In this work, we study the linear temporal stability of a unidirectional Poiseuille blood flow under the influence of the magnetic parameter and the Reynolds number. Blood is modeled as a spatially inhomogeneous fluid, with its viscosity dependent on the red blood cell concentration. We focus on small blood vessels, such as the terminal branches of arteries, arterioles, and small veins. For the basic flow, assumed to be laminar and two-dimensional, the basic velocity profiles were obtained numerically, including the magnetic effect, using the BVP4C method in Matlab. Stability analysis was performed using classical linear temporal analysis in terms of normal modes, leading to a fourth-order eigenvalue problem solved numerically using the Chebyshev spectral collocation method. The results indicate that the flow is unconditionally unstable. However, increasing the magnetic parameter or the Reynolds number, coupled with decreasing red blood cell concentration, reduces the degree of flow instability. The stabilization of blood flow in microvessels is essential for protecting the vascular walls.
References
[1]
Haynes, R.H. (1960) Physical Basis of the Dependence of Blood Viscosity on Tube Radius. American Journal of Physiology-Legacy Content, 198, 1193-1200. https://doi.org/10.1152/ajplegacy.1960.198.6.1193
[2]
Moyers-Gonzalez, M., Owens, R.G. and Fang, J. (2008) A Non-Homogeneous Constitutive Model for Human Blood. Part 1. Model Derivation and Steady Flow. Journal of Fluid Mechanics, 617, 327-354. https://doi.org/10.1017/s002211200800428x
[3]
Ascolese, M., Farina, A. and Fasano, A. (2019) The F？hr？us-Lindqvist Effect in Small Blood Vessels: How Does It Help the Heart? Journal of Biological Physics, 45, 379-394. https://doi.org/10.1007/s10867-019-09534-4
[4]
Fournier, R.L. (2012) Basic Transport in Biomedical Engineering. CRC. https://archive.org/details/basictransportph0000four_c7g8_3ed/page/468/mode/1up
[5]
Anand, M. and Rajapola, K. (2005) A Note on the Flows of Inhomogeneous Fluids with Shear-Dependent Viscosities. Archives of Mechanics, 57, 417-428. https://am.ippt.gov.pl/index.php/am/article/view/v57p417/pdf
[6]
Hickox, C.E. (1971) Instability Due to Viscosity and Density Stratification in Axisymmetric Pipe Flow. The Physics of Fluids, 14, 251-262. https://doi.org/10.1063/1.1693422
[7]
Preziosi, L., Chen, K. and Joseph, D.D. (1989) Lubricated Pipelining: Stability of Core-Annular Flow. Journal of Fluid Mechanics, 201, 323-356. https://doi.org/10.1017/s0022112089000960
[8]
Anand, M., Kiranmai, P. and Garimella, S.M. (2024) Stability of Fully Developed Pipe Flow of a Shear-Thinning Fluid That Approximates the Response of Viscoplastic Fluids. Applications in Engineering Science, 19, Article 100191. https://doi.org/10.1016/j.apples.2024.100191
[9]
Moyers-Gonzalez, M., Owens, R.G. and Fang, J. (2008) A Non-Homogeneous Constitutive Model for Human Blood. Part 1. Model Derivation and Steady Flow. Journal of Fluid Mechanics, 617, 327-354. https://doi.org/10.1017/s002211200800428x
[10]
Fang, J. and Owens, R.G. (2006) Numerical Simulations of Pulsatile Blood Flow Using a New Constitutive Model. Biorheology, 43, 637-660. https://doi.org/10.1177/0006355x2006043005001
[11]
Owens, R.G. (2006) A New Microstructure-Based Constitutive Model for Human Blood. Journal of Non-Newtonian Fluid Mechanics, 140, 57-70. https://doi.org/10.1016/j.jnnfm.2006.01.015
[12]
Fusi, L. (2018) Two-Dimensional Thin-Film Flow of an Incompressible Inhomogeneous Fluid in a Channel. Journal of Non-Newtonian Fluid Mechanics, 260, 87-100. https://doi.org/10.1016/j.jnnfm.2018.07.001
[13]
Massoudi, M., Kim, J. and Antaki, J.F. (2012) Modeling and Numerical Simulation of Blood Flow Using the Theory of Interacting Continua. International Journal of Non-Linear Mechanics, 47, 506-520. https://doi.org/10.1016/j.ijnonlinmec.2011.09.025
[14]
Fusi, L., Farina, A., Rosso, F. and Rajagopal, K. (2019) Thin-Film Flow of an Inhomogeneous Fluid with Density-Dependent Viscosity. Fluids, 4, Article 30. https://doi.org/10.3390/fluids4010030
[15]
Fasano, A. and Sequeira, A. (2017) Hemomath: The Mathematics of Blood. Springer. https://link.springer.com/book/10.1007/978-3-319-60513-5
[16]
Khan, A., Zaman, G., Li, Y., Ahmad, S. and Hussain, A. (2017) The Oscillatory Motion of Oldroyd-B Fluid by Incorporating Some of the Mechanical Factors. Journal of Applied Mathematics and Physics, 5, 2402-2410. https://doi.org/10.4236/jamp.2017.512196
[17]
Papalexandris, M.V. (2004) A Two-Phase Model for Compressible Granular Flows Based on the Theory of Irreversible Processes. Journal of Fluid Mechanics, 517, 103-112. https://doi.org/10.1017/s0022112004000874
[18]
Varsakelis, C. and Papalexandris, M.V. (2011) Low-Mach-Number Asymptotics for Two-Phase Flows of Granular Materials. Journal of Fluid Mechanics, 669, 472-497. https://doi.org/10.1017/s0022112010005173
[19]
Goldsmith, H.L. and Marlow, J.C. (1979) Flow Behavior of Erythrocytes. II. Particle Motions in Concentrated Suspensions of Ghost Cells. Journal of Colloid and Interface Science, 71, 383-407. https://doi.org/10.1016/0021-9797(79)90248-0
[20]
Moyers-Gonzalez, M.A. and Owens, R.G. (2010) Mathematical Modelling of the Cell-Depleted Peripheral Layer in the Steady Flow of Blood in a Tube. Biorheology, 47, 39-71. https://doi.org/10.3233/bir-2010-0558
[21]
Dimakopoulos, Y., Kelesidis, G., Tsouka, S., Georgiou, G.C. and Tsamopoulos, J. (2015) Hemodynamics in Stenotic Vessels of Small Diameter under Steady State Conditions: Effect of Viscoelasticity and Migration of Red Blood Cells. Biorheology, 52, 183-210. https://doi.org/10.3233/bir-14033
[22]
Coupier, G., Kaoui, B., Podgorski, T. and Misbah, C. (2008) Non-Inertial Lateral Migration of Vesicles in Bounded Poiseuille Flow. Physics of Fluids, 20, Article 111702. https://doi.org/10.1063/1.3023159
[23]
Fusi, L. and Farina, A. (2020) Linear Stability Analysis of Blood Flow in Small Vessels. Applications in Engineering Science, 1, Article 100002. https://doi.org/10.1016/j.apples.2020.100002
[24]
Lihonou, T.F., Monwanou, A.V., Miwadinou, C.H. and Orou, J.B.C. (2022) Active Control of the Instability of a Dynamic Laminar Boundary Layer on a Flat Impermeable Plate Subjected to a Magnetic Field. Indian Journal of Physics, 96, 3591-3601. https://doi.org/10.1007/s12648-022-02284-5
[25]
John, M.O., Obrist, D. and Kleiser, L. (2015) Stabilizing a Leading-Edge Boundary Layer Subject to Wall Suction by Increasing the Reynolds Number. Procedia IUTAM, 14, 394-402. https://doi.org/10.1016/j.piutam.2015.03.065
[26]
Lebbal, S. (2022) Dynamics of Pulsed Flows in Deformable Channels. University of Lyon. (In Fran？ais) https://theses.hal.science/tel-03577495v1
[27]
Hund, S.J., Kameneva, M.V. and Antaki, J.F. (2017) A Quasi-Mechanistic Mathematical Representation for Blood Viscosity. Fluids, 2, 10-36. https://doi.org/10.3390/fluids2010010
[28]
Hatschek, E. (1920) Eine Reihe von abnormen Liesegang’schen Schichtungen. Kolloid-Zeitschrift, 27, 225-229. https://doi.org/10.1007/bf01427700
[29]
Cokelet, G.R. (1963) The Rheology of Human Blood. Doctoral Dissertation, Massachusetts Institute Technology. https://dspace.mit.edu/entities/publication/ca65fc8d-5307-4f81-a2af-283b46698659
[30]
Nubar, Y. (1967) Effect of Slip on the Rheology of a Composite Fluid: Application to Blood. Biorheology, 4, 133-150. https://doi.org/10.3233/bir-1967-4401
[31]
Shampine, L., Kierzenka, J. and Reichelt, M. (2000) Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c. Tutorial Notes, 1-27. https://share.google/kQ4RcRzPotyDwbDTA
[32]
Jamir, T. and Konwar, H. (2022) Effects of Radiation Absorption, Soret and Dufour on Unsteady MHD Mixed Convective Flow Past a Vertical Permeable Plate with Slip Condition and Viscous Dissipation. Journal of Heat and Mass Transfer Research, 9, 155-168. https://doi.org/10.22075/jhmtr.2023.28693.1399
[33]
Lihonou, T.F., Laouer, A., Mathos, K.P. and Yombouno, F.M. (2025) Analysis of Heat and Mass Transfer in MHD Free Convection with Chemical Reaction Effects on a Moving Vertical Porous Plate. Open Journal of Fluid Dynamics, 15, 87-115. https://doi.org/10.4236/ojfd.2025.152006
[34]
Lihonou, T.F., Diakite, M., Laouer, A., Segning, H., Mathos, K.P. and Camara, N. (2025) Analysis of Heat and Mass Transfer in MHD Forced Convection with Chemical Parameter Effects on a Horizontal Porous Plate. Brazilian Journal of Physics, 55, Article No. 275. https://doi.org/10.1007/s13538-025-01907-6
[35]
Laouer, A., Mezaache, E.H. and Laouar, S. (2016) Influence of Surface Mass Transfer on the Stability of Forced Convection Flow over a Horizontal Flat Plate. Computational Thermal Sciences: An International Journal, 8, 355-369. https://doi.org/10.1615/computthermalscien.2016016415
Weideman, J.A. and Reddy, S.C. (2000) A MATLAB Differentiation Matrix Suite. ACM Transactions on Mathematical Software, 26, 465-519. https://doi.org/10.1145/365723.365727
[38]
Peyret, R. (2002) Spectral Methods for Incompressible Viscous Flow. Springer.
[39]
Driscoll, T.A., Hale, N. and Trefethen, L.N. (2014) Chebfun Guide. Pafnuty Publications.
[40]
Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1988) Spectral Methods in Fluid Dynamics. Springer. https://link.springer.com/book/10.1007/978-3-642-84108-8
[41]
Motsa, S.S., Marewo, G.T., Sibanda, P. and Shateyi, S. (2011) An Improved Spectral Homotopy Analysis Method for Solving Boundary Layer Problems. Boundary Value Problems, 2011, Article No. 3. https://doi.org/10.1186/1687-2770-2011-3
