Abstract
In this article, we have discussed blood flow in the nano-Prandtl fluid flow analysis in tapered stenosed arteries. The occurrence of nanoparticle fraction and heat transfer was found. Gravitational effects were also considered because the tube was taken vertically upward. Homotopy perturbation method was used to find the analytical solution of coupled nonlinear differential equations. Physical features have been discussed through graphs of concentration profile σ, velocity profile w, resistance impedance λ, temperature profile θ, wall shear stress S r z , wall shear stress at the stenosis throat τ s , and the stream lines.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Background
Blood flow in the artery has some important aspects due to engineering as well as medical application points of view. The hemodynamic behavior of the blood flow is influenced by the presence of arterial stenosis. If stenosis is present in the artery, normal blood flow is disturbed. Thurston and Chien et al. [1, 2] present the viscoelastic properties of the blood. According to them, the arterial configuration is closely connected to blood flow. Arteries which are basically considered as living tissues need a supply of metabolites including oxygen and removal of waste products. Aroesty and Gross [3] have discussed the pulsatile flow of blood in the small blood vessels. Chaturani and Ponnalagar Samy [4] reported the theory of Aroesty and Gross [3] and studied the pulsatile flow of blood in stenosed arteries modeling blood as Casson fluid. Scott Blair and Spanner [5] discussed that blood as Casson fluid is valid for moderate shear rate, and the validity of Casson and Herschel-Bulkley theory for blood flow is the same. In another article, Siddiqui et al. [6] discussed Casson fluid in arterial stenosis. Pulsatile flow of blood for a modified second-grade fluid model is presented by Massoudi and Phuoc [7].
Mekheimer and El Kot [8] examined the micropolar fluid model for axisymmetric blood flow through an axially nonsymmetric but radially symmetric mild stenosis tapered artery. Mandal [9] presented unsteady flow analysis for blood by treating blood as a non-Newtonian fluid through tapered arteries with stenosis. He discussed the numerical solution for the flow equations. Varshney et al. [10] considered the generalized power law fluid model for blood flow in the artery considering multiple stenosis. They present a numerical study under the action of a transverse magnetic field. A power law fluid model for blood flow through a tapered artery with stenosis is recently developed by Nadeem et al. [11]. In another article, Nadeem and Akbar [12] revisited the flow analysis of Nadeem et al. [11] for Jeffrey fluid. Mustafa et al. [13] make the analysis of blood flow for the generalized Newtonian fluid through a couple of irregular arterial stenosis. Blood flow with an irregular stenosis in the presence of magnetic field has been touched by Abdullah et al. [14].
Nanofluids are the fluids of nanometer-sized particles of metals, oxides, carbides, nitrides, or nanotubes. Nowadays, nanofluids, among researchers, are considered an active area of research. In fact, nanofluids are a suspension of nanosized solid particles in a base fluid. The nanofluids have high thermal conductivity as compared to the base fluid. Nanofluids basically increase heat transfer rate. Recent articles on nanofluids have been cited [15–20].
The main theme of the present article is to discuss the nanofluid analysis for steady blood flow of the Prandtl model with stenosis. To the best of the authors’ knowledge, blood flow analysis for nanofluids has not been investigated so far. We arranged this article in the following manner. The ‘Methods’ section consists of mathematical formulation and the solution expressions for velocity, temperature, nanoparticle, resistance impedance, wall shear stress, and shearing stress at the stenosis throat. The ‘Results and discussion’ section analyzes the salient features of the problem by graphical illustration.
Methods
Formulation of the problem
Consider the flow of incompressible Prandtl fluid lying in a tube having the length L. We are considering the cylindrical coordinate system in such a way that , , and are the velocity components in , , and directions. The governing equations of the steady incompressible Prandtl fluid are as follows [11]:
In the presented equations, describes the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid, as the Brownian diffusion coefficient, as the thermophoretic diffusion coefficient, as the coefficient of thermal expansion, and as the coefficient of thermal expansion with nanoconcentration.
The geometry of stenosis is defined as follows [8]:
with
where Q(z) be the radius of the tapered arterial section, Q0 be the radius of the non-tapered arterial section, ζ be the tapering parameter, J1 be the stenosis length, and J0 indicates its place. The parameter η is defined as follows:
where δ denotes the maximum height of the stenosis located as follows:
Non-dimensional variables are as follows:
Making use of Equation 10 and after taking the condition, we get the following equation:
Equations 1 to 4, for the case of mild stenosis , take the the following form:
In the above equations, N t ,N b ,B r , and G r are the defined thermophoresis parameter as Brownian motion parameter, as local nanoparticle Grashof number, and as local temperature Grashof number, respectively. The boundary conditions are as follows:
where
and
where ϕ is the tapered angle and defined for the non-tapered artery (ϕ=0), for converging tapering (ϕ<0), and for diverging tapering (ϕ>0), as described in Figure 1.
Solution of the problem
Homotopy perturbation solution
The homotopy perturbation method suggests that we may write Equations 13, 14, and 15 as follows [21, 22]:
taking the following initial guesses:
We define
Putting Equations 23 to 26 into Equations 19 to 21 and taking Ķ →1, the following form for temperature profile, concentration profile, velocity profile, and pressure gradient are written as follows:
Pressure drop (Δ p=p at z=0 and Δ p=−p and z=L) through the stenosis between the regions z=0 and z=L computed from Equation 30 can be written as follows:
Resistance impedance
Using Equation 31, the expression for resistance impedance is given as follows:
where
Expression for the wall shear stress
Dimensionless shear stress is defined as follows:
or
Using Equation 27, we obtain the following:
The shearing stress at the stenosis throat located at is defined as follows:
or
The final expression for λ, S r z , and τ s can be defined as follows:
in which
Results and discussion
The quantitative results of the α, β, n, δ, N t ,a n d N b for diverging tapering, converging tapering, and non-tapered arteries are observed physically in Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24. Variations of velocity profile for α, β, n, δ, N t , and N b for the cases of converging tapering, diverging tapering, and non-tapered arteries are displayed in Figures 2, 3, 4, 5, 6, and 7. In Figures 2, 3, 4, 5, 6, and 7, it is analyzed that with an increase in the thermophoresis parameter N t and stenosis shape n, the velocity profile decreases, while velocity profile increases with an increase in the maximum height of the stenosis δ, Brownian motion parameter N b , and Prandtl parameters α and β. It is also seen that for the case of converging tapering, it has a larger value as compared with the case of diverging tapering and non-tapered arteries. Figures 8, 9, 10, 11, 12, and 13 depict how the converging tapering, diverging tapering, and non-tapered arteries influence the wall shear stress S r z . It is observed that with an increase in stenosis shape n, thermophoresis parameter N t , stenosis height δ, and Prandtl parameter β, the shear stress decreases, while it increases with an increase in Prandtl parameter α and Brownian motion parameter N b . Figures 14 and 15 depict variations of the shearing stress at the stenosis throat τ s with δ. In these figures, it is analyzed that the shearing stress at the stenosis throat decreases with an increase in β and increases with an increase in α. Figures 16 and 17 show variations of concentration profile for the Brownian motion parameter N b and thermophoresis parameter N t . It is observed that with an increase in the Brownian motion parameter N b , the concentration profile increases, while it decreases with an increase in the thermophoresis parameter N t and the concentration profile gives a larger value for the converging tapering artery. Figures 18 and 19 depict variations of the temperature profile for the Brownian motion parameter N b and thermophoresis parameter N t . It is observed that with an increase in the Brownian motion parameter N b and the thermophoresis parameter N t , the temperature profile decreases. Figures 20, 21, 22, 23, and 24 describe the impedance resistance increases for non-tapered, diverging tapering, and converging tapering arteries when we increase the Prandtl parameters, α, and β, and Brownian motion parameter N b , while it decreases with an increase in thermophoresis parameters, N t and n.
Trapping
Trapping phenomena can be analyzed in Figures 24, 25, 26, 27, 28, 29, 30, 31. It is analyzed that with an increase in flow rate F, the number of trapping bolus increases. We also observed that with an increase in β, the size of trapping bolus increases, while the number of trapping bolus increases with an increase in α. It is also observed that with an increase in the Brownian motion parameter N b , the number of trapping bolus increases, while with an increase in the thermophoresis parameter N t the number of trapping bolus decreases. It is also analyzed that with local nanoparticle Grashof numbers B r and G r , the size of trapping bolus decreases.
Conclusions
The main points of the study that were examined are as follows:
-
1.
It is analyzed that with an increase in the thermophoresis parameter N t and stenosis shape n, the velocity profile decreases, while the velocity profile increases with an increase in the maximum height of the stenosis δ, the Brownian motion parameter N b , and Prandtl parameters α and β.
-
2.
It is analyzed that with an increase in stenosis shape n, the thermophoresis parameter N t , stenosis height δ, and Prandtl parameter β, the shear stress decreases, while it increases with an increase in the Brownian motion parameter N b and Prandtl parameter α.
-
3.
It is analyzed that the shearing stress at the stenosis throat decreases with an increase in β and increases with an increase in α.
-
4.
It is observed that with an increase in the Brownian motion parameter N b , the concentration profile increases, while it decreases with an increase in the thermophoresis parameter N t .
-
5.
It is observed that with an increase in the Brownian motion parameter, the temperature profile decreases, while with an increase in the thermophoresis parameter, the temperature profile increases.
-
6.
It is observed that with an increase in the Brownian motion parameter N b and the thermophoresis parameter N t , the temperature profile decreases.
-
7.
It is analyzed that with an increase in flow rate F, Prandtl parameter α, and Brownian motion parameter N b , the number of trapping bolus increases, while it decreases with an increase in the thermophoresis parameter N t .
-
8.
It is also analyzed that the size of trapping bolus decreases with an increase in the local nanoparticle Grashof numbers B r and G r , while the size of trapping bolus increases with an increase in the Prandtl parameter β.
References
Thurston GB: Viscoelasticity of human blood. Biophys. J 1972, 12: 1205–1217. 10.1016/S0006-3495(72)86156-3
Chien S, King RG, Skalak R, Usami S, Copley AL: Viscoelastic properties of human blood and red cell suspension. Biorheology 1975, 12: 341–346.
Aroesty J, Gross JF: Pulsatile flow in small vessels – I Casson theory. Biorheology 1972, 9: 33–42.
Chaturani P, Ponnalagar Samy R: Pulsatile flow of a Casson fluid through stenosed arteries with application to blood flow. Biorheology 1986, 23: 499–511.
Scott Blair GW, Spanner DC: An Introduction to Biorheology. Amsterdam: Elsevier Scientific Publishing Company; 1974.
Siddiqui SU, Verma NK, Mishra S, Gupta RS: Mathematical modelling of pulsatile flow of Casson’s fluid in arterial stenosis. Appl. Math. Comput 2009, 210: 1–10. 10.1016/j.amc.2007.05.070
Massoudi M, Phuoc TX: Pulsatile flow of blood using a modified second-grade fluid model. Comput. Math. Appl 2008, 56: 199–211. 10.1016/j.camwa.2007.07.018
Mekheimer KHS, El Kot MA: The micropolar fluid model for blood flow through a tapered artery with a stenosis. Acta Mech Sin 2008, 24: 637–644. 10.1007/s10409-008-0185-7
Mandal PK: An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis. Int. J. Non-Linear Mech 2005, 40: 151–164. 10.1016/j.ijnonlinmec.2004.07.007
Varshney G, Katiyar VK, Kumar S: Effect of magnetic field on the blood flow in artery having multiple stenosis: a numerical study. Int. J. Eng. Sci. Tech 2010, 2: 67–82.
Nadeem S, Sher Akbar N, Hayat T, Hendi AA: Power law fluid model for blood flow through a tapered artery with a stenosis. Appl. Math. Comput 2011, 217: 7108–7116. 10.1016/j.amc.2011.01.026
Nadeem S, Sher Akbar N: Jeffrey fluid model for blood flow through a tapered artery with a stenosis. J. Mech. Med. Biol 2011, 11: 529–545. 10.1142/S0219519411003879
Mustafa N, Mandal PK, Abdullah I, Amin NS, Hayat T: Numerical Simulation of generalized Newtonian blood flow past a couple of irregular arterial stenosis. Numer. Meth. Partial Diff. Eqs 2011, 27: 960–981. 10.1002/num.20563
Abdullah I, Amin NS, Hayat T: Magnetohydrodynamic effects on blood flow through an irregular stenosis. Int. J. Number. Method Fluids 10.1002/fld.2436
Choi SUS: Enhancing thermal conductivity of fluids with nanoparticles. In Siginer, DA, Wang, HP (eds) Developments and Applications of Non-Newtonian Flows, vol. 66, pp. 99–105,. New York: ASME; 1995.
Nadeem S, Lee C: Boundary layer flow of nanofluid over an exponentially stretching surface. Nanoscale Res. Lett 2012, 7: 94. 10.1186/1556-276X-7-94
Khan WA, Pop I: Boundary-layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transfer 2010, 53: 2477–2483. 10.1016/j.ijheatmasstransfer.2010.01.032
Sher Akbar N, Nadeem S: Endoscopic effects on the peristaltic flow of a nanofluid. Commun. Theor. Phys 2011, 56: 761–768. 10.1088/0253-6102/56/4/28
Sher Akbar N, Nadeem S, Hayat T, Hendi AA: Peristaltic flow of a nanofluid in a non-uniform tube. Heat Mass Transfer 2012, 48: 451–459. 10.1007/s00231-011-0892-7
Sher Akbar N, Nadeem S, Hayat T, Hendi AA: Peristaltic flow of a nanofluid with slip effects. Meccanica 2012, 5: 1283–1294.
He JH: Homotopy perturbation technique, a new nonlinear analytical technique. Comput. Methods. Appl 2003, 135: 73–79. 10.1016/S0096-3003(01)00312-5
He JH: Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons Fractals 2005, 26: 695–700. 10.1016/j.chaos.2005.03.006
Acknowledgments
The corresponding author is thankful to Quaid-i-Azam University for providing URF to complete this research.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors - SN, SI, and NSA - have contributed in all the sections in the manuscript. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Nadeem, S., Ijaz, S. & Akbar, N.S. Nanoparticle analysis for blood flow of Prandtl fluid model with stenosis. Int Nano Lett 3, 35 (2013). https://doi.org/10.1186/2228-5326-3-35
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/2228-5326-3-35