Skip to main content
SpringerLink
Log in

Analytical study of micropolar fluid flow through porous layered microvessels with heat transfer approach

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

The transport theory of three-layered fluid flow and heat transfer aspects in porous layered tubes is considered in the present work to study the flow of microlevel fluids through porous layered microvessels. The transportation of energy through porous media and the applications associated with heat transfer in physiological aspects are analyzed. Blood is considered as three-layered liquid model in which the core and peripheral regions of the tube are occupied by micropolar and Newtonian fluids, respectively. A thin glycocalyx layer near the wall is considered that represents the porous region due to the deposition of carbohydrates, fibrous tissues or macromolecules inside the interior surface of the tube wall. Analytical expressions for the various flow quantities like velocity, temperature profile, flow rate, flow impedance and additional quantities like hematocrit and Fahraeus effect are obtained and the impacts of various parameters like heat transfer and porous layer parameters are analyzed pictorially for two different formulations (no-spin and no-couple stress conditions). A noteworthy observation is that the impact of no-couple stress condition is relatively more significant in flow quantities, hematocrit and Fahraeus effect than the no-spin condition at the interface. The motivational work of the blood flow through porous blood vessels by selecting the micropolar fluid for the microlevel effects of the molecules may leave a significant impact in the treatment of the various diseases in medical sciences.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Abbreviations

\(\widetilde{}\) :

Represents the dimensional quantities

r, z :

Radial and longitudinal coordinate, respectively

\(h_1, h\) :

Core and plasma layer thickness, respectively

B :

Magnetic field

g :

Gravitational force

\(w_\mathrm{{M}}, w_\mathrm{{N}}, w_\mathrm{{B}}\) :

Axial velocities

\(W_0, W\) :

Characteristic and average (mean) velocity of the fluid, respectively

\(K_0\) :

Ratio of thermal conductivities

\(p_\mathrm{{M}}, p_\mathrm{{N}}, p_\mathrm{{B}}\) :

Pressures

\(p_\mathrm{{s}}\) :

Pressure gradient

k :

Permeability of the porous medium

\(R_1, R_2, R_3\) :

Radii of core, intermediate and peripheral regions, respectively

\(R_3\) :

Radius of the tube

Gr:

Grashof number

N :

Coupling parameter

\(N_1\) :

Radiation parameter

n :

Micro-scale parameter

H :

Hartmann number

\(H_1^2\) :

Magnetic number

\(J_0, J_1\) :

Bessel functions

Ht:

Hematocrit

Fe:

Fahraeus effect

\(H_\mathrm{{T}}\) :

Tube hematocrit

\(H_\mathrm{{D}}\) :

Discharge hematocrit

\(W_\mathrm{{rbc}}\) :

Velocity of RBCs

\(C_v(r)\) :

Concentration profile

\(c_v\) :

Constant in concentration relation

\(\text {SFM, TFM}\) :

Single and two-fluid model, respectively

\(\text {NS, NCS}\) :

No-spin and no-couple stress conditions, respectively

\(\text {PW}\) :

Porous region near the tube wall

\(\text {Region-I}\) :

Core region \((0< r\le R_1)\)

\(\text {Region-II}\) :

Intermediate region \((R_1\le r\le R_2)\)

\(\text {Region-III}\) :

Brinkman region \((R_2\le r\le 1)\)

\(\phi \) :

Azimuthal angle

\(\phi _\mathrm{{M}}\) :

Parameter (couple stress)

\(\Omega _\mathrm{{M}}\) :

Angular velocity

\(\alpha _\mathrm{{M}}, \alpha _\mathrm{{N}}\) :

Mean absorption coefficients

\(\sigma _\mathrm{{M}}, \sigma _\mathrm{{N}}\) :

Electrical conductivities

\(\rho _0\) :

Density ratio

\(\alpha _0\) :

Ratio of mean absorption coefficients

\(\sigma _0\) :

Ratio of electrical conductivities

\(\theta \) :

Dimensionless temperature

\(\theta _\mathrm{{M}}, \theta _\mathrm{{N}}, \theta _\mathrm{{B}}\) :

Temperatures

\(\rho _\mathrm{{M}}, \rho _\mathrm{{N}}\) :

Densities of blood

\(\alpha _\mathrm{{p}}\) :

Porosity parameter

\(\beta _\mathrm{{S}}\) :

Stress jump parameter

\(\kappa _\mathrm{{M}}\) :

Rotational viscosity

\(\mu _\mathrm{{M}}, \mu _\mathrm{{N}}\) :

Constant viscosity coefficients

\(\mu _\mathrm{{E}}\) :

Effective viscosity coefficient of porous medium

\(\mu _\mathrm{{R}}\) :

Viscosity ratio (i.e., \(\widetilde{\mu }_\mathrm{{N}}/\widetilde{\mu }_\mathrm{{M}}\))

\(\lambda _1\) :

Viscosity ratio parameter [i.e., square root of \((\widetilde{\mu }_\mathrm{{E}}/\widetilde{\mu }_\mathrm{{N}})\)]

p:

Represents the porosity of medium (for \(\alpha _\mathrm{{p}}\))

s:

Represents steady flow value (for \(p_\mathrm{{s}}, Q_\mathrm{{s}}, \lambda _\mathrm{{s}}\))

E:

Represents effective viscosity in porous region (for \(\mu _\mathrm{{E}}\))

M:

Represents for micropolar fluid (for \(w_\mathrm{{M}}, \theta _\mathrm{{M}}, \tau _\mathrm{{M}}, \alpha _\mathrm{{M}}, K_\mathrm{{M}}, \mu _\mathrm{{M}}, \rho _\mathrm{{M}}, p_\mathrm{{M}}\))

N:

Represents for Newtonian fluid (for \(w_\mathrm{{N}}, \theta _\mathrm{{N}}, \tau _\mathrm{{N}}, \alpha _\mathrm{{N}}, K_\mathrm{{N}}, \mu _\mathrm{{N}}, \rho _\mathrm{{N}}, p_\mathrm{{N}}\))

B:

Represents for Brinkman region (for \(w_\mathrm{{B}}, p_\mathrm{{B}}\))

References

  1. D.F. Young, Fluid mechanics of arterial stenosis. J. Biomech. Eng. 101, 157–175 (1979)

    Google Scholar 

  2. D.S. Sankar, K. Hemalatha, A non-Newtonian fluid flow model for blood flow through a catheterized artery-steady flow. Appl. Math. Model. 31, 1847–1864 (2007)

    Google Scholar 

  3. D.S. Sankar, K. Hemalatha, Pulsatile flow of Herschel–Bulkley fluid through stenosed arteries—a mathematical model. Int. J. Nonlinear Mech. 41, 979–990 (2006)

    Google Scholar 

  4. S.U. Siddiqui, N.K. Verma, S. Mishra, R.S. Gupta, Mathematical modelling of pulsatile flow of Casson’s fluid in arterial stenosis. Appl. Math. Comput. 210, 1–10 (2009)

    Google Scholar 

  5. A.C. Burton, Physiology and Biophysics of the Circularion (Year Book Medical Publishers, Chicago, 1965)

    Google Scholar 

  6. Y.C. Fung, Biomechanics. Appl. Mech. Rev. 21, 1–20 (1968)

    Google Scholar 

  7. T. Ariman, On the analysis of blood flow. J. Biomech. 4, 185–192 (1971)

    Google Scholar 

  8. G. Bugliarello, J.W. Hayden, High-speed microcinematographic studies of blood flow in vitro. Science 138, 981–983 (1962)

    Google Scholar 

  9. A.C. Eringen, Simple microfluids. Int. J. Eng. Sci. 2, 205–217 (1964)

    Google Scholar 

  10. A.C. Eringen, Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)

    Google Scholar 

  11. T. Ariman, M.A. Turk, N.D. Sylvester, Microcontinuum fluid mechanics—a review. Int. J. Eng. Sci. 11, 905–930 (1973)

    Google Scholar 

  12. T. Ariman, M.A. Turk, N.D. Sylvester, Applications of microcontinuum fluid mechanics. Int. J. Eng. Sci. 12, 273–293 (1974)

    Google Scholar 

  13. R. Devanathan, S. Parvathamma, Flow of micropolar fluid through a tube with stenosis. Med. Biol. Eng. Comput. 21, 438–445 (1983)

    Google Scholar 

  14. Kh. S. Mekheimer, M.A. Kot, The micropolar fluid model of blood flow through a tapered artery with a stenosis. Acta Mech. Sin. 24, 637–644 (2008)

  15. D. Yu. Khanukaeva, A.N. Filippov, P.K. Yadav, A. Tiwari, Creeping flow of micropolar fluid parallel to the axis of cylindrical cells with porous layer. Eur. J. Mech. B Fluids 76, 73–80 (2019)

  16. D. Yu. Khanukaeva, A.N. Filippov, P.K. Yadav, A. Tiwari, Creeping flow of micropolar fluid through a swarm of cylindrical cells with porous layer (membrane). J. Mol. Liq. 294, 111558 (2019)

  17. G. Bugliarello, J. Sevilla, Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes. Biorheology 7, 85–107 (1970)

    Google Scholar 

  18. P. Chaturani, V.S. Upadhya, On micropolar fluid model for blood flow through narrow tubes. Biorheology 16, 419–428 (1979)

    Google Scholar 

  19. J.B. Shukla, R.S. Parihar, S.P. Gupta, Effects of peripheral layer viscosity on blood flow through the artery with mild stenosis. Bull. Math. Biol. 42, 797–805 (1980)

    Google Scholar 

  20. J.C. Misra, S.K. Ghosh, Flow of a Casson fluid in a narrow tube with a side branch. Int. J. Eng. Sci. 38, 2045–2077 (2000)

    Google Scholar 

  21. A.E. Medvedev, V.M. Fomin, Two-phase blood-flow model in large and small vessels. Dokl. Phys. 56, 610–613 (2011)

    Google Scholar 

  22. A. Tiwari, S.S. Chauhan, Effect of varying viscosity on two-fluid model of blood flow through constricted blood vessels: a comparative study. Cardiovasc. Eng. Technol. 10, 155–172 (2019)

    Google Scholar 

  23. H. Darcy, Les Fontaines Publiques De La Ville De Dijon (Dalmont, Paris, 1856)

    Google Scholar 

  24. H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34 (1947)

    Google Scholar 

  25. H.C. Brinkman, On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. A 1, 81–86 (1947)

    Google Scholar 

  26. J.A. Ochoa-Tapia, S. Whitaker, Momentum transfer at the boundary between a porous medium and a homogeneous fluid-I. Theoretical development. Int. J. Heat Mass Transf. 38, 2635–2646 (1995)

    Google Scholar 

  27. J.A. Ochoa-Tapia, S. Whitaker, Momentum transfer at the boundary between a porous medium and a homogeneous fluid-II. Comparison with experiment. Int. J. Heat Mass Transf. 38, 2647–2655 (1995)

    Google Scholar 

  28. R.K. Dash, K.N. Mehta, G. Jayaraman, Casson fluid flow in a pipe filled with a homogeneous porous medium. Int. J. Eng. Sci. 34, 1145–1156 (1996)

    Google Scholar 

  29. C. Desjardins, B.R. Duling, Microvessel hematocrit: measurement and implications for capillary oxygen transport. Am. Physiol. Soc. 252, H494–H503 (1987)

    Google Scholar 

  30. C. Desjardins, B.R. Duling, Heparinase treatment suggests a role for the endothelial cell glycocalyx in regulation of capillary hematocrit. Am. Physiol. Soc. 258, H647–H654 (1990)

    Google Scholar 

  31. T.W. Secomb, R. Hsu, A.R. Pries, A model for red blood cell motion in glycocalyx-lined capillaries. Am. Physiol. Soc. 274, H1016–H1022 (1998)

    Google Scholar 

  32. N.C. Sacheti, P. Chandran, B.S. Bhatt, R.P. Chhabra, Steady creeping motion of a liquid bubble in an immiscible viscous fluid bounded by a vertical porous cylinder of finite thickness. Adv. Stud. Theor. Phys. 2, 243–260 (2008)

    Google Scholar 

  33. S. Deo, A.N. Filippov, A. Tiwari, S. Vasin, V. Starov, Hydrodynamic permeability of aggregates of porous particles with an impermeable core. Adv. Colloid Interface Sci. 164, 21–37 (2011)

    Google Scholar 

  34. A. Tiwari, S. Deo, Pulsatile flow in a cylindrical tube with porous walls: applications to blood flow. J. Porous Media 16, 335–340 (2013)

    Google Scholar 

  35. A.A. Hill, B. Straughan, Poiseuille flow in a fluid overlying a porous medium. J. Fluid Mech. 603, 137–149 (2008)

    Google Scholar 

  36. C. Boodoo, B. Bhatt, D. Comissiong, Two-phase fluid flow in a porous tube: a model for blood flow in capillaries. Rheol. Acta 52, 579–588 (2013)

    Google Scholar 

  37. B.D. Sharma, P.K. Yadav, A two-layer mathematical model of blood flow in porous constricted blood vessels. Transp. Porous Media 120, 239–254 (2017)

    Google Scholar 

  38. A. Tiwari, S.S. Chauhan, Effect of varying viscosity on a two-layer model of the blood flow through porous blood vessels. Eur. Phys. J. Plus 134, 41 (2019)

    Google Scholar 

  39. A. Tiwari, S.S. Chauhan, Effect of varying viscosity on two-fluid model of pulsatile blood flow through porous blood vessels: a comparative study. Microvasc. Res. 123, 99–110 (2019)

    Google Scholar 

  40. A. Tiwari, S.S. Chauhan, Effect of varying viscosity on two-layer model of pulsatile flow through blood vessels with porous region near walls. Transp. Porous Media 129, 721–741 (2019)

    Google Scholar 

  41. S. Jaiswal, P.K. Yadav, A micropolar–Newtonian blood flow model through a porous layered artery in the presence of a magnetic field. Phys. Fluids 31, 071901 (2019)

    Google Scholar 

  42. A.J. Chamkha, T. Grosan, I. Pop, Fully developed free convection of a micropolar fluid in a vertical channel. Int. Commun. Heat Mass Transf. 29, 1119–1127 (2002)

    Google Scholar 

  43. A.J. Chamkha, T. Grosan, I. Pop, Fully developed mixed convection of a micropolar fluid in a vertical channel. Int. J. Fluid Mech. Res. 30, 251–263 (2003)

    Google Scholar 

  44. A. Ogulu, T.M. Abbey, Simulation of heat transfer on an oscillatory blood flow in an indented porous artery. Int. Commun. Heat Mass Transf. 32, 983–989 (2005)

    Google Scholar 

  45. J. Prakash, A. Ogulu, A study of pulsatile blood flow modeled as a power law fluid in a constricted tube. Int. Commun. Heat Mass Transf. 34, 762–768 (2007)

    Google Scholar 

  46. K. Hooman, H. Gurgenci, A theoretical analysis of forced convection in a porous-saturated circular tube: Brinkman–Forchheimer model. Transp. Porous Media 69, 289–300 (2007)

    Google Scholar 

  47. J.C. Misra, A. Sinha, G.C. Shit, Flow of a biomagnetic viscoelastic fluid: application to estimation of blood flow in arteries during electromagnetic hyperthermia, a therapeutic procedure for cancer treatment. Appl. Math. Mech. 31, 1405–1420 (2010)

    Google Scholar 

  48. S. Nadeem, N.S. Akbar, M. Hameed, Peristaltic transport and heat transfer of a MHD Newtonian fluid with variable viscosity. Int. J. Numer. Methods Fluids 63, 1375–1393 (2010)

    Google Scholar 

  49. Kh. S. Mekheimer, Y. Abd Elmaboud, Simultaneous effects of variable viscosity and thermal conductivity on peristaltic flow in a vertical asymmetric channel. Can. J. Phys. 92, 1541–1555 (2014)

  50. M. Sheikholeslami, M. Hatami, D.D. Ganji, Micropolar fluid flow and heat transfer in a permeable channel using analytical method. J. Mol. Liq. 194, 30–36 (2014)

    Google Scholar 

  51. N.S. Akbar, D. Tripathi, O.A. Bég, Variable-viscosity thermal hemodynamic slip flow conveying nanoparticles through a permeable-walled composite stenosis artery. Eur. Phys. J. Plus 132, 294 (2017)

    Google Scholar 

  52. G.J. Reddy, M. Kumar, B. Kethireddy, A.J. Chamkha, Colloidal study of unsteady magnetohydrodynamic couple stress fluid flow over an isothermal vertical flat plate with entropy heat generation. J. Mol. Liq. 252, 169–179 (2018)

    Google Scholar 

  53. T. Elnaqeeb, N.A. Shah, Kh. S. Mekheimer, Hemodynamic characteristics of gold nanoparticle blood flow through a tapered stenosed vessel with variable nanofluid viscosity. Bionanoscience 9, 245–255 (2019)

  54. A.J. Chamkha, On laminar hydromagnetic mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions. Int. J. Heat Mass Transf. 45, 2509–2525 (2002)

    Google Scholar 

  55. A.J. Chamkha, Non-Darcy fully developed mixed convection in a porous medium channel with heat generation/absorption and hydromagnetic effects. Numer. Heat Transfer Part A Appl. 32, 653–675 (1997)

    Google Scholar 

  56. A.J. Chamkha, Unsteady laminar hydromagnetic flow and heat transfer in porous channels with temperature-dependent properties. Int. J. Numer. Methods Heat Fluid Flow 11, 430–448 (2001)

    Google Scholar 

  57. A.J. Chamkha, Double-diffusive convection in a porous enclosure with cooperating temperature and concentration gradients and heat generation or absorption effects. Numer. Heat Transfer Part A Appl. 41(1), 65–87 (2002)

    Google Scholar 

  58. D. Srinivasacharya, M. Shiferaw, Magnetohydrodynamic flow of a micropolar fluid in a circular pipe with hall effects. ANZIAM J. 51, 277–285 (2009)

    Google Scholar 

  59. J.C. Umavathi, A.J. Chamkha, A. Mateen, A. Al-Mudhaf, Unsteady two-fluid flow and heat transfer in a horizontal channel. Heat Mass Transf. 42, 81–90 (2005)

    Google Scholar 

  60. R. Ponalagusamy, R. Tamil Selvi, Influence of magnetic field and heat transfer on two-phase fluid model for oscillatory blood flow in an arterial stenosis. Meccanica 50, 927–943 (2015)

    Google Scholar 

  61. J.P. Kumar, J.C. Umavathi, A.J. Chamkha, I. Pop, Fully-developed free-convective flow of micropolar and viscous fluids in a vertical channel. Appl. Math. Model. 34, 1175–1186 (2010)

    Google Scholar 

  62. A.J. Chamkha, Flow of two-immiscible fluids in porous and nonporous channels. J. Fluids Eng. 122, 117–124 (1999)

    Google Scholar 

  63. J.C. Umavathi, A.J. Chamkha, A. Mateen, A. Al-Mudhaf, Unsteady oscillatory flow and heat transfer in a horizontal composite porous medium channel. Nonlinear Anal. Model. Control 14, 397–415 (2009)

    Google Scholar 

  64. J.C. Umavathi, A.J. Chamkha, K.S.R. Sridhar, Generalized plain Cauette flow and heat transfer in a composite channel. Transp. Porous Media 85, 157–169 (2010)

    Google Scholar 

  65. A.J. Chamkha, Unsteady flow of a dusty conducting fluid through a pipe. Mech. Res. Commun. 21, 281–288 (1994)

    Google Scholar 

  66. A.J. Chamkha, Hydromagnetic two-phase flow in a channel. Int. J. Eng. Sci. 33, 437–446 (1995)

    Google Scholar 

  67. A.J. Chamkha, Unsteady laminar hydromagnetic fluid–particle and heat transfer in channels and circular pipes. Int. J. Heat Fluid Flow 21, 740–746 (2000)

    Google Scholar 

  68. A.J. Chmakha, M.A. Al-Subaie, Hydromagnetic buoyancy-induced flow of a particulate suspension through a vertical pipe with heat generation or absorption effects. Turk. J. Eng. Environ. Sci. 33, 127–134 (2009)

    Google Scholar 

Download references

Acknowledgements

Authors acknowledge their sincere thanks to Department of Science and Technology (DST), New Delhi, India for providing support under FIST grant (SR/FST/MSI-090/2013(C)).

Author information

Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani, Rajasthan, 333 031, India

    Ashish Tiwari, Pallav Dhanendrakumar Shah & Satyendra Singh Chauhan

Authors
  1. Ashish Tiwari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pallav Dhanendrakumar Shah
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Satyendra Singh Chauhan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Satyendra Singh Chauhan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tiwari, A., Shah, P.D. & Chauhan, S.S. Analytical study of micropolar fluid flow through porous layered microvessels with heat transfer approach. Eur. Phys. J. Plus 135, 209 (2020). https://doi.org/10.1140/epjp/s13360-020-00128-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-020-00128-x

Search

Navigation