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Mathematical analysis of non-Newtonian blood flow in stenosis narrow arteries.

Sriyab S - Comput Math Methods Med (2014)

Bottom Line: When skin friction and resistance of blood flow are normalized with respect to Newtonian blood in stenosis artery, the results present the effect of non-Newtonian blood.The resistance of blood flow (when normalized by non-Newtonian blood in normal artery) increases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length.The resistance of blood flow (when normalized by Newtonian blood in stenosis artery) decreases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand.

ABSTRACT
The flow of blood in narrow arteries with bell-shaped mild stenosis is investigated that treats blood as non-Newtonian fluid by using the K-L model. When skin friction and resistance of blood flow are normalized with respect to non-Newtonian blood in normal artery, the results present the effect of stenosis length. When skin friction and resistance of blood flow are normalized with respect to Newtonian blood in stenosis artery, the results present the effect of non-Newtonian blood. The effect of stenosis length and effect of non-Newtonian fluid on skin friction are consistent with the Casson model in which the skin friction increases with the increase of either stenosis length or the yield stress but the skin friction decreases with the increase of plasma viscosity coefficient. The effect of stenosis length and effect of non-Newtonian fluid on resistance of blood flow are contradictory. The resistance of blood flow (when normalized by non-Newtonian blood in normal artery) increases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length. The resistance of blood flow (when normalized by Newtonian blood in stenosis artery) decreases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length.

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Stenosis geometries for different values of the stenosis shape parameter m.
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fig3: Stenosis geometries for different values of the stenosis shape parameter m.

Mentions: Since the blood flow in narrow arteries is slow, the magnitude of the inertial forces is negligible, and the inertial terms in the momentum equations are neglected. The radial component of momentum equation is ignored because the considered flow is unidirectional. Therefore, the axial component of momentum equation is simplified to the following:(1)−dPdz=1rdrτdr,where P is the pressure and τ is the shear stress. The K-L model that is a relationship between shear and strain rate is defined as follows:(2)−dudr=f(τ)=2k22−4k1(τc−τ)−2k2k22−4k1(τc−τ)4k12,τ≥τc0,τ<τc,where u is the velocity of blood in the axial direction, τc is the yield stress, k1 is plasma viscosity, and k2 is a parameter constant in K-L model. In this work, the geometry of segment of the narrow artery with mild bell-shaped stenosis is shown in Figure 1 and defined as follows:(3)Rz=R01−ae−bz2,where R(z) is the radius of artery in the stenosis region and R0 is radius of normal artery. Note that a is a nondimensional parameter of stenosis height, defined as a = δ/R0, where δ is depth of stenosis. Parameter b is a nondimensional parameter that is the length of the stenosis in the segment of narrow artery, defined as b = m2/L02, where m is the stenosis shape. When parameter a is variable and b is constant, the R/R0 marginally increase along the z-axis with decrease of a (Figure 2). On the other hand, keep a as constant and b as variable (for different values of m and fixed value of L0 = 1.5); it is noticed that the width of the stenosis (R/R0) increases with increase in value of m (Figure 3).


Mathematical analysis of non-Newtonian blood flow in stenosis narrow arteries.

Sriyab S - Comput Math Methods Med (2014)

Stenosis geometries for different values of the stenosis shape parameter m.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4281458&req=5

fig3: Stenosis geometries for different values of the stenosis shape parameter m.
Mentions: Since the blood flow in narrow arteries is slow, the magnitude of the inertial forces is negligible, and the inertial terms in the momentum equations are neglected. The radial component of momentum equation is ignored because the considered flow is unidirectional. Therefore, the axial component of momentum equation is simplified to the following:(1)−dPdz=1rdrτdr,where P is the pressure and τ is the shear stress. The K-L model that is a relationship between shear and strain rate is defined as follows:(2)−dudr=f(τ)=2k22−4k1(τc−τ)−2k2k22−4k1(τc−τ)4k12,τ≥τc0,τ<τc,where u is the velocity of blood in the axial direction, τc is the yield stress, k1 is plasma viscosity, and k2 is a parameter constant in K-L model. In this work, the geometry of segment of the narrow artery with mild bell-shaped stenosis is shown in Figure 1 and defined as follows:(3)Rz=R01−ae−bz2,where R(z) is the radius of artery in the stenosis region and R0 is radius of normal artery. Note that a is a nondimensional parameter of stenosis height, defined as a = δ/R0, where δ is depth of stenosis. Parameter b is a nondimensional parameter that is the length of the stenosis in the segment of narrow artery, defined as b = m2/L02, where m is the stenosis shape. When parameter a is variable and b is constant, the R/R0 marginally increase along the z-axis with decrease of a (Figure 2). On the other hand, keep a as constant and b as variable (for different values of m and fixed value of L0 = 1.5); it is noticed that the width of the stenosis (R/R0) increases with increase in value of m (Figure 3).

Bottom Line: When skin friction and resistance of blood flow are normalized with respect to Newtonian blood in stenosis artery, the results present the effect of non-Newtonian blood.The resistance of blood flow (when normalized by non-Newtonian blood in normal artery) increases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length.The resistance of blood flow (when normalized by Newtonian blood in stenosis artery) decreases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand.

ABSTRACT
The flow of blood in narrow arteries with bell-shaped mild stenosis is investigated that treats blood as non-Newtonian fluid by using the K-L model. When skin friction and resistance of blood flow are normalized with respect to non-Newtonian blood in normal artery, the results present the effect of stenosis length. When skin friction and resistance of blood flow are normalized with respect to Newtonian blood in stenosis artery, the results present the effect of non-Newtonian blood. The effect of stenosis length and effect of non-Newtonian fluid on skin friction are consistent with the Casson model in which the skin friction increases with the increase of either stenosis length or the yield stress but the skin friction decreases with the increase of plasma viscosity coefficient. The effect of stenosis length and effect of non-Newtonian fluid on resistance of blood flow are contradictory. The resistance of blood flow (when normalized by non-Newtonian blood in normal artery) increases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length. The resistance of blood flow (when normalized by Newtonian blood in stenosis artery) decreases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length.

Show MeSH
Related in: MedlinePlus