages of the cardiac cycle. It is a hemodynamically critical parameter, playing a vital role in the interactions of blood cells with the artery wall and the chances of their damage [yap2012experimental,yap2012experimental2]. An increased WSS level can induce mechanical damage to red blood cells (hemolysis) and activate platelets, potentially promoting thrombotic events. On the other extreme, low or oscillatory WSS regions can also promote platelet adhesion, protein accumulation, and aggregation, thereby increasing the risk of thrombus or clot formation. Therefore, characterising the WSS distribution near and downstream of prosthetic valves is essential for evaluating the thromboembolic safety of the device. At the early systolic phase (T1 stage) of the cardiac cycle, a distinct `M'-shaped high-WSS region emerges near the hinge area of the valve, particularly within the valve housing. The magnitude of this stress region is notably higher under the slip boundary condition (sub-Fig. 13(b)) than under the no-slip condition (sub-Fig. 13(a)). As the cardiac cycle advances to the peak systolic T2 phase, the WSS further increases in the hinge and housing regions. Additionally, considerable finite WSS values start to appear within the sinus region, especially near its distal end, as observed in sub-Figs. 13(e) and(f) for the no-slip and slip conditions, respectively. During the deceleration phase (T3 stage), WSS magnitudes decline in the valve housing area but remain comparatively higher for the slip condition. At this stage, the high WSS regions shift downstream and become more prominent in the distal sinus region. Under defective valve conditions (e.g., with leaflet dysfunction), the WSS distribution alters significantly. The magnitude increases throughout the cardiac cycle, regardless of the stage. In the valve housing, the previously localised high-WSS patch observed in fully functional conditions expands significantly. Moreover, the increased WSS regions extend downstream beyond the sinus, with their extent and intensity increasing progressively from the T1 to the T3 stages of the cardiac cycle. All in all, the WSS exhibits substantial spatial and temporal variation across the cardiac cycle. Its magnitude and distribution are strongly influenced by the valve's functional condition and, to a lesser extent, by the nature of the leaflet boundary, i.e., slip or no-slip. These insights are essential for optimising valve design and predicting sites susceptible to thrombogenesis or mechanical damage.

begin{table*} \caption{Parameter values acquired for different dysfunctions, with both no-slip (NS) and slip (S) type boundary conditions (BC) implemented on the surface of the heart valve leaflets. The three time instances reported here are: (mid-acceleration phase), (peak systolic phase), and (mid-deceleration phase).} \begin{center} \scalebox{0.9}{ \begin{tabular}{cccccc} \hline Parameter & Time instance & Type of BC & defect defect defect \\ ()}} & \multirow{2}{*}{T1} & NS & 42.866 & 45.244 & 50.443 \& & S & 42.779 & 45.188 & 50.936 \\[2pt] & \multirow{2}{*}{T2} & NS & 9.938 & 17.593 & 31.299 \& & S & 9.842 & 17.069 & 31.848 \\[2pt] & \multirow{2}{*}{T3} & NS & 71.353 & 70.164 & 68.281 \& & S & 71.292 & 69.679 & 68.591 \\ [5pt] \multirow{6}{*}{\parbox{1.3cm}{ }} & \multirow{2}{*}{T1} & NS & 57.203 & 109.919 & 198.163 \& & S & 51.291 & 261.704 & 737.050 \\[2pt] & \multirow{2}{*}{T2} & NS & 131.391 & 316.748 & 565.004 \& & S & 138.591 & 733.930 & 1987.188 \\[2pt] & \multirow{2}{*}{T3} & NS & 15.106 & 50.312 & 72.813 \& & S & 19.133 & 139.235 & 295.450 \\[5pt] \multirow{6}{*}{\parbox{1.3cm}{ \\ ()}} & \multirow{2}{*}{T1} & NS & 42.563 & 93.169 & 402.758 \& & S & 66.947 & 109.429 & 420.918 \\[2pt] & \multirow{2}{*}{T2} & NS & 167.334 & 244.113 & 763.748 \& & S & 203.547 & 269.177 & 742.943 \\[2pt] & \multirow{2}{*}{T3} & NS & 39.707 & 103.536 & 224.890 \& & S & 71.931 & 117.621 & 289.028 \\[5pt] \multirow{6}{*}{\parbox{1.3cm}{ \\ ()}} & \multirow{2}{*}{T1} & NS & 4.407 & 4.540 & 4.718 \& & S & 4.406 & 4.541 & 4.758 \\[2pt] & \multirow{2}{*}{T2} & NS & 5.529 & 6.005 & 6.541 \& & S & 5.553 & 5.984 & 6.682 \\[2pt] & \multirow{2}{*}{T3} & NS & 3.066 & 3.522 & 4.788 \& & S & 3.094 & 3.591 & 4.849 \\[5pt] \multirow{6}{*}{\parbox{1.3cm}{BDI ()}} & \multirow{2}{*}{T1} & NS & 0.063 & 0.094 & 0.177 \& & S & 0.058 & 0.077 & 0.123 \\[2pt] & \multirow{2}{*}{T2} & NS & 0.161 & 0.322 & 0.693 \& & S & 0.148 & 0.231 & 0.450 \\[2pt] & \multirow{2}{*}{T3} & NS & 0.071 & 0.125 & 0.283 \& & S & 0.071 & 0.109 & 0.250 \\[2pt] \hline \end{tabular}} \label{table:Pulsatile} \end{center}\end{table*}

Although the surface distributions of different quantities, such as velocity magnitude, WSS, etc., provide key insights related to blood flow characteristics around the BMHV, it is the surface-averaged metrics that have greater clinical relevance. These averaged values are particularly useful for evaluating the influence of valve leaflet surface properties (i.e., no-slip vs. slip conditions) and varying degrees of valve dysfunction on the overall hemodynamic performance of a BMHV. Accordingly, Table 1 presents several clinically important, post-processed parameters under two leaflet surface boundary conditions and three functional states (0%, 50%, and 100%) of the valve across three stages of the cardiac cycle, i.e., T1, T2, and T3. First of all, the pressure drop magnitude is hardly affected by the valve leaflet surface condition, i.e., whether it is slip or no-slip. It is minimal in the T2 stage, whereas maximal in the T3 stage of the cardiac cycle, irrespective of the leaflet surface condition. As the defect severity in the heart valve increases, the pressure drop rises in the T1 and T2 stages, while remaining nearly unchanged in the T3 stage. The values of drag forces () acting on the heart valve leaflets are seen to be higher in slip conditions than in the no-slip conditions, and the difference between the two conditions increases drastically as the valve becomes dysfunctional. For instance, it is around 5% for a fully functional valve, which increases to around 70% for a 100% defective valve at the same peak systolic phase, i.e., at . The maximum WSS is highest during the time instance T2 and is consistently higher under the slip condition compared to the no-slip condition. For instance, in the case of a fully functional valve (0% defect), the maximum WSS under slip conditions is approximately 17% higher than under no-slip. As valve dysfunction increases, maximum WSS values increase for both leaflet surface boundary conditions; however, the relative difference between slip and no-slip cases diminishes at all stages of the cardiac cycle. Interestingly, while the maximum WSS shows clear differences between slip and no-slip conditions, the surface-averaged WSS values remain nearly the same across these cases. The highest and lowest averaged WSS values occur during the T2 and T3 stages, respectively. The BDI, a parameter linked to shear-induced trauma and hemolysis, shows a strong dependence on valve functionality. It increases progressively with increasing valve defects across all cardiac stages. In all scenarios, the slip condition results in a lower BDI compared to the no-slip condition. This difference becomes more pronounced as valve damage increases. For example, while the difference in BDI between slip and no-slip surfaces is around 8% for a fully functional valve, it increases to approximately 35% under complete dysfunction conditions, i.e., 100% defective case during T2 stage of the cardiac cycle. Overall, these surface-averaged parameters provide a more integrated understanding of the hemodynamic performance and blood compatibility of artificial heart valves under various clinical conditions.

This study conducted extensive computational fluid dynamics numerical simulations to investigate how the slip conditions induced through surface modifications of a BMHV affect the hemodynamics compared to the no-slip conditions. The analysis was performed for physiologically realistic pulsatile flow conditions and for both fully functional and dysfunctional conditions of the BMHV. The results were discussed and compared between slip and no-slip conditions in terms of surface variations of velocity magnitude contours, Q-criteria, von Mises stresses, etc., as well as in terms of parameters including WSS, drag coefficient, pressure drop, and BDI. It has been found that the influence of slip conditions originated due to the surface superhydrophobicity on the hemodynamics strongly depends on the time instance of the cardiac cycle and valve functional state. For instance, at (i.e., mid-acceleration phase of the cardiac cycle), any difference was hardly present in the velocity magnitude contours between the slip and no-slip conditions. However, as the cycle progressed to the systolic peak (i.e., ), the central jet of blood formed between two leaflets extended beyond the valve housing region to the sinus region in the case of no-slip conditions, whereas it was mostly confined in the valve housing region for the slip case. On the contrary, the lateral jet formed between the top leaflet and valve housing wall was found to be larger for the slip case compared to the no-slip one when there was a 100% defect present in the heart valve. Furthermore, more small-scale vortical structures were seen to be present around the leaflet surfaces in the case of no-slip conditions than in the slip conditions during the mid-acceleration phase () of the cardiac cycle. However, as the blood flow rate increased and reached to the T2 stage, the slip case exhibited more vortical structures between the two leaflets, suggesting the presence of more swirling motion in this region than the no-slip case. The kymograph of velocity fluctuations at a probe line between the two leaflets demonstrated that the no-slip condition can uphold the velocity fluctuation for a longer time than the slip condition. The pressure drop was seen to be hardly changed by the slip and no-slip conditions, although it was substantially changed by the valve functional conditions. The maximum values of WSS were higher for the slip condition, although the surface-averaged values remained almost the same in both cases. However, the BDI was seen to be always lower for the slip case compared to the no-slip case, and the difference between the two increased progressively as the valve became increasingly dysfunctional.

Overall, it can be concluded that although the slip condition induced by surface superhydrophobicity causes some localised surface variations in the flow field, the global surface and time-averaged hemodynamic metrics, such as WSS, pressure drop, and BDI, exhibit hardly any noticeable differences between slip and no-slip conditions in the case of a fully functional BMHV. However, as the valve starts to deteriorate in its function, the impact of slip conditions becomes more significant. In particular, the BDI shows a notable decrease, likely due to changes in the distribution and magnitude of von Mises stresses on the valve leaflet surfaces. Therefore, the beneficial effects of slip conditions become more pronounced for a dysfunctional heart valve than for a fully functional one, at least from the perspectives of hemodynamic metrics, although the value of drag forces acting on the surface of heart valve leaflets increases drastically under these surface conditions.

bibliographystyle{sn-basic}\bibliography{References}