Mathematical Analysis of Casson Fluid Model for Blood Rheology in Stenosed Narrow Arteries

Theflowof blood through a narrow arterywith bell-shaped stenosis is investigated, treating blood asCasson fluid. Present results are compared with the results of the Herschel-Bulkley fluidmodel obtained byMisra and Shit (2006) for the same geometry. Resistance to flow and skin friction are normalized in two different ways such as (i) with respect to the same non-Newtonian fluid in a normal artery which gives the effect of a stenosis and (ii) with respect to the Newtonian fluid in the stenosed artery which spells out the non-Newtonian effects of the fluid. It is found that the resistance to flow and skin friction increase with the increase of maximum depth of the stenosis, but these flow quantities (when normalized with non-Newtonian fluid in normal artery) decrease with the increase of the yield stress, as obtained by Misra and Shit (2006). It is also noticed that the resistance to flow and skin friction increase (when normalized with Newtonian fluid in stenosed artery) with the increase of the yield stress.


Introduction
The study of the fluid dynamical aspects of blood flow through a stenosed artery is useful for the fundamental understanding of circulatory disorders.Stenosis in an artery is the narrowing of the blood flow area in the artery by the development of arteriosclerosis plaques due to the deposits of fats, cholesterol, and so forth on the inner wall of the artery.This leads to an increase in the resistance to flow and associated reduction in blood supply in the downstream which leads to serious cardiovascular diseases such as myocardial infarction and cerebral strokes [1][2][3].
Blood shows Newtonian fluid's character when it flows through larger diameter arteries at high shear rates, but it exhibits a remarkable non-Newtonian behavior when it flows through small diameter arteries at low shear rates [4,5].Moreover, there is an increase in viscosity of blood at low rate s of shear as the red blood cells tend to aggregate into the Rouleaux form [6]. Rouleaux form behaves as a semi-solid along the center forming a plug flow region.In the plug flow region, we have a flattened parabolic velocity profile rather than the parabolic velocity profile of a Newtonian fluid.This behavior can be modeled by the concept of yield stress.The yield stress for blood depends strongly on fibrinogen concentration and is also dependent on the hematocrit.The yield stress values for normal human blood is between 0.01 and 0.06 dyn/cm 2 [7].
Casson fluid model is a non-Newtonian fluid with yield stress which is widely used for modeling blood flow in narrow arteries.Many researchers have used the Casson fluid model for mathematical modeling of blood flow in narrow arteries at low shear rates.It has been demonstrated by Blair [8] and Copley [9] that the Casson fluid model is adequate for the representation of the simple shear behavior of blood in narrow arteries.Casson [10] examined the validity of Casson fluid model in his studies pertaining to the flow characteristics of blood and reported that at low shear rates the yield stress for blood is nonzero.It has been established by Merrill et al. [11] that the Casson fluid model predicts satisfactorily the flow behaviors of blood in tubes with the diameter of 130-1000 m.
Charm and Kurland [12] pointed out in their experimental findings that the Casson fluid model could be the best representative of blood when it flows through narrow arteries at low shear rates and that it could be applied to human blood at a wide range of hematocrit and shear rates.Blair and Spanner [13] reported that blood behaves like a Casson fluid in the case of moderate shear rate flows, and it is appropriate to assume blood as a Casson fluid.Aroesty and Gross [14] have developed a Casson fluid theory for pulsatile blood flow through narrow uniform arteries.Chaturani and Samy [15] analyzed the pulsatile flow of Casson fluid through stenosed arteries using the perturbation method.
Herschel-Bulkley fluid is also a non-Newtonian fluid with yield stress which is more general in the sense that it contains two parameters such as the yield stress and power law index, whereas the Casson fluid has only one parameter which is the yield stress.Herschel-Bulkley fluid's constitutive equation can be reduced to the constitutive equations of Newtonian, Power law, and Bingham fluid models by taking appropriate values to the parameters.Chaturani and Ponnalagar Samy [16] analyzed the steady flow of Herschel-Bulkley fluid for blood flow through cosine-shaped stenosed arteries.
Misra and Shit [17] analyzed the steady flow of Herschel-Bulkley fluid for blood flow in narrow arteries with bellshaped mild stenosis.The mathematical modeling of Casson fluid model for steady flow of blood in narrow arteries with bell-shaped mild stenosis was not studied by anyone so far, according to the knowledge of the authors.Hence, in the present study, a mathematical model is developed to analyze the blood flow at low shear rates in narrow arteries with mild bell-shaped stenosis, treating blood as Casson fluid model.The results of the present study are compared with the results of Misra and Shit [17], and Chaturani and Ponnalagar Samy [16] and some possible clinical applications to the present study are also given.

Formulation
Let us consider an axially symmetric, laminar, steady, and fully developed flow of a non-Newtonian incompressible viscous fluid (blood) in the axial direction () through a circular artery with bell-shaped mild stenosis.The non-Newtonian behavior of the flowing blood is characterized by Casson fluid model.The artery wall is assumed to be rigid (due to the presence of the stenosis) and the artery is assumed to be long enough so that the entrance and end effects can be neglected in the arterial segment under study.A cylindrical polar coordinate system (, , ) is used to analyze the blood flow, where  and  are the variables taken in the radial and axial directions, respectively, and  is the azimuthal angle.The geometry of the arterial segment with mild constriction is shown in Figure 1.
Since the blood flow in narrow arteries is slow, the magnitude of the inertial forces is negligibly small, and thus the inertial terms in the momentum equations are neglected.Since the considered flow is unidirectional and is in the axial direction, the radial component of the momentum equation is ignored.The axial component of the momentum equation is simplified to the following: where  is the shear stress and  is the pressure.The constitutive equation (relationship between the shear stress and strain rate) of Casson fluid model is defined as follows: where  is the velocity of blood in the axial direction,   is the yield stress, and  is the viscosity coefficient of Casson fluid.The geometry of the segment of the narrow artery with mild bell-shaped stenosis is mathematicaly defined as follows: where  0 is the radius of the normal artery, () is the radius of the artery in the stenosis region,  is the depth of the stenosis at its throat,  is a parametric constant, and  characterizes the relative length of the constriction, defined as  =  0 / 0 .Equation (3) can be rewritten as where  = / 0 and  =  2  2 / 2 0 .Note that  and  are the parameters in the nondimensional form corresponding to the maximum projection of the stenosis at its throat and variable length of the stenosis in the segment of the narrow artery under study, respectively.Equation (3) spells out that the bellshaped geometry has the advantage of having two parameters such as  and  compared to cosine-curve shaped geometry which has only one parameter, namely, the maximum depth of the stenosis [15].In the bell-shaped stenosis geometry, by keeping  as variable and  as constant, one can generate arteries with different stenosis lengths with the same maximum depth of the stenosis , and also by keeping  as variable and  as constant one can generate arteries with different maximum depths with the same length of the stenosis.The different stenosis shapes obtained by varying the stenosis height  with 30% stenosis that is,  0 = 1.5, are shown in   Equations ( 1) and (2) have to be solved with the help of the following no slip boundary condition: and the regularity condition

Method of Solution
Integrating (1) and then using (6), we get From ( 7), the skin friction   is obtained as where  = ().
The volumetric flow rate  is as follows: where  and   are given by ( 7) and ( 8), respectively.Substituting (2) into ( 9), we get Integrating (10) and then simplifying, one can get Since   /  ≪ 1, neglecting the term involving (  /  ) 4 in (11), we get the expression for flow rate as Using ( 8) in ( 12), we get Neglecting the term involving  2  in ( 13), we get Integrating ( 14) along the length of the artery and using the conditions that  =  1 at  = − and  =  2 at  = , we obtain Simplifying (15), one can obtain the following expression for pressure drop: 3.1.Resistance to Flow.The resistance to flow  is defined as follows: The resistance to flow for Casson fluid in a stenosed artery is obtained as follows: In the absence of any constriction ( = 0 and  =  0 ), the resistance to flow (in the normal artery)   is given by the following: The expression for resistance to flow in the dimensionless form is obtained as follows: where It is noted that  1 measures the relative resistance in a stenosed artery compared to normal artery.Substituting the expression for / 0 from (4), the integrals  1 ,  2 , and  3 are reduced to the following: Using a two-point Gauss quadrature formula, the integrals in (21) are evaluated as follows: If we want to compare the resistance to flow for different fluid models, we have to normalize it with respect to resistance to flow    of Newtonian fluid in normal artery, and the respective expression for Casson fluid model is obtained as follows: where which is obtained from (18) with  = 0, () =  0 , and   = 0.
3.2.Skin Friction.From ( 8) and ( 14), the expression for the skin friction is obtained as follows: In the absence of any constriction when (() =  0 ), the expression for the skin friction becomes the following: The nondimensional form of skin friction with effects on stenosis is defined as the ratio between the skin friction in the stenosed artery and skin friction in the normal artery.From ( 25) and ( 26), the skin friction with effects on stenosis is obtained as The nondimensional form of skin friction with effects on the non-Newtonian behavior of blood is defined as the ratio between the skin friction of the non-Newtonian fluid in the stenosed artery and the skin friction of the Newtonian fluid in the same stenosed artery.The expression for skin friction with effects on the non-Newtonian behavior of blood is obtained as follows: where

Numerical Simulations of the Results
The objective of this study is to discuss the effects of various parameters on the physiologically important flow quantities such as skin friction, resistance to flow, and flow rate.The following parameters with their ranges mentioned as : 0-1, : 2.0-7.0 (CP)  /sec −1 ,   : 0.0-0.5 dyne/cm 2 ,  = 2,  = 5 cm, and  0 = 0.40 are used to evaluate the expressions of these flow quantities and get data for plotting the graphs.30% stenosis are computed in Table 1.It is observed that the skin friction increases very slightly with the increase of the viscosity coefficient.
In order to compare our results with those of Misra and Shit [17], the variations of skin friction  1 with axial distance for different fluids are shown in Figure 6 for 30% of stenosis with  = 4 and   = 0.05.It is observed that the plot of the skin friction of Casson fluid model lies between those of the Herschel-Bulkley fluid model with  = 1 and  = 1.05.Here, the skin friction is normalized with respect to the normal artery with the same fluid, as was done by Misra and Shit [17].
The mathematical form of cosine curve-shaped geometry for stenosis given by Chaturani and Ponnalagar Samy [16] is reproduced as follows: For computation, values of the parameters are taken as  = 10,  = 3.5, and  0 = 3.The variations of skin friction  1 with axial distance for Casson fluid model with cosine-and bell-shaped stenosis geometries with 30% stenosis and with  = 4 and   = 0.05 are shown in Figure 7 (the data for plotting the graph is computed from (27)).It is found that the skin friction in cosine curveshaped stenosed arteries is considerably higher than that in bell-shaped stenosed arteries.

Effects of Non-Newtonian Behavior on Skin-Friction.
We can study the effects of different non-Newtonian fluids if we normalize it with respect to Newtonian fluid as done  Bell-shaped geometry by [15,16,[18][19][20].In this case, the following results are in agreement with the results given by these authors but are opposite in nature to those given in Section 4.1.1except for the variations of skin friction with axial distance for different values of stenosis length as given in Figure 8.The variations of skin friction  2 with axial distance for different values of  0 / and for a given  = 4 and   = 0.05 are illustrated in Figure 9.It is observed that the skin friction increases considerably with the increase in the stenosis length.
Figure 10   Figure 12 depicts the variations of skin friction  2 with axial distance for different values of viscosity coefficient  for   = 0.05 and 30% of stenosis.One can notice that the skin friction decreases slightly as viscosity increases.
Figure 13 shows the variations of skin friction  2 with axial distance for different fluid models using cosine-shaped stenosis with 30% of stenosis, yield stress   = 0.05, and  = 4. Figure 15 shows the variations of skin friction  2 with axial distance for Casson fluid model with cosine-and bellshaped geometries with 30% stenosis,  = 4, and   = 0.05 (data computed using (28)).It is seen that the cosine-shaped stenosis has a greater width than that of bell-shaped stenosis as depicted in Figure 8.

It is observed that the skin friction of
Misra and Shit [17] have analyzed the variations of skin friction with axial distance for different values of power law index and yield stress and reported that the skin friction decreases with increase in yield stress.If normalized with Newtonian fluid, the skin friction increases considerably when the yield stress increases with axial distance for 30% stenosis,  = 4 as displayed in Figure 16.It is observed that the flow resistance decreases considerably when the length of the stenosis increases.Also, the variations of resistance to flow  2 with stenosis height for different values of yield stress and for  = 4 and  0 / = 0.3 is shown in Figure 19.It is clear that the resistance to flow increases significantly with the increase in the yield stress.

Flow Rate. Figure 20 depicts the variation of flow rate with axial distance for different values of viscosity coefficient
and yield stress   with 30% of stenosis.One can notice that the flow rate decreases significantly with the increase of either the viscosity coefficient or the yield stress.It is also observed that the flow rate decreases very significantly (nonlinearly) at the throat of the stenosis for lower values of either the yield stress or viscosity coefficient and slowly (almost linearly) for higher values of either the yield stress or viscosity coefficient.

Physiological Application.
To highlight some possible clinical applications of the present study, the data used by Sankar [21] (the radii of different arteries and flow rate) are used to compute the physiologically important flow quantities such as resistance to flow and skin friction.The values of the parameters are taken as  = 3.5 and   = 0.04.The dimensionless resistance to flow  2 and skin friction  2 are  computed for arteries with different radii and with  = 2 and  0 / = 1 from ( 23) and (28) (normalized with Newtonian fluid in stenosed and are presented in Table 2.It is noted that the resistance to flow increases, considerably when the stenosis height increases and skin friction significantly increases with the increase of stenosis height.The percentages of increase in resistance to flow and skin friction over that for uniform diameter tube (no stenosis) and for arteries with different radii are computed in Table 3.One can observe that the percentages of increase in resistance      a given set of values of the parameters, the percentage of increase in resistance to flow in the case of bell-shaped stenosed artery is slightly lower than that of cosine curve-shaped stenosed artery.This is physically verified with the depth of the stenosis for cosine-shaped stenosis (shown in Figure 8) being more; the resistance to flow is higher in this type of stenosis geometry compared to the bell-shaped stenosis geometry.

Conclusion
The present study analyzed the steady flow of blood in a narrow artery with bell-shaped mild stenosis, treating blood as Casson fluid, and the results are compared with the results of Misra and Shit for, Herschel-Bulkley fluid model [17] and also with the results of Chaturani and Ponnalagar Samy [16] (for blood flow in cosine curve-shaped stenosed arteries, treating blood as Herschel-Bulkley fluid model).The main findings of the present mathematical analysis are as follows: (ii) The effect of stenosis on blood flow is that the resistance to flow increases when either the stenosis length or depth increases.
(iii) The effect of non-Newtonian fluid on blood flow is that the resistance to flow increases significantly with the increase of yield stress, but it decreases when either the stenosis length or depth increases.
(iv) The percentage of increase in resistance to flow in the case of bell-shaped stenosed artery is slightly lower than that of the cosine curve-shaped stenosed artery in the case of normalization with respect to Newtonian fluid.
(v) Flow rate decreases with the increase of the yield stress and viscosity coefficient.
Hence, in view of the results obtained, we conclude that the present study may be considered as an improvement in the studies of the mathematical modeling of blood flow in narrow arteries with mild stenosis.Radial coordinate : Axial coordinate : Radial velocity  0 : Radius of the normal artery (): Radius of the artery in the stenosed portion : Half-length of segment of the narrow artery  0 : Half-length of the stenosis : Pressure : Viscosity coefficient.

Figure 1 :
Figure 1: Geometry of the arterial segment with stenosis.

Figure 2 :
Figure 2: Stenosis geometries for different values of stenosis height .

Figure 2 .
Figure2.The percentage of stenosis is given by  0 / × 100.In the present study, we have taken  = 5 cm and  = 2 as taken by Misra and Shit[17].Figure3(a) depicts the shapes of stenoses with different lengths by fixing  = 0.2 and keeping  as variable (for different values of stenosis the length  0 with fixed value of  = 2).Figure 3(b) shows the shapes of stenoses with different lengths by fixing  = 0.2 and varying the values of  (for different values of  and with a fixed value of  0 = 1.5).It is noticed that the width of the stenosis decreases with the increase in the values of .Equations (1) and (2) have to be solved with the help of the following no slip boundary condition:

Figure 3 (
Figure2.The percentage of stenosis is given by  0 / × 100.In the present study, we have taken  = 5 cm and  = 2 as taken by Misra and Shit[17].Figure3(a) depicts the shapes of stenoses with different lengths by fixing  = 0.2 and keeping  as variable (for different values of stenosis the length  0 with fixed value of  = 2).Figure 3(b) shows the shapes of stenoses with different lengths by fixing  = 0.2 and varying the values of  (for different values of  and with a fixed value of  0 = 1.5).It is noticed that the width of the stenosis decreases with the increase in the values of .Equations (1) and (2) have to be solved with the help of the following no slip boundary condition:

Figure 3 :
Figure 3: (a) Shapes of the arterial stenosis for different values of the stenosis length  0 .(b) Shapes of the arterial stenosis for different values of the stenosis shape parameter .

5 HFigure 6 :
Figure 6: Variations of skin friction  1 with axial distance for different fluids.

Figure 7 :
Figure 7: Variations of skin friction  1 for Casson fluid with axial distance for different arterial geometries.

Figure 11 .
Figure 11.It is clear that the skin friction increases marginally when the yield stress increases.Figure12depicts the variations of skin friction  2 with axial distance for different values of viscosity coefficient  for   = 0.05 and 30% of stenosis.One can notice that the skin friction decreases slightly as viscosity increases.Figure13shows the variations of skin friction  2 with axial distance for different fluid models using cosine-shaped stenosis with 30% of stenosis, yield stress   = 0.05, and  = 4.It is observed that the skin friction of Casson fluid model lies between those of the Herschel-Bulkley fluid model with  = 0.95 and  = 1.The variations of skin friction  2 with axial distance for different fluid models using bell-shaped stenosis with 30% of stenosis, yield stress   = 0.05, and  = 4 are shown in Figure14.It is observed that the skin friction of Casson fluid model lies between those of the Herschel-Bulkley fluid model with  = 0.95 and  = 1.Figure15shows the variations of skin friction  2 with axial distance for Casson fluid model with cosine-and bellshaped geometries with 30% stenosis,  = 4, and   = 0.05 (data computed using (28)).It is seen that the cosine-shaped stenosis has a greater width than that of bell-shaped stenosis as depicted in Figure8.Misra and Shit[17] have analyzed the variations of skin friction with axial distance for different values of power law index and yield stress and reported that the skin friction decreases with increase in yield stress.If normalized with Newtonian fluid, the skin friction increases considerably when the yield stress increases with axial distance for 30% stenosis,  = 4 as displayed in Figure16.
Figure 11.It is clear that the skin friction increases marginally when the yield stress increases.Figure12depicts the variations of skin friction  2 with axial distance for different values of viscosity coefficient  for   = 0.05 and 30% of stenosis.One can notice that the skin friction decreases slightly as viscosity increases.Figure13shows the variations of skin friction  2 with axial distance for different fluid models using cosine-shaped stenosis with 30% of stenosis, yield stress   = 0.05, and  = 4.It is observed that the skin friction of Casson fluid model lies between those of the Herschel-Bulkley fluid model with  = 0.95 and  = 1.The variations of skin friction  2 with axial distance for different fluid models using bell-shaped stenosis with 30% of stenosis, yield stress   = 0.05, and  = 4 are shown in Figure14.It is observed that the skin friction of Casson fluid model lies between those of the Herschel-Bulkley fluid model with  = 0.95 and  = 1.Figure15shows the variations of skin friction  2 with axial distance for Casson fluid model with cosine-and bellshaped geometries with 30% stenosis,  = 4, and   = 0.05 (data computed using (28)).It is seen that the cosine-shaped stenosis has a greater width than that of bell-shaped stenosis as depicted in Figure8.Misra and Shit[17] have analyzed the variations of skin friction with axial distance for different values of power law index and yield stress and reported that the skin friction decreases with increase in yield stress.If normalized with Newtonian fluid, the skin friction increases considerably when the yield stress increases with axial distance for 30% stenosis,  = 4 as displayed in Figure16.

Figure 10 :Figure 11 :
Figure 10: Variations of skin friction  2 with axial distance for different values of yield stress   .

2 Figure 12 : 2 Figure 13 :
Figure 12: Variations of skin friction  2 with axial distance for different values of viscosity .

2 Figure 14 :
Figure 14: Variations of skin friction  2 with axial distance for different fluid models with bell-shaped stenosis.

Figure 15 :
Figure 15: Variations of skin friction  2 of Casson fluid model with axial distance for different geometries.

Figure 16 :
Figure 16: Variations of skin friction with axial distance for different values of power index and yield stress   .
Shear stress   : Yield stress for Casson fluid   : Skin friction in stenosed artery  1 : Nondimensional skin friction normalized with non-Newtonian fluid  2 : Nondimensional skin friction normalized with Newtonian fluid   : Skin friction in normal artery normalized with non-Newtonian fluid    : Skin friction in normal artery normalized with Newtonian fluid : Stenosisheight : Flow resistance  1 : Nondimensional flow resistance normalized with non-Newtonian fluid  2 : Nondimensional flow resistance normalized with Newtonian fluid   : Flow resistance in normal artery normalized with non-Newtonian fluid    : Flow resistance in normal artery normalized with Newtonian fluid : Volumetricflowrate :

Table 1 :
Variations of the skin friction  1 with axial distance for different values of viscosity with   = 0.05 and  0 / = 0.3.

Table 2 :
Estimates of resistance to flow  2 and skin friction (dimensionless)  2 in arteries with different radii with  0 / = 1.

Table 4 .
It is recorded that the resistance to flow in bell-shaped stenosed artery increases considerably when the length of the stenosis increases, and also it significantly increases when stenosis height increases.But it marginally decreases with the increase of the stenosis length parameter .It is also found that for

Table 3 :
Estimates of the percentage of increase in resistance to flow  2 and skin friction  2 for arteries with different radii with  0 / = 1 and   = 0.04 dyn cm −2 .

Table 4 :
Estimates of percentage of increase in resistance to flow  2 for arteries with different radii and with (i) bell-shaped stenosis (ii) cosine curve-shaped stenosis.