**American Journal of Applied Mathematics and Statistics**

## Multiple Stenotic Effect on Blood Flow Characteristics in Presence of Slip Velocity

Department of Mathematics, Shyampur Siddheswari Mahavidyalaya, Howrah, IndiaAbstract | |

1. | Introduction |

2. | Mathematical Formulation |

3. | Numerical Discussions |

4. | Conclusions |

References |

### Abstract

The aim of the present analysis is to study the effect of slip velocity on blood flow through an arterial tube in presence of multiple stenosis. The effects of length of stenosis, shape parameter, parameter γ on resistance to flow and shear stress have been incorporated here. The results have been shown in graphical form and discussed.

**Keywords:** resistance to flow, wall shear stress, stenosis, shape parameter, Herschel-Bulkley flui.

**Copyright**© 2016 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Arun Kumar Maiti. Multiple Stenotic Effect on Blood Flow Characteristics in Presence of Slip Velocity.
*American Journal of Applied Mathematics and Statistics*. Vol. 4, No. 6, 2016, pp 194-198. http://pubs.sciepub.com/ajams/4/6/5

- Maiti, Arun Kumar. "Multiple Stenotic Effect on Blood Flow Characteristics in Presence of Slip Velocity."
*American Journal of Applied Mathematics and Statistics*4.6 (2016): 194-198.

- Maiti, A. K. (2016). Multiple Stenotic Effect on Blood Flow Characteristics in Presence of Slip Velocity.
*American Journal of Applied Mathematics and Statistics*,*4*(6), 194-198.

- Maiti, Arun Kumar. "Multiple Stenotic Effect on Blood Flow Characteristics in Presence of Slip Velocity."
*American Journal of Applied Mathematics and Statistics*4, no. 6 (2016): 194-198.

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### At a glance: Figures

### 1. Introduction

As blood is a suspension of fluid particles, such as erythrocytes, leukocytes and platelets in an aqueous solution, blood behaves neither Newtonian nor non-Newtonian fluid. The behaviour of blood is almost Newtonian at high shear rate, while at low shear rate the blood exhibits yield stress and then blood behaves as a non-newtonian fluid ^{[1, 2]}). Many biomedical researchers (Eckstein et. al. ^{[3]}, Choung et. al. ^{[4]}, Fung ^{[5]}, Sparks ^{[6]}, Tilles et. al. ^{[7]}, Goldsmith ^{[8]}, Carr ^{[9]}) have analysed various types of mathematical models to study the blood flow characteristics for better understanding the physiological systems of human body. A large number of researchers (Feng et. al. ^{[10]}, Jilma ^{[11]}, Haldar ^{[12]}, Liepsch ^{[13]}) have presented various types of mathematical models by considering blood as Newtonian or non-Newtonian fluid to study the flow characteristics of blood. Presently in many cases it has been noticed that cardiovascular diseases are responsible for the death of the people. From various types of medical literature it is clear to us that around 80% of total death of people are due to the malfunction of blood flow characteristics. Blood flow characteristics mainly depends on the arterial diseases, such as stenosis and aneurysm. Stenosis is a serious arterial disease causes serious cardiovascular disorders, by reducing the supply of blood. Though actual formation of stenosis is somewhat unclear to us, it is well-known that deposition of fatty substances like lipids, cholesterol in the inner wall of the artery and unnatural growth of the connective tissue may be responsible for the formation of stenosis. Many biomedical researchers (Jung et. al. ^{[14]}, Mousa ^{[15]}, Sugihara Seki ^{[16]}, Cokelet ^{[17]}, Das et. al. ^{[18]}) have studied the non-Newtonian behaviour of blood to discuss the various aspects of blood flow. Various mathematical models have been developed by many Mathematicians (Young ^{[19]}, Shukla ^{[20]}, RamchandraRao and Devanathan ^{[21]}, Hall ^{[22]}, Manton ^{[23]}, Smith ^{[24]}, Duck ^{[25]}) to study the steady and unsteady flow of blood through channels with variable cross section. The steady flow of blood through stenotic arterial tube has been studied by many Mathematicians (Misra and Chakraborty ^{[26]}, Padmanavan ^{[27]}, Mehrotha et. al. ^{[28]}) by considering blood as non-Newtonian fluid, in presence of mild stenosis. In a recent paper Siddiqui ^{[29]}, Biswas and Laskar ^{[30]} have been developed mathematical models to study the blood flow characteristics through a stenosed arterial segment under slip condition. But they have considered the effect of single stenosis. But since stenosis may be developed in series or in overlapping form, in the present analysis I have considered the effect of multiple stenosis on blood flow by considering blood as Herschel-Bulkley type non- Newtonian fluid in presence of slip velocity on the arterial wall.

### 2. Mathematical Formulation

Let us consider the steady one dimensional laminar axially symmetric and radially non-symmetric constricted artery. The mathematical expression for stenosis is given by

(1) |

Where is the radius of the tube in the stenotic region; the radius of the artery outside the stenotic region; is a shape parameter determining stenosis shape; the length of the stenosis, indicates its location, is the number of stenosis in the artery, is a parametric constant.

(2) |

Where is the maximum height of the stenosis at

(3) |

**Fig**

**ure**

**1**. Geometry of arterial stenosis

The equation governing the flow of blood is given by

(4) |

The boundary conditions are,

(i) is finite at (regular condition)

(ii) at (slip condition).

The relationship between shear stress and shear rate for Herschel-Bulkley fluid is given by,

(5) |

For simplicity the Herschel-Bulkley index is taken as . Therefore,

(6) |

Integrating (4) and using the boundary condition (i) we get,

(7) |

The skin friction is given by

(8) |

The volumetric flow rate is given by

(9) |

when the boundary condition (ii) is used, which can be written as

(10) |

When replacing by we get,

(11) |

From which we get,

(12) |

Since we get

(13) |

Integrating with respect to z with the condition that at and at

(14) |

Resistance to flow is given by

(15) |

Let us take

(16) |

Thus we can write

(17) |

(18) |

Where

(19) |

If there is no stenosis, i.e., in the normal condition

(20) |

Therefore dimensionless resistance to flow is given by

(21) |

Where

(22) |

Now

(23) |

Wall shear stress in normal situation can be written as

(24) |

Thus the wall shear stress ratio can be obtained as

(25) |

The wall shear stress ratio at the midpoint of stenosis is given by

(26) |

### 3. Numerical Discussions

To illustrate the flow behaviour, the results are shown graphically with the help of MATLAB-7.6. The effects of various parameters on resistance to flow and wall shear stress are calculated here.

Figure 2- Figure 6 show the variation of resistance to flow for different values of stenosis length, slip velocity, yield stress and shape parameters It is observed from the figures that increases with the increase of stenosis size and stenosis length for all other constant values of the parameters, but the reverse effectoccurs when the slip velocity and shape parameters and increase. It is also clear from the figures that increases with the increase of up to the value 0.18 of and then decreases.

**Figure 2.**Variation of non-dimensional resistance to flow for different values of stenosis length

**Fig**

**ure**

**3.**Variation of non-dimensional resistance to flow for different values of slip velocity

**Fi**

**gure**

**4.**Variation of non-dimensional resistance to flow for different values of yield stress

**Fig**

**ure**

**5.**Variation of non-dimensional resistance to flow with the variation of shape parameter

**Fig**

**ure**

**6.**Variation of non-dimensional resistance to flow with the variation of parameter

Figure 7-Figure 8 illustrate the effect of slip velocity and yield stress on wall shear stress. It is found that increases with the increase of slip velocity and yield stress as* z* increases.

**Fig**

**ure**

**7.**Variation of non-dimensional wall shear stress for different values of slip velocity

**Fig**

**ure**

**8.**Variation of non-dimensional wall shear stress for different values of yield stress

Figure 9 reveals that the variation of wall shear stress ratio at the midpoint of stenosis. It is found that increases with the increase of slip velocity.

**Fig**

**ure**

**9.**Fluctuation of non-dimensional wall shear stress at the midpoint of stenosis for different values of slip velocity

### 4. Conclusions

The analytic expression for resistance to flow with multiple stenosis situated at equal distances has been incorporated here, which has definite effect in various types of cardiovascular diseases, like stroke, hypertension, brain haemorrhage etc. It is observed that growing of has small variations for different value of stenosis shape parameter. This study may be effective for invention of new diagnostic tools in medical sciences.

### References

[1] | Y. C. Fung. Biodynamics. New York: Springer- Verlag, 1984. | ||

In article | |||

[2] | Y. C. Fung. Biomechanics: Circulation. Springer Science and Business Media, 1996. | ||

In article | |||

[3] | E. C. Eckstein and F. Belgacem. “Model of platelet transport in flowing blood with drift and diffusion terms,” Biophysical journal, vol. 60, no. 1, pp. 53-69, 1991. | ||

In article | View Article | ||

[4] | C. J. Choung and Y. C. Fung. “Residual stress in arteries,” Journal of Biomechanics, vol. 180, no. 2, pp. 189-192, 1986. | ||

In article | View Article | ||

[5] | Y. C. Fung. “A hemodynamic analysis of coronary capillary blood flow based on anatomic and distensibility data,” American Journal of Physiology-Heart and Circulatory Physiology, vol. 277, no. 6, pp. H2158-H2166, 1999. | ||

In article | |||

[6] | K. D. Sparks. “Platelet concentrationprofiles in blood flow through capillary tubes,” Ph.D. dissertation, MS thesis, University of Miami, Coral Gables, FL, 1983. | ||

In article | |||

[7] | A. W. Tilles and E. C. Eckstein. “The near-wall excess of platelet-sized particles in blood flow: its dependence on hematocrit and wall shear rate,” Microvascular research, vol. 33, no. 2, pp. 211-223, 1987. | ||

In article | View Article | ||

[8] | H. Goldsmith. “Red cell motions and wall interactions in tube flow.” in Federation Proceedings, vol. 30, no. 5, 1971, p. 1578. | ||

In article | PubMed | ||

[9] | R. Carr. “Estimation of hematocrit profile symmetry recovery length downstream from a bifurcation.” Biorheology, vol. 26, no. 5, pp. 907-920, 1988. | ||

In article | |||

[10] | J. Feng and S. Weinbaum. “Lubrication theory in highly compressible porous media: the mechanics of skiing, from red cells to humans,” Journal of Fluid Mechanics, vol. 422, pp. 281-317, 2000. | ||

In article | View Article | ||

[11] | B. Jilma. “Synergistic antiplatelet effects of clopidogrel and aspirin detected with the pfa-100 in stroke patients.” Stroke; a journal of cerebral circulation, vol. 34, no. 4, pp. 849-854, 2003. | ||

In article | View Article | ||

[12] | K. Haldar. “Effects of the shape of stenosis on the resistance to blood flow through an artery,” Bulletin of Mathematical Biology, vol. 47, no. 4, pp. 545-550, 1985. | ||

In article | View Article PubMed | ||

[13] | D. Liepsch. “An introduction to biofluid mechanicsbasic models and applications,” Journal of biomechanics, vol. 35, no. 4, pp. 415-435, 2002. | ||

In article | View Article | ||

[14] | H. Jung, J. W. Choi, and C. G. Park. “Asymmetric flows of nonnewtonian fluids in symmetric stenosed artery,” Korea-Australia Rheology Journal, vol. 16, no. 2, pp. 101-108, 2004. | ||

In article | |||

[15] | S. A. Mousa. “Antiplatelet therapies: from aspirin to gpiib/iiia-receptor antagonists and beyond,” Drug discovery today, vol. 4, no. 12, pp. 552-561, 1999. | ||

In article | View Article | ||

[16] | M. Sugihara-Seki. “Motion of a sphere in a cylindrical tube filled with a brinkman medium,” Fluid dynamics research, vol. 34, no. 1, pp. 59-76, 2004. | ||

In article | View Article | ||

[17] | G. R. Cokelet. The rheology of human blood. Englewood Cliffs: Prentice Hall: In Biomechanics, Ed. By Y. C. Fung et. al., Vol. 63, 1972. | ||

In article | PubMed | ||

[18] | B. Das, P. Johnson, and A. Popel. “Effect of nonaxisymmetric hematocrit distribution on non-newtonian blood flow in small tubes,” Biorheology, vol. 35, no. 1, pp. 69-87, 1998. | ||

In article | View Article | ||

[19] | D. Young. “Fluid mechanics of arterial stenoses,” Journal of Biomechanical Engineering, vol. 101, no. 3, pp. 157-175, 1979. | ||

In article | View Article | ||

[20] | J. Shukla, R. Parihar, and S. Gupta. “Effects of peripheral layer viscosity on blood flow through the artery with mild stenosis,” Bulletin of Mathematical Biology, vol. 42, no. 6, pp. 797-805, 1980. | ||

In article | View Article PubMed | ||

[21] | A. R. Rao and R. Devanathan. “Pulsatile flow in tubes of varying crosssections,” Zeitschrift für angewandte Mathematik und Physik ZAMP, vol. 24, no. 2, pp. 203-213, 1973. | ||

In article | |||

[22] | P. Hall. “Unsteady viscous flow in a pipe of slowly varying crosssection,” Journal of Fluid Mechanics, vol. 64, no. 02, pp. 209-226, 1974. | ||

In article | View Article | ||

[23] | M. Manton. “Low reynolds number flow in slowly varying axisymmetric tubes,” Journal of Fluid Mechanics, vol. 49, no. 03, pp. 451-459, 1971. | ||

In article | View Article | ||

[24] | F. Smith. “Flow through constricted or dilated pipes and channels: Part 2,” The Quarterly Journal of Mechanics and Applied Mathematics, vol. 29, no. 3, pp. 365-376, 1976. | ||

In article | View Article | ||

[25] | P. Duck. “Separation of jets or thermal boundary layers from a wall,” The Quarterly Journal of Mechanics and Applied Mathematics, vol. 363, no. 2, p. 33, 1976. | ||

In article | |||

[26] | J. Misra and S. Chakravarty. “Flow in arteries in the presence of stenosis,” Journal of Biomechanics, vol. 19, no. 11, pp. 1907-1918, 1986. | ||

In article | View Article | ||

[27] | N. Padmanabhan and R. Devanathan. “Mathematical model of an arterial stenosis, allowing for tethering,” Medical and Biological Engineering and Computing, vol. 19, no. 4, pp. 385-390, 1981. | ||

In article | View Article PubMed | ||

[28] | R. Mehrotra, G. Jayaraman, and N. Padmanabhan. “Pulsatile blood flow in a stenosed artery: Theoretical model,” Medical and Biological Engineering and Computing, vol. 23, no. 1, pp. 55-62, 1985. | ||

In article | View Article PubMed | ||

[29] | S. Siddiqui, N. Verma, and R. Gupta. “A mathematical model for pulsatile flow of herschel-bulkley fluid through stenosed arteries,” e-Journal of Science and Technology (e-JST), vol. 4, no. 5, pp. 49-66, 2010. | ||

In article | |||

[30] | D. Biswas and R. B. Laskar. “Steady flow of blood through a stenosed artery: A non-newtonian fluid model,” Assam University Journal of Science and Technology, vol. 2, no. 1, pp. 144-154, 2011. | ||

In article | |||