### Geometry formulation of arteries

This study aimed to understand the change in velocity distribution, pressure drop, and wall shear stress in the stenosed artery, artery with single and double stenosis at different area reduction. To do this, first, the geometry of three different types of arteries, i.e., normal artery without stenosis, artery with single stenosis and artery with double stenosis (Fig. 1) was made. Later geometry for different area reduction at the stenosis region was made.

The artificial geometrical model of the blood vessel with stenosis was created by using the following formula of cosine curve [7],

$$ \frac{r(Z)}{R}=1-{\delta}_c\left[1+\cos \left(\frac{Z\pi}{D}\right)\right],\kern0.75em -D\le Z\le D $$

(1)

This equation was modified for double stenosis as follows:

$$ \frac{r(Z)}{R}=\Big\{1-{\delta}_c\left[1+\cos \left(\frac{Z\pi}{D}\right)\right],{\kern0.75em }_{5D\le Z\le 7D}^{-D\le Z\le D} $$

(2)

In eq.1, *r* and *Z* are the radial and axial coordinates respectively; *R* and *D* are the radius and diameter of the un-stenosed vessel respectively. The percentage of the stenosis was controlled by the parameter δ_{c}. The constrictions of the artery followed in the cosine curve with the reduction of the area. For single stenosis, the area was reduced as 60% and 75%. For double stenosis, the area of first and second stenosis was reduced as 60% and 60%, 60% and 75%, 75% and 60%, and 75% and 75%. This smooth reduction of the cross-sectional area produced the inside of the vessel by using eq. 1. This provides a fairly accurate representation of the biological form of arterial stenosis. This was employed previously in theoretical calculations by Deshpande et al. [24] and Din et al. [25]. The entire span of the biological form of the model was taken as 540 mm (27*D*) [24], where diameter *D* = 20 mm and the length of the upstream, downstream, and stenosed zone are taken as 4*D*, 21*D*, and *2D*, respectively for single stenosis. 4D, 2D, 4D, 2D, and 15D are the length of upstream, first stenosis, first downstream, second stenosis, and second downstream respectively. In most of the cases, blood flow through different models was considered as incompressible and Newtonian-homogeneous fluid [23] with a density (*ρ*) of j1060 kg/m^{3}. The constant dynamic viscosity (*μ*) was taken as of 3.71 × 10^{−3} Pa s whereas in this study non-Newtonian model has been used.

The Reynolds averaged Navier–Stokes equations (RANS equations) were considered as the governing equations for blood flow motion. The RANS equations are time-averaged equations of motion for fluid flow. The equations are generally used to describe the turbulent flows and the idea is to Reynolds decomposition where an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities [26]. RANS models employ an empirical closure hypothesis to compute the components of the Reynolds stress tensor [27]. Classification of RANS models is based on the number of additional differential transport equations required to determine turbulence quantities [28]. After applying the Reynolds time-averaging techniques, the Reynolds averaged Navier–Stokes (RANS) are obtained as tensor form [29] as:

$$ \frac{\partial {\mathrm{u}}_{\mathrm{i}}}{\partial {\mathrm{x}}_{\mathrm{i}}}=0 $$

(3)

$$ \frac{\partial }{\partial t}\left(\rho {u}_i\right)+\frac{\partial }{\partial {x}_j}\left(\rho {u}_i{u}_j\right)=-\frac{\partial p}{\partial {x}_i}+\frac{\partial }{\partial {x}_j}\left[\mu \left(\frac{\partial {u}_i}{\partial {x}_j}+\frac{\partial {u}_j}{\partial {x}_i}\right)\right]+\frac{\partial {\tau}_{ij}}{\partial {x}_j} $$

(4)

In eqs. 2 and 3, *x*_{i} = (*x*, *y*, *z*) are the Cartesian coordinate systems, *u*_{i} is the mean velocity components, *ρ* is the density, *p* is the pressure, and *τ*_{ij} are the Reynolds stress (wall shear stress). The Boussinesq hypothesis is employed to model the Reynolds stress *τ*_{ij} for our current simulations. The Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuations in fluid momentum [30]. Reynolds stress model can successfully capture the turbulence characteristics [31]. Recently, a theoretical basis for determination of the Reynolds stress in canonical flows has been presented [32]. It is based on the turbulence momentum balance for a control volume moving at the local mean flow speed [33]. Therefore, it constitutes a Lagrangian analysis for the momentum transport, which happens to contain the Reynolds stress term [33].

This is given as follows:

$$ {\tau}_{ij}=-\rho \left\langle u{\hbox{'}}_iu{\hbox{'}}_j\right\rangle ={\mu}_t\left(\frac{\partial {u}_i}{\partial {x}_j}+\frac{\partial {u}_j}{\partial {x}_i}\right)-\frac{2}{3}\rho k{\delta}_{ij} $$

(5)

In eq. 4, *u*'_{i} are the fluctuating velocity components and \( k=\frac{1}{2}\left\langle u{\hbox{'}}_iu{\hbox{'}}_j\right\rangle \) is the turbulent kinetic energy. The turbulent eddy viscosity is denoted as *μ*_{t} which is obtained by employing the standard *k*-*ω* model of Wilcox. The eddy viscosity is modeled as:

$$ {\mu}_t=\frac{\rho k}{\omega } $$

(6)

where *ω* is the specific dissipation rate.

The following equation of Wilcox [18] is solved to obtain *k* and *ω*:

$$ \frac{\partial k}{\partial t}+\frac{\partial k\left\langle {u}_j\right\rangle }{\partial {x}_j}=-\frac{1}{\rho}\left\langle \rho u{\hbox{'}}_iu{\hbox{'}}_j\right\rangle \frac{\partial \left\langle {u}_i\right\rangle }{\partial {x}_j}-{\beta}^{\ast } k\omega +\frac{\partial }{\partial {x}_j}\left[\frac{1}{\rho}\left(\mu +{\sigma}^{\ast }{\mu}_t\right)\frac{\partial k}{\partial {x}_j}\right] $$

(7)

$$ \frac{\partial \omega }{\partial t}+\frac{\partial \omega \left\langle {u}_j\right\rangle }{\partial {x}_j}=-{\alpha}_1\frac{\omega }{\rho k}\left\langle \rho u{\hbox{'}}_iu{\hbox{'}}_j\right\rangle \frac{\partial \left\langle {u}_i\right\rangle }{\partial {x}_j}-{\beta \omega}^2+\frac{\partial }{\partial {x}_j}\left[\frac{1}{\rho}\left(\mu +{\sigma \mu}_t\right)\frac{\partial \omega }{\partial {x}_j}\right] $$

(8)

where *σ*^{*} = 0.5, *ß*^{*} = 0.072, *σ* = 0.5, *α*_{1}= 1.0, *ß* = 0.072

In computational fluid dynamics, the k–omega (*k–ω*) turbulence model is a common two-equation turbulence model, that is used as an approximation for the Reynolds-averaged Navier–Stokes equations. The model attempts to predict turbulence by two partial differential equations for two variables, *k* and *ω*, with the first variable being the turbulence kinetic energy (*k*) while the second (*ω*) is the specific rate of dissipation. Details of the derivation of the model (eqs. 7 and 8) are given in Wilcox [34] and Wilcox [35]. The detailed descriptions of these turbulent models are explained in Varghese et al. [36].

When blood is treated as non-Newtonian fluid and then the viscosity of blood can be calculated from different models such as the Power-law model, Cross model, and Carreau model. In this study, the well-known Carreau model was used with parameters verified by previous studies [20]. The Carreau model is defined as:

$$ \mu \left(\left|\overset{\bullet }{\gamma}\right|\right)={\mu}_{\infty }+\left({\mu}_0-{\mu}_{\infty}\right){\left[1+{\left(\lambda \overset{\bullet }{\gamma}\right)}^2\right]}^{\left(n-1\right)/2} $$

(9)

where *μ*_{∞} (0.00345 Pa) is the infinite shear viscosity, *μ*_{0} (0.056 Pa) is the blood viscosity at zero shear rate, γ is the instantaneous shear rate, *λ* (3.313) is the time constant which is associated with the viscosity that changes with shear rate and *n* is the power-law index.

A total of seven inflexible and solid circular model arteries were used as the model artery with different symmetric stenosis. Firstly, the geometry of an inflexible and solid circular model artery without stenosis was developed. Secondly, the geometry of two arteries with single stenosis (i.e., 60% and 75% area reduction) was developed. Thirdly, the geometry of four arteries with double stenosis (i.e., 60% and 60%, 60% and 75%, 75% and 75%, and 75% and 60% area reduction) was developed.

### Boundary conditions and computational procedure

No-slip boundary condition with zero velocity (*u*_{i} = 0) relative to the boundary along with a pulsatile velocity profile has been imposed at the inlet of the model. The pulsatile velocity profile is computed with the following equation:

$$ {\displaystyle \begin{array}{l}{V}_{inlet}(t)=\left\{\begin{array}{l}0.5\sin \left[4\pi \left(1+0.0160236\right)\right]:0.5n<t\le 0.5n+0.218\\ {}0.1:0.5n+0.218<t\le 0.5\left(n+1\right)\end{array}\right.\\ {}n=0,1,2,.\dots \dots \dots \end{array}} $$

(10)

where V (0.5) Vis the bulk stream-wise velocity related to the Reynolds number, Re *= ρVD/μ*Re = ρVD/μ of the blood flow. Inside the blood-vessel, a proportion of the forward spiral velocity (Ω) was calculated by using the following equation:

$$ \Omega =\frac{V}{R}C $$

(11)

A constant *C= 1/6* was used to limit the magnitude of the spiral speed.

The outlet of the model has been treated as a pressure outlet and setting for the gauge pressure to become 13332 Pa as the systolic and diastolic pressure of a healthy human is around 15999 Pa (i.e., 120 mmHg) and 10666 Pa (i.e., 80 mmHg), respectively. Thus, the average pressure of the two phases, we use 100 mmHg (around 13332 Pascal). All simulations were performed with the commercially available computational fluid dynamics (CFD) software Fluent [37]. This software uses finite volume method for the discretization of the flow governing equations. The finite volume method evaluates partial differential equations in the form of algebraic equations. Pressure based solver was used to solve the flow equations with the implicit formulation method. Besides, the semi-implicit method for pressure-linked equation scheme for pressure-velocity coupling was used. In the spatial discretization process, the least squares based cell scheme was used for the gradient and bounded central differencing scheme was used for momentum. Bounded second-order implicit scheme was used for transient formulation, while the second-order accurate scheme was used for the Poisson-like pressure equation. The minimum time-step size used for the simulation was 1 × 10^{−2} s with 10000 numbers of total time-steps. The maximum iterations were 20 per each time step to collect the statistical data. The inlet boundary conditions for the stream-wise velocity was written in C-language using the interface of user defined function of Fluent and linked with the solver. The solution process was initiated using arbitrary values of the velocity components and *k-ω*, and their residuals are monitored at every iteration. The magnitude of the residuals dropped gradually, which is a strong indicator of stable and accurate solutions. The iteration process was stopped when the residuals are leveled off at 10^{−4} and the final converged solutions are achieved.