Simulation studies on the hemodynamic model for blood flow

Considering that the blood is viscous fluid and the flow through a circular cylindrical tube of radius a. In the view of Navier-Stokes equation or equivalently the second law of Newton for fluid flow, the model is proposed [2]; (1) utt(x,t)−v02uxx(x,t)=12u(x,t)2tt+s uxxtt(x,t)+βv02uxx(x,t), {u_{tt}}(x,t) - v_0^2{u_{xx}}(x,t) = \frac{1}{2}{\left( {u{{(x,t)}^2}} \right)_{tt}} + s\, {u_{xxtt}}(x,t) + \beta v_0^2{u_{xx}}(x,t), where v0=ηE2ρ1/2 {v_0} = {\left( {\frac{{\eta E}}{{2\rho }}} \right)^{1/2}} and s=ηρwR02ρ s = \frac{{\eta {\rho _w}{R_0}}}{{2\rho }} are determined. v₀ is the phase velocity of nonlinear waves and depends on wall incompressibility (η), Young modulus (E) and density of blood (ρ), whereas sdepends on wall incompressibility (η), density of the artery wall (ρ_w), equilibrium for artery of radius (R₀) and density of blood (ρ). β = α − 2 and α is the coefficient of nonlinear elasticity. As seen all parameters vary from individual to individual. Equation (1) is proposed for the nonlinear coefficient of elasticity of the artery wall and for dispersion connected to the inertial effects of the wall. Equation (1) is generally reduced to KdV equation via the reductive perturbation method and rescaling of coordinates [2, 33, 34] and the solutions obtained via known numerical or analytical methods [2]. Now, the classical wave transformation ξ = x − vt, v ≠ 0 is used to reduce (1) to NODE and it is obtained (2) v−1+βv02u″−v2u′2−v2uu″−sv2u(4)=0, \left( {v - \left( {1 + \beta } \right){v_0}^2} \right)u'' - {v^2}{\left( {u'} \right)^2} - {v^2}uu'' - s{v^2}{u^{(4)}} = 0, being u′=dudξ u' = \frac{{du}}{{d\xi }} . Firstly, (2) is integrated twice respect to ξ and integration constants are assumed zero. One reads the equation (2) as below (3) v−1+βv02u−v22u2−sv2u″=0. \left( {v - \left( {1 + \beta } \right){v_0}^2} \right)u - \frac{{{v^2}}}{2}{u^2} - s{v^2}u'' = 0.

For (3), the exact solution is assumed as follows uξ=g0+g1zξ+g2z2ξ, u\left( \xi \right) = {g_0} + {g_1}z\left( \xi \right) + {g_2}{z^2}\left( \xi \right), where g₀, g₁, g₂ are parameters and z (ξ) is a solution of BDE z′ (ξ) = Pz (ξ) + Qz² (ξ), P and Q are considered as constant functions. Therefore, substituting the assumptions in (3), a nonlinear algebraic equation system is obtained. The solution sets of the nonlinear algebraic system include trivial and constant solutions, the meaningful solution is given below; zξ=−β+1α2+v2−β+1α2+v2C1expτ+vQsα2β+1−v2, z\left( \xi \right) = \frac{{ - \left( {\beta + 1} \right){\alpha ^2} + {v^2}}}{{\left( { - \left( {\beta + 1} \right){\alpha ^2} + {v^2}} \right){C_1}\exp \left( \tau \right) + vQ\sqrt {s\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right)} }}, where g0=−2α2β+1−v2v2 {g_0} = - \frac{{2\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right)}}{{{v^2}}} , g1=−12α2β+1−v2Qv2P {g_1} = - \frac{{12\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right)Q}}{{{v^2}P}} , g2=−12α2β+1−v2Q2v2P2 {g_2} = - \frac{{12\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right){Q^2}}}{{{v^2}{P^2}}} , P=±sα2β+1−v2vs P = \pm \frac{{\sqrt {s\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right)} }}{{vs}} , α2β+1−v2≠0 \left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right) \ne 0 , τ=sα2β+1−v2vsξ \tau = \frac{{\sqrt {s\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right)} }}{{vs}}\xi . When all parameters are substituted into the assumption of solution, the exact solution as the travelling wave is obtained. The 3D-plot of the exact solution for Q = 1, s = 2, v = 0.5, β = 3.55, α = 0.1, C₁ = 0.1 is given in Figure 1. Additionally, for the same parameter’s values, 2D-plot is given at t = 0, t = 0.2, t = 0.4, t = 0.6, t = 0.8, respectively, in Figure 2.

Fig. 1

Fig. 2

Considering that the blood is viscous fluid and the flow through a circular cylindrical tube of radius a. In the view of Navier-Stokes equation or equivalently the second law of Newton for fluid flow, the model is proposed [2]; (4) utt(x,t)−v02uxx(x,t)=12u(x,t)2tt+s uxxtt(x,t)+βv02uxx(x,t), {u_{tt}}(x,t) - {v_0}^2{u_{xx}}(x,t) = \frac{1}{2}{\left( {u{{(x,t)}^2}} \right)_{tt}} + s {u_{xxtt}}(x,t) + \beta {v_0}^2{u_{xx}}(x,t), being v0=ηE2ρ1/2 {v_0} = {\left( {\frac{{\eta E}}{{2\rho }}} \right)^{1/2}} and s=ηρwR02ρ s = \frac{{\eta {\rho _w}{R_0}}}{{2\rho }} are determined. v₀ is the phase velocity of nonlinear waves and depends on wall incompressibility (η), Young modulus (E) and density of blood (ρ), whereas sdepends on wall incompressibility (η), density of the artery wall (ρ_w), equilibrium for artery of radius (R₀) and density of blood (ρ). β = α − 2 and α is the coefficient of nonlinear elasticity. As seen all parameters vary from individual to individual. (1) is proposed for the nonlinear coefficient of elasticity of the artery wall and for dispersion connected to the inertial effects of the wall. (1) is generally reduced to KdV equation via the reductive perturbation method and rescaling of coordinates [2,33,34] and the solutions obtained via known numerical or analytical methods [2]. Now, the classical wave transformation ξ = x − vt, v ≠ 0 is used to reduce (1) to NODE and it is found (5) v−1+βv02u″−v2u′2−v2uu″−sv2u(4)=0, \left( {v - \left( {1 + \beta } \right){v_0}^2} \right)u'' - {v^2}{\left( {u'} \right)^2} - {v^2}uu'' - s{v^2}{u^{(4)}} = 0, where u′=dudξ u' = \frac{{du}}{{d\xi }} . Firstly, (2) is integrated twice respect to ξ and integration constants are assumed zero (6) v−1+βv02u−v22u2−sv2u″=0. \left( {v - \left( {1 + \beta } \right){v_0}^2} \right)u - \frac{{{v^2}}}{2}{u^2} - s{v^2}u'' = 0.

For (3), the exact solution is assumed as follows uξ=g0+g1zξ+g2z2ξ, u\left( \xi \right) = {g_0} + {g_1}z\left( \xi \right) + {g_2}{z^2}\left( \xi \right), where g₀, g₁, g₂ are parameters and z (ξ) is a solution of BDE z′ (ξ) = Pz (ξ) + Qz² (ξ), P and Q are considered as constant functions. Therefore, substituting the assumptions in (3), a nonlinear algebraic equation system is obtained. The solution sets of the nonlinear algebraic system include trivial and constant solutions, the meaningful solution is given below. zξ=−β+1α2+v2−β+1α2+v2C1expχ+vQsα2β+1−v2, z\left( \xi \right) = \frac{{ - \left( {\beta + 1} \right){\alpha ^2} + {v^2}}}{{\left( { - \left( {\beta + 1} \right){\alpha ^2} + {v^2}} \right){C_1}\exp \left( \chi \right) + vQ\sqrt {s\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right)} }}, where g0=−2α2β+1−v2v2 {g_0} = - \frac{{2\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right)}}{{{v^2}}} , g1=−12α2β+1−v2Qv2P {g_1} = - \frac{{12\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right)Q}}{{{v^2}P}} , g2=−12α2β+1−v2Q2v2P2 {g_2} = - \frac{{12\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right){Q^2}}}{{{v^2}{P^2}}} , P=±sα2β+1−v2vs P = \pm \frac{{\sqrt {s\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right)} }}{{vs}} , α2β+1−v2≠0 \left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right) \ne 0 , χ=sα2β+1−v2vsξ \chi = \frac{{\sqrt {s\left( {{\alpha ^2}\left( {\beta + 1} \right) - {v^2}} \right)} }}{{vs}}\xi . When all parameters are substituted into the assumption of solution, the exact solution as the travelling wave is obtained. The 3D-plot of the exact solution for Q = 1, s = 2, v = 0.5, β = 3.55, α = 0.1, C₁ = 0.1 is given in Figure 3. Additionally, for the same parameter’s values, 2D-plot is given at t = 0, t = 0.2, t = 0.4, t = 0.6, t = 0.8, respectively, in Figure 4.

Fig. 3

Fig. 4

The authors have no competing interests to declare that are relevant to the content of this article.

No funding was received to assist with the preparation of this manuscript.

Z.P.I.-Conceptualization, Methodology, Software, Writing-Review Editing, Formal Analysis, Validation, Writing-Original Draft. The author read and approved the final submitted version of this manuscript.

The author deeply appreciate the anonymous reviewers for their helpful and constructive suggestions, which can help further improve this paper.

All data that support the findings of this study are included within the article.

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.