Numerical simulation of unsteady micropolar hemodynamics in a tapered catheterized artery with a combination of stenosis and aneurysm

Abstract

The unsteady flow characteristics of blood are analyzed through a catheterized stenotic artery with post-stenotic dilatation. A rigid tube with a pair of abnormal wall segments in close proximity to each other is employed to geometrically simulate the diseased artery. A micropolar fluid model is used to capture the rheological characteristics of the streaming blood in the annulus. The mild stenosis approximation is employed to derive the governing flow equation which is then solved using a robust finite difference method. Particular attention is paid to the effects of geometrical parameters of the arterial wall and rheological parameters of the blood on axial velocity, flow rate, resistance impedance and wall shear stress. The global behavior of blood is also analyzed through instantaneous pattern of streamlines.

Appendix

The explicit numerical scheme which is forward in time and central in space [27] is validated using a variational finite element method (FEM). The finite element method is a powerful technique for solving partial differential equations as well as integral equations and utilizes numerical integration, rather than differentiation as with difference methods. The whole domain is delineated into smaller elements (subdomains) of finite dimensions called “finite elements.” The collection of elements is called the finite element mesh or grid. From the mesh, a typical element is isolated and the variational formulation of the given problem over the typical element is constructed. An approximate solution of the variational problem is assumed, and the element equations are derived by substituting this solution in the governing differential equations. This process generates an element matrix, which is called a stiffness matrix and is constructed by using element interpolation functions. The algebraic equations so obtained are assembled by imposing the inter-element continuity conditions. This yields a large number of algebraic equations defining the global finite element model, which governs the whole domain. The essential and natural boundary conditions are imposed on the assembled equations. The assembled equations so obtained can be solved by any “matrix” numerical technique, e.g., Gaussian elimination method, Householder’s approach and LU Decomposition method, as elaborated by Bathe [6]. Numerous nonlinear micro- and nanoscale biofluid mechanics problems have been successfully addressed in recent years with variational FEM including biomagnetic micropolar convection flow in tissue [2], deoxygenated blood flows [8], pulsating drug dispersion [3], nanobiopolymer manufacture [37] and nanopharmacodynamics [36]. In the present transient problem, the linear momentum Eq. (28) and angular momentum Eq. (29) contain the two dependent variables, w and v, and the independent variables, x and t. Discretization is performed separately in the time domain (t) and in the spatial domain (x). Quadratic elements are used. The variational form associated with Eqs. (28)–(29) over a typical quadratic element is constructed, and arbitrary test functions invoked. Interpolation functions are applied for the dependent variables in the form:

$$ w = \sum\limits_{j = 1}^{2} {w_{j} \varPsi_{j} } \quad {\text{and}}\quad v = \sum\limits_{j = 1}^{2} {v_{j} \varPsi_{j} } $$

(42)

With w ₁ = w ₂ = Ψ _i , i = 1,2. The shape elements for a typical quadratic element (x _e , x _e+1 ) in the space variable take the form [9]:

$$ \varPsi_{1}^{e} = \frac{{[x_{e + 1} - x_{e} - 2x][x_{e + 1} - x]}}{{[x_{e + 1} - x_{e} ]^{2} }} $$

(43)

$$ \varPsi_{2}^{e} = \frac{{4[x - x_{e} ][x_{e + 1} - x]}}{{[x_{e + 1} - x_{e} ]^{2} }} $$

(44)

$$ \varPsi_{3}^{e} = \frac{{[x_{e + 1} - x_{e} - 2x][x - x_{e} ]}}{{[x_{e + 1} - x_{e} ]^{2} }} $$

(45)

Valid for $x_{e} \le x \le x_{e + 1}$. A similar procedure is used for the time domain. The finite element model of the equations thus formed is given in matrix-vector form by;

$$ \left[ \begin{aligned} \left[ {K^{11} } \right]\,\,\left[ {K^{12} } \right]\, \hfill \\ \left[ {K^{21} } \right]\,\,\left[ {K^{22} } \right] \hfill \\ \end{aligned} \right]\left[ \begin{aligned} \left\{ w \right\} \hfill \\ \left\{ v \right\} \hfill \\ \end{aligned} \right] = \left[ \begin{aligned} \left\{ {\delta_{1} } \right\} \hfill \\ \left\{ {\delta_{2} } \right\} \hfill \\ \end{aligned} \right] $$

(46)

Here [K ^mn] and [δ ^m] (m, n = 1, 2) are the components of the stiffness matrix and displacement vector and are lengthy integral expressions, which are omitted for conservation of space here. Further details are readily available in the articles of Bég and co-workers [2, 3, 8, 36, 37]. The entire flow domain is divided into 600 quadratic elements. Approximately 1800 linear equations are generated. Following the assembly of all element equations, a large-order matrix is generated. The nonlinear algebraic system of equations is solved iteratively. An accuracy of 0.00001 is used. A convergence criterion based on the relative difference between the current and previous iterations is employed. When these differences reach to desired accuracy, the solution is assumed to have converged and the iterative process is terminated. Two-point Gaussian quadrature is implemented for solving the integrations. The FEM algorithm has been executed in MATLAB running on an Octane SGI desktop workstation and takes 25 s on average. Comparisons of the FEM and FDM solutions are documented in Tables 1, 2 and 3 for axial velocity at two different locations, for both diverging and converging arteries in micropolar blood flow (M ≠ 0). Excellent correlation is obtained testifying to the validity of the FDM computations, the latter which are used in all graphical illustrations.

Table 1 Numerical values of axial velocity at a cross section z = 0.7 corresponding to the critical height of the stenosis for k = 0.1, $\delta$ = 0.1, t = 0.3

Table 2 FDM numerical values of axial velocity at a cross section z = 2 corresponding to the critical height of the post-stenotic dilated region (aneurysm) for k = 0.1, $\delta$ = 0.1, t = 0.3

Table 3 FEM computations for axial velocity at a cross section z = 2 corresponding to the critical height of the post-stenotic dilated region (aneurysm) for k = 0.1, $\delta$ = 0.1, t = 0.3