1 Introduction

The presence of constriction (medically called stenosis) in the lumen of an artery disturbs the normal blood flow and causes arterial diseases (myocardial infarction, hypertension, and cerebral strokes). It is believed that hydrodynamic factors (e.g., wall shear stress) play a pivotal role in the development and progression of arterial stenosis. It is further evident that the investigation of blood flow in tapered arteries could play an important role in the fundamental understanding, diagnosis, and treatment of many cardiovascular diseases (Chaturani and Ponnalagarsamy, 1983; Dwivedi et al., 1982; Ponnalagarsamy and Kawahara, 1989). Looking at the immense importance of the fundamental understanding of blood flow, the objective of this analysis is motivated to provide a generalized model of blood and obtain some information about the flow.

Chakravarthy and Mandal (2000), How and Black (1987), Mandal (2005), Oka (1973), and Oka and Murata (1969) studied the flow of blood through tapered arteries by treating blood as a Newtonian fluid, Bingham plastic fluid, Power-law fluid, and Casson fluid, and they found the relationship between the flow rate and pressure drop. Scott Blair and Spanner (1974) suggested that blood obeys Casson’s model only for moderate shear rate flows, and that there is no difference between Casson’s and Herschel-Bulkley plots over the range where Casson’s plot is valid (for blood). Furthermore, Sacks et al. (1963) have experimentally pointed out that blood shows behavior characteristic of a combination of Bingham plastic and pseudoplastic fluid–Herschel-Bulkley fluid with the fluid behavior index greater than unity. In view of the experimental observation (Sacks et al., 1963) and a suggestion made in Scott Blair and Spanner (1974), it is pertinent to consider the behavior of blood as a Herschel-Bulkley fluid. Based on the foregoing views, it is worthwhile to describe a model taking the factors of pulsatility, nonuniform cross-section of a tube, and non-Newtonian character into the present analysis and study the flow characteristics.

The studies performed by Chaturani and Ponnalagarsamy (1986), Aroesty and Gross (1972a,b), Sankar and Hemalatha (2006, 2007) and Dash et al. (1999) reveal that they adopted a standard perturbation technique and obtained an approximate solution in which the flow characteristics are expressed as asymptotic natures in powers of the Womersley number α (Womersley, 1955). Rohlf and Tenti (2001) argued that the use of the Womersley number as a perturbation parameter to produce approximate solutions of the pulsatile flow of non-Newtonian fluid is not suitable; and considering the Reynolds number (ɛ) as a perturbation parameter, they made an attempt to validate their results through their perturbation theory in comparison with a numerical integration of the full mathematical model for blood flow.

It is further observed that the Womersley number α is not dependent on the flow velocity, and consequently, the same value of α can represent massively different flow conditions; hence, the Womersley number α is not an appropriate perturbation parameter. Sankar and Hemalatha (2006, 2007) and Sankar (2011) analyzed the pulsatile flow of Herschel-Bulkley fluid through the arteries. It is pertinent to point out here that the analytic expressions for flow variables such as velocity, wall shear stress, and flow rate obtained by Sankar and Hemalatha (2006, 2007), Sankar (2011), and by the present investigation, respectively, are entirely different. The reason is that they neglected the higher-order terms in the binomial expansion of the relationship between the velocity gradient and the shear stress involved the constitutive equation of Herschel-Bulkley fluid. Hence, the results obtained by Sankar and Hemalatha (2006, 2007) and Sankar (2011) do not represent the actual behavior of biorheological flow characteristics. Also, they have not derived the analytic expression for flow resistance. In view of this circumstance, we sought approximate solutions to the nonlinearity of the equation of motion (involved in this analysis) in terms of the Reynolds number (ɛ) as the perturbation parameter and derived the analytic expressions for velocity profile, wall shear stress, flow rate, and mean flow resistance. Further, an effort has been made to investigate the influence of the tapered angle on the flow-resistance and stream-line patterns.

The combined effects of the non-Newtonian behavior of the blood, taper angle, and the axial distance have been discussed. Section 2 of this chapter deals with the general mathematical formulation of the problem, in which the equation of motion using Herschel– Bulkley’s constitutive equation and the nondimensionalization procedure are given. In section 3, the analytic expressions for flow quantities such as velocity, flow rate, wall shear stress, and flow resistance are obtained. Comparison between the velocities obtained by the present mathematical model and the experimental observation is made, and the effects of parameters of the present study on wall shear stress and flow resistance are discussed and analyzed in section 4. Finally, in section 5, the main conclusions of the work are drawn and the future research works have been indicated.

2 Formulation of the problem

Consider a laminar, pulsatile, and fully developed flow of blood (i.e., Herschel-Bulkley fluid) in the z* -direction through a slightly tapered artery, as shown in Figure 3.1. The wall profile of the flow geometry may mathematically be described as

$R^{*}\left(z^{*}\right)=R_0^{*}-z^{*}tan\varphi ,$

where R*(z*) is the radius of the tube in the tapered section, R₀* is the radius of the tube in the normal region, z* is the axial direction, and ϕ is the angle of taper. Further, L₁* represents the axial distance of the cross section between z* = 0 and the cone apex, and L₂* indicates the axial distance of any cross section at z* from the apex. We shall take cylindrical coordinate system (r*, ϕ*, z*), whose origin is located on the tube axis.

Let us introduce the following nondimensional variables:

$r=\frac{r^{*}}{R_0^{*}},z=\frac{z^{*}}{R_0^{*}},t=\frac{t^{*}}{T^{*}},u=\frac{u^{*}}{U_0^{*}},\theta =\frac{{\tau }_y^{*}}{\left({\mu }^{*}\frac{U_0^{*}}{R_0^{*}}\right)},\tau =\frac{{\tau }^{*}}{\left({\mu }^{*}\frac{U_0^{*}}{R_0^{*}}\right)},p=\frac{p^{*}}{\left({\mu }^{*}\frac{U_0^{*}}{R_0^{*}}\right)},$

where T* is the characteristic time; μ* the Newtonian viscosity; U₀* the characteristic velocity, which is expressed by the relation R₀* = U₀* T*; u* the axial component of the velocity; t* the time; r* the radial direction; τ_y* the yield stress; p* the fluid pressure; and τ* the shear stress. (* over a letter denotes the dimensional form of the corresponding quantity.) Further, θ is the dimensionless yield stress.

Based on the discussions made by Oka (1973), Oka and Murata (1969) and Ponalagusamy (1986), the radial velocity is negligibly small and can be neglected for a low Reynolds number flow through a tapered tube with an angle of taper up to 2⁰. Since the lumen radius is very small in comparison with the wavelength of the pressure wave, equation of motion in the radial direction reduced to $-\frac{\partial p}{\partial r}=0,$ and hence the fluid pressure becomes a function of the axial distance (z) and time (t).

Keeping these in mind, and using nondimensional variables, the momentum equations governing the flow are given as

$\begin{array}{l}\varepsilon \frac{\partial u}{\partial t}=\beta q\left(z\right)f\left(t\right)+\frac{1}{r}\frac{\partial }{\partial r}\left[r\tau \right],\\ \end{array}$

$\frac{\partial u}{\partial r}=\frac{1}{k}{\left[1-\frac{\theta }{\left|\tau \right|}\right]}^n{\left|\tau \right|}^{n-1}\tau ,\mathit{if}\left|\tau \right|\ge \theta$

$=0,\mathit{if}\left|\tau \right|\le \theta ,$

where

$k=\left(\frac{k^{*}}{{\mu }^{*}}\right){\left(\frac{R_0^{*}}{U_0^{*}{\mu }^{*}}\right)}^{n-1},\varepsilon =U_0^{*}{R}_0^{*}\frac{{\rho }^{*}}{{\mu }^{*}},\beta =\frac{q_0^{*}{R_0^{*}}^2}{{\mu }^{*}{U}_0^{*}},q\left(z\right)=-\frac{\partial p}{\partial z}=\frac{q^{*}\left(z^{*}\right)}{q_0^{*}}\text{.}$

In that expression, $f\left(t\right)=$ 1 + A sin(Ωt), k* is the consistency index of blood, n is the Power-law index, q*(z*) is the pressure gradient in the tapered arterial region, and q₀* is the constant pressure gradient in the normal tube region. Eqs. (3.4) and (3.5) are reduced to that for a Bingham fluid, when $n=1.0$, to that for a Power-law fluid when θ = 0.0, and to that for a Newtonian fluid when $n=1.0$ and θ = 0.0. Here, Ω = $\frac{{\omega }^{*}{R}_0^{*}}{U_0^{*}}$ is the Strouhal number, where ω* is the frequency of the oscillations of the flow. The dimensionless parameter ɛ is the Reynolds number of the flow. Taking $\beta =1$, characteristic velocity U₀* is expressed as

$U_0^{*}=\frac{q_0^{*}{R_0^{*}}^2}{{\mu }^{*}}$

Consistency then requires that the time scale be derived as

$T^{*}=\frac{{\mu }^{*}}{q_0^{*}{R}_0^{*}}$

The geometry of the tapered tube in dimensionless form is given by

$R\left(z\right)=1-ztan\varphi$

The boundary conditions in dimensionless form are

$\left(\mathrm{i}\right)\tau \mathrm{is}\mathrm{finite}\mathrm{at}r=0\mathrm{and}\left(\mathrm{i}\mathrm{i}\right)u=0\mathrm{at}r=R\left(z\right)$

The volumetric flow rate Q (t) is given by

$Q\left(t\right)=2{\displaystyle \underset{0}{\overset{R\left(z\right)}{\int }}ru\left(r,z,t\right)dr}$

As mentioned elsewhere (Ponnalagarsamy and Kawahara, 1989), we take

$R\left(z\right)=1-z\varphi \mathrm{and}z=L_1-L_2$

3 Solution

The flow variable G is assumed to have the following form:

$G\left(r,z,t\right)=G_0\left(r,z,t\right)+\varepsilon G_1\left(r,z,t\right)+O\left({\varepsilon }^2\right),$

where G(r, z, t) refers to the velocity and shear stress. In what follows, for convenience, we write only function notation deleting its variables. Substituting Eqs. (3.4)–(3.5) into Eq. (3.3) and integrating twice with the help of the boundary conditions [Eq. (3.9)], the analytic expression for velocity distribution may be obtained as follows:

$\begin{array}{l}u=\frac{{\left(qf\right)}^n{\left(1-z\varphi \right)}^{n+1}}{k\left(n+1\right)2^n}[{\left(1-S\right)}^{n+1}-{\left(r/\left(1-z\varphi \right)-S\right)}^{n+1}+\left\{\mathit{n\varepsilon A}\mathrm{\Omega }cos\left(\mathrm{\Omega }t\right){\left(1-z\varphi \right)}^{n+1}/2kf\right\}{\left(qf\right)}^{n-1}.\\ \left[{\left(1-S\right)}^n\left(n+S\right)\right\{\frac{{\left(1-S\right)}^{n+1}-{\left(\frac{r}{\left(1-z\varphi \right)}-S\right)}^{n+1}}{n+1}+\left({\left(1-S\right)}^n-{\left(\frac{r}{\left(1-z\varphi \right)}-S\right)}^n\right)/n\\ \\ -\frac{n}{\left(n+1\right)\left(n+3\right)}\left(1-{\left(\frac{r}{\left(1-z\varphi \right)}\right)}^{2n+2}\right)\}+\frac{S\left(4n^3+8n^2-6n-6\right)}{\left(n+2\right)\left(n+3\right)\left(2n+1\right)}\left(1-{\left(\frac{r}{\left(1-z\varphi \right)}\right)}^{2n+1}\right)\\ \\ -2S^2{\displaystyle \sum _{j=0}^{2n-1}\frac{{\left(-1\right)}^j\left(2n\right)c_j{S}^j}{\left(2n-j\right)}}\left(1-{\left(\frac{r}{\left(1-z\varphi \right)}\right)}^{2n-j}\right)\left\{\frac{\left(2n+1\right)\left(2n+2\right)}{\left(j+1\right)\left(j+2\right)\left(n+3\right)}-\frac{{\left(2n+1\right)}^2}{\left(j+1\right)\left(n+2\right)}+1\right\}\\ \\ +\frac{6{\left(-1\right)}^{2n}}{\left(n+2\right)\left(n+3\right)}{S}^{2n+2}log\left(\frac{r}{\left(1-z\varphi \right)}\right)\left]\right]\end{array}$

Multiplying Eq. (3.13) by r and integrating with respect to r, the analytic expression for a stream function ψ(r, z) may be obtained. Invoking Eq. (3.10) into Eq. (3.13), after tedious manipulation, the analytic expression for flow rate is obtained as

$\begin{array}{l}Q\left(t\right)=\frac{{\left(qf\right)}^n{\left[\left(1-z\varphi \right)\right]}^{3+n}}{2^{n-1}k\left(n+1\right)\left(n+2\right)\left(n+3\right)}[{\left(1-S\right)}^{n+1}\left\{\left(n+2\right)\left(n+1\right)+2S\left(n+1+S\right)\right\}\\ +\frac{\mathit{n\varepsilon A}\mathrm{\Omega }cos\left(\mathrm{\Omega }t\right)k{\left\{1-z\varphi \right\}}^2}{2^{2n-2}f}{\left[q{f}_1\left(1-z\varphi \right)\right]}^{n-1}[{\left(1-S\right)}^{2n}\left(n+S\right)\\ \left\{\left(n+2\right)+3S+\frac{6S^2}{n\left(n+1\right)}\left(1+S\right)\right\}-n\left(1-S^{2n+4}\right)+\frac{S\left(4n^3+8n^2-6n-6\right)}{\left(2n+3\right)}\\ \left(1-S^{2n+3}\right)-3S^{2n+2}\left(1-S^2\right)-2S^2{\displaystyle \sum _{j=0}^{2n-1}\frac{{\left(-1\right)}^j\left(2n\right)c_j{S}^j}{\left(2n+2-j\right)}}\{\frac{\left(2n+1\right)\left(2n+2\right)\left(n+2\right)}{\left(j+1\right)\left(j+2\right)}\\ -\frac{\left(n+3\right){\left(2n+1\right)}^2}{\left(j+1\right)}+\left(n+2\right)\left(n+3\right)\left(1-S^{2n+2-j}\right)\left\}\right]],\end{array}$

where $S=2\theta /\left\{q{f}_1\left(1-z\varphi \right)\right\}\text{.}$

It is of interest to note that the Womersley number α is obtained as

$\alpha ={\left(\varepsilon \mathrm{\Omega }\right)}^{\frac{1}{2}}=R_0^{*}{\left(\frac{w^{*}}{{\gamma }^{*}}\right)}^{\frac{1}{2}}$, where γ* is the kinematic viscosity.

The steady flow rate Q_s is expressed as

$\begin{array}{l}{Q}_s=\frac{q^n{\left\{\left(1-z\varphi \right)\right\}}^{3+n}}{2^{n-1}k\left(n+1\right)\left(n+2\right)\left(n+3\right)}[{\left\{1-\frac{2\theta }{q\left(1-z\varphi \right)}\right\}}^{n+1}\\ \left\{\left(n+2\right)\left(n+1\right)+\left\{\frac{4\theta }{q\left(1-z\varphi \right)}\right\}\left\{n+1+\frac{2\theta }{q\left(1-z\varphi \right)}\right\}\right\}]\text{.}\end{array}$

The shear stress on the wall τ_w is a physiologically important quantity, given by

${\tau }_w=\frac{qf\left(1-z\varphi \right)}{2}\left[1+\frac{\varepsilon A\mathrm{\Omega }cos\left(\mathrm{\Omega }t\right){\left\{1-z\varphi \right\}}^{n+1}{\left\{\frac{qf}{2}\right\}}^{n-1}}{2^{n+1}\left(n+1\right)\left(n+2\right)\left(n+3\right)f}.\left\{1-S^n\right\}\left\{\begin{array}{l}n\left(n+1\right)\left(n+2\right)+\\ 3n\left(n+1\right)S+6n{S}^2+6S^3\end{array}\right\}\right]\text{.}$

It is pertinent to mention here that Chaturani and Ponnalagarsamy (1986) explained the method of calculating the value of the steady pressure gradient q(z) for any value of θ using Eq. (3.14).

The flow resistance λ is defined as

$\lambda =\frac{f\left(t\right)\Delta p}{Q\left(t\right)},$

where Δp is the pressure drop.

The mean flow resistance over the period of the flow cycle is defined as

$\overline{\lambda }=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(\frac{p_0-p_1}{Q\left(t\right)}\right)f\left(t\right)dt}\text{.}$

For any value of S, one can numerically compute the values of the flow resistance λ and the mean flow resistance $\overline{\lambda }$, respectively, from Eqs. (3.17) and (3.18) for different values of the parameters involved in this investigation. It is mathematically and physiologically important to obtain an analytic expression for the mean flow resistance. Eqs. (3.14) and (3.18) reveal that it is not possible to obtain the analytic expression for the mean flow resistance for any value of S. For small values of S [with fluid having low yield stress value (θ), such as blood (Sacks et al., 1963)], the analytic expression for the mean flow resistance from Eqs. (3.14) and (3.18) may be obtained as

$\overline{\lambda }=\left(\frac{n}{3\varphi }\right){\left(2^{n}k\left(n+3\right)\right)}^{\frac{1}{n}}\overline{Q}\left\{{\left(\frac{1}{1-z\varphi }\right)}^{3/n}-1\right\}+\left\{\frac{2\left(n+3\right)\theta Q^{*}}{\varphi \left(n+2\right)}\right\}log\left(\frac{1}{1-z\varphi }\right),$

where

$\overline{Q}=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{2\pi }{\int }}\frac{dt}{{\left[Q\left(t\right)\right]}^{1-\frac{1}{n}}}}\mathrm{and}{Q}^{*}=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{2\pi }{\int }}\frac{dt}{Q\left(t\right)}}\text{.}$

The values of $\overline{Q}$ and Q* are to be numerically computed after computing the value of steady pressure gradient q(z) using Eq. (3.15) for different values of the parameter involved in the present paper.

4 Discussion

The results of a comparison between the velocity profiles for Newtonian, Power-law, Bingham plastic, and Herschel-Bulkley fluids obtained from the present study and the experimental data (Ponalagusamy, 1986) is shown in Figure 3.2. The graph reveals that blood behaves more like a Herschel-Bulkley fluid than like Bingham plastic, Power-law, and Newtonian fluids.

It is of interest to note from Figure 3.3 that the flow rate decreases with the increase in the axial distance (Z) and taper angle. The percentage of decrease in the flow rate as the value of Z increases is found to be higher for higher values of the taper angle.

The analytic expression for wall shear stress is derived and its variation with the axial distance and the taper angle has been studied (see Figure 3.4). This graph shows that as the taper angle increases, the rate of increase in the wall shear stress with respect to an increase in the axial distance Z is found to be very significant. Another pivotal result is concerning the variation of mean flow resistance ($\overline{\lambda }$) with respect to the taper for different values as the axial distance (Z) and it is shown in Figure 3.5. The mean flow resistance increases as the value of the taper angle increases. The main effect of pulsatility on the flow is the phase lag between the pressure gradient and flow rate and wall shear stress. It can be seen from the present analysis that the phase lag between pressure gradient and flow rate (or wall shear stress) has been found to be 2.03 deg, and its value is unaltered as the values of axial distance and taper angle increase or decrease. It may be important to note that many standard results regarding steady and uniform tube flow of Power-law, Bingham, and Newtonian fluids can be obtained as special cases in the present investigation.

The influence of the taper angle (ϕ) on the stream-line pattern has been analyzed for a given value of $n=1.4,K=1.2,\theta =0.1,A=0.5,{\alpha }^2=0.5,Q_s=1.0$ and $\mathrm{\Omega }t={88}^{\circ }$, and shown in Figure 3.6. It is observed that the value of stream function decreases as the taper angle (ϕ) increases.

5 Conclusion

The main objective of this investigation was studying the problem of pulsatile flow of blood (Herschel-Bulkley fluid) through a tapered arterial stenosis. A perturbation technique was adopted to study the flow. The analytical expressions for velocity, flow rate, wall shear stress, and mean flow resistance were obtained, and the results depicted in graphs. Using the finite volume technique, the quasi-steady, nonlinear, coupled, implicit system of differential equations has been solved numerically and the axial velocity computed. It is verified that the error between the axial velocities obtained by the present perturbation method and the numerical technique becomes less than 1.052% for the values of α² between 0.0 and 1.0. Furthermore, the error becomes more than 9.0% when α² is greater than 2.10.

When the flow characteristics are expressed in terms of α², the present perturbation results coincide with the results found in other papers (Sankar and Hemalatha, 2006, 2007). Hence, our predictions coincide with theirs and further comparison is unnecessary, so long as α² < 1.0. Further, their predictions are valid when the Reynolds number is small (< 10) since the Strouhal number Ω is unity in their analyses. But our approach is applicable even to larger blood vessel and moderate Reynolds numbers. One of the most remarkable merits of the present perturbation scheme is that it is very suitable to any mathematical models of blood flow in tubes with uniform and nonuniform cross sections compared to the models developed by others (Chaturani and Ponnalagarsamy, 1986; Aroesty and Gross, 1972a, 1972b; Sankar and Hemalatha, 2006, 2007; Sankar, 2011).

The theoretically computed velocity profiles were compared with the experimental data, and it was observed that blood behaves like a Herschel-Bulkley fluid rather than Power-law or Bingham fluid. The increase in the taper angle (ϕ) leads to a decrease in the flow rate, wall shear stress, and resistance to flow. It is evident that for abnormal hearts, an increase in shear stress on the blood vessel could be very dangerous, as it can result in paralysis or ultimate death. The resistance to flow is one of the physiologically important flow variables to be investigated because it indicates whether the required amount of blood supply to vital organs is ensured (Ponalagusamy, 2007, 2012; Chaturani and Ponnalagarsamy, 1984).

It is well established that hemodynamic factors (such as wall shear stress, flow resistance) play a key role in the development and progression of arterial diseases (Fry, 1973). Caro et al. (1971) experimentally demonstrated that during the initial stage of arterial disease, there may be an important intercorrelation between atherogenesis and detailed characteristics of blood flow through the damaged, diseased, or otherwise affected artery. Keeping in view the importance of hemodynamic and rheologic factors in the understanding of blood flow and arteriosclerostic diseases, it may be said that the important results obtained in this analysis could be helpful to acquire knowledge regarding the characteristics of blood flow. Hence, the present investigation could be useful for analyzing the blood flow through a tube of nonuniform cross section, which in turn could lead to the development of new diagnostic tools for the effective treatment of patients suffering from cancer, hypertension, myocardial infarction, stroke, and paralysis.

Zamir (2000) pointed out that the oscillatory nature of pulsatile flow of blood prompts other forces, apart from driving and retarding forces in the case of steady flow, and other variables and the heat and mass transport through endothelial cells lying in the inner layer of the vessel wall is very much altered when viscoelastic properties of blood and its vessel wall have been taken into account. When artery walls are viscoelastic, a 10% variation in the artery radius over a cardiac cycle is typically observed and the shear stress at the wall is primarily affected by the radial wall motion in comparison with that of rigid arteries. Bugliarello and Sevilla (1970) and Bugliarello and Hayden (1963) have experimentally observed that there exists a cell-free plasma layer near the wall when blood flows through arteries. It is well understood that blood consists of a suspension of a variety of cells. Hookes et al. (1972) pointed out that the microrotation and spinning velocity of blood cells increase flow resistance and wall shear stress. In view of their experiments and the aforementioned arguments, it is preferable to represent the flow of blood through arteries with their viscoelastic nature by a two-layered model instead of one layer and the rheology of blood as a micropolar viscoelastic fluid while investigating the realistic mathematical model on investigating blood flow. Hence, a modest effort will be made to investigate the problem of blood flow by incorporating the factors mentioned in this chapter (two or three factors at a time, since it is impossible to consider all the factors simultaneously) and the numerical findings will be published in the future.