Skip to main content
SpringerLink
Log in

Analytical Study of the Effect of Variable Viscosity and Heat Transfer on Two-Fluid Flowing through Porous Layered Tubes

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

The proposed study is an attempt to perceive theoretically the heat transfer phenomenon in the flow of temperature-dependent viscous blood through microvessels internally surrounded by a thin layer of endothelial glycocalyx at the wall. While flowing through microvessels, the blood separates into erythrocytes suspended fluid and cell-depleted fluid into core and peripheral regions respectively. Therefore, to best represent the flow of human blood in microvessels, it has been modeled as a two-fluid. Erythrocytes appearing in the core stimulates the non-Newtonian behavior of the fluid is manifested here by Herschel-Bulkley fluid with temperature-dependent viscosity. The plasma surrounded over the blood cells in the peripheral layer is expressed as a Newtonian fluid with constant viscosity. An added advantage of utilizing the Brinkman-Forchheimer equation to govern the flow through the layer of endothelial glycocalyx (EGL) is that it is credible for both small and large Darcy numbers (permeability). Linear approximation of the Reynolds, viscosity model is exercised to obtain the analytical solutions for the governing equations of Herschel-Bulkley fluid flowing through the core region. In the non-porous peripheral region, the analytical solutions have been obtained for Newtonian fluid with constant viscosity directly and in the porous peripheral region, the Brinkman-Forchheimer equation is solved using regular perturbation for large Darcy number and singular perturbation with a matched asymptotic condition for small Darcy number. Analytical expressions for the velocity, flow rate, flow impedance, and temperature field have been obtained for the different regions. Graphical analysis revealing significant results regarding the variable viscosity, thermal conductivity, Grashof number, Forchheimer number, Richardson number, and permeability on the hemodynamical variables are conducted and results are discussed in detail. The study concludes that an EGL adjacent to the vessel wall increase the resistance to blood flow. The notable discovery of the study is that the temperature parameters influence all the quantities and therefore establish that the temperature-dependent viscosity plays a vital role in medical treatments involving temperature variation such as chemotherapy.

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

Similar content being viewed by others

Abbreviations

r, z :

Radial and axial coordinates, respectively

\(h_1, h\) :

Core and peripheral region thickness, respectively

\(w_H, w_N, w_B\) :

Axial velocities in core, intermediate and porous regions, respectively

\(R_p, w_p\) :

Radius and velocity of the fluid in plug core region, respectively

\(W_0\) :

Characteristic velocity

n :

Herschel-Bulkley fluid parameter

\(p_H, p_N, p_B\) :

Pressures in core, intermediate and porous regions, respectively

\(T_H, T_N, T_B\) :

Temperatures in core, intermediate and porous regions, respectively

\(K_H, K_N\) :

Thermal conductivities in core and plasma regions, respectively

\(K_0\) :

Thermal conductivity ratio

\(Q_H, Q_N\) :

Constant heat absorptions in core and plasma regions, respectively

\(p_s\) :

Pressure gradient

\(Q_s\) :

Rate of fluid flow (volumetric flow rate)

\(R_1, R_2, R_3\) :

Tube radius in core, intermediate and porous regions, respectively

\(T_w, T_\infty\) :

Temperature at wall and ambient temperature, respectively

Gr :

Grashof number (free convection parameter)

Re :

Reynolds number

Ri :

Richardson number (i.e., \(Ri=Gr/Re^2\))

F :

Forchheimer number

S :

Dimensionless shape parameter in porous medium defined in Eq. (4)

\(C_F\) :

Inertial coefficient

k :

Permeability of the porous medium (i.e., Darcy number Da)

g :

Gravitational force

\(\text {PW}\) :

Porous region near the vessel wall

\(\text {NF, PL}\) :

Newtonian and Power-law fluids, respectively

\(\text {BP, HB}\) :

Bingham-plastic and Herschel-Bulkley fluids, respectively

\(\text {TFM}\) :

Two-fluid model

\(\text {SDN, LDN}\) :

Small Darcy number and large Darcy number, respectively

Phase-I:

Core region (i.e., \(0<\widetilde{r}\le \widetilde{R}_1\))

Phase-II:

Intermediate region (i.e., \(\widetilde{R}_1\le \widetilde{r}\le \widetilde{R}_2\))

Phase-III:

Porous region (i.e., \(\widetilde{R}_2\le \widetilde{r}\le \widetilde{R}_3\))

\(\phi\) :

Azimuthal angle

\(\alpha\) :

Viscosity index

\(\alpha _p\) :

Porosity parameter

\(\tau _H\) :

Shear stress of Herschel-Bulkley fluid

\(\tau _y\) :

Yield stress

\(\theta\) :

Dimensionless yield stress

\(\theta _H, \theta _N, \theta _B\) :

Temperatures in core, intermediate and porous regions, respectively

\(\rho _H, \rho _N\) :

Density of Herschel-Bulkley and Newtonian fluids, respectively

\(\rho _0\) :

Density ratio

\(\gamma _1,\gamma _2\) :

Mean absorption coefficients

\(\gamma _0\) :

Mean absorption coefficients ratio

\(\gamma\) :

Coefficient of the volume expansion due to the temperature

\(\tau _w\) :

Shear stress at wall

\(\mu (\theta _H)\) :

Variable viscosity of core region fluid

\(\mu _H, \mu _N\) :

Constant viscosity coefficients of HB and NF, respectively

\(\mu _E\) :

Effective viscosity of the porous medium

\(\lambda _1\) :

Viscosity ratio parameter (\(\lambda _1^2=\widetilde{\mu }_E/\widetilde{\mu }_N\))

\(\lambda _s\) :

Flow resistance (impedance)

p :

Plug flow value (for \(w_p, R_p\))

s :

Steady flow value (for \(p_s, Q_s, \lambda _s\))

H :

Symbolizes for Herschel-Bulkley fluid (for \(w_H, \theta _H, \tau _H, Q_H, K_H, \mu _H, \rho _H, p_H\))

N :

Symbolizes for Newtonian fluid (for \(w_N, \theta _N, Q_N, K_N, \mu _N, \rho _N, p_N\))

B :

Symbolizes for Brinkman-Forchheimer region (for \(w_B, p_B\))

w :

Value at vessel wall (for \(\tau _w, T_w\))

y :

Value at yield stress (for \(\tau _y\))

References

  • Abd Elmaboud, Y., Mekheimer, Kh.S., Emam, T.G.: Numerical examination of gold nanoparticles as a drug carrier on peristaltic blood flow through physiological vessels:Cancer therapy treatment. BioNanoScience 9, 952-965 (2019)

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

    Article  Google Scholar 

  • Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34 (1949)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Bush, A.W.: Perturbation Methods for Engineers and Scientists. CRC Press, Boca Raton (1992)

    Book  Google Scholar 

  • Chamkha, A.J.: Non-darcy fully developed mixed convection in a porous medium channel with heat generation/absorption and hydromagnetic effects. Num. Heat Transfer Part A: Appl. 32(6), 653–675 (1997)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Chamkha, A.J., Dogonchi, A.S., Ganji, D.D.: Magneto-hydrodynamic flow and heat transfer of a hybrid nanofluid in a rotating system among two surfaces in the presence of thermal radiation and Joule heating. AIP Adv. 9, 025103 (2019)

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  • Fujiwara, H., Ishikawa, T., Lima, R., Matsuki, N., Imai, Y., Kaji, H., Nishizawa, M., Yamaguchi, T.: Red blood cell motions in high-hematocrit blood flowing through a stenosed microchannel. J. Biomech. 42, 838–843 (2009)

    Article  Google Scholar 

  • Galdi, G.P., Rannacher, R., Robertson, A.M., Turek, S.: Hemodynamical Flows: Modeling, Analysis and Simulation, Birkhauser Verlag, Basel- Boston-Berlin 37, (2008)

  • Garoosi, F., Garoosi, S., Hooman, K.: Numerical simulation of natural convection and mixed convection of the nanofluid in a square cavity using Buongiorno model. Powder Technol. 268(1), 279–292 (2014)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Hooman, K.: A perturbation solution for forced convection in a porous-saturated duct. J. Comput. Appl. Math. 211, 57–66 (2008)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Lih, M.M.: Transport Phenomena in Medicine and Biology, \(1^{st}\) Edition John Wiley, New York (1975)

  • Mahmoudi, A.H., Hooman, K.: Effect of a discrete heat source location on entropy generation in mixed convective cooling of a nanofluid inside the ventilated cavity. Int. J. Exergy 13(3), 299–319 (2013)

    Article  Google Scholar 

  • Majee, S., Shit, G.C.: Numerical investigation of MHD flow of blood and heat transfer in a stenosed arterial segment. J. Magnet. Magnet. Mater. 424, 137–147 (2017)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Misra, J.C., Sinha, A., Shit, G.C.: 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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Nayfeh, A.H.: Problems in Perturbation, 2\(^{nd}\) Ed. Wiley, New York (1993)

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Ogulu, A., Bestman, A.R.: Deep heat muscle treatment- a mathematical model - I. Acta Phys. Hungarica 73, 3 (1993)

    Article  Google Scholar 

  • Ponalagusamy, R., Tamil Selvi, R.: 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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Raza, J., Mebarek-Oudina, F., Chamkha, A.J.: Magnetohydrodynamic flow of molybdenum disulfide nanofluid in a channel with shape effects. Multi Discip. Model. Mater. Struct. 15(4), 737–757 (2019)

    Article  Google Scholar 

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

    Google Scholar 

  • Saini, A.K., Chauhan, S.S., Tiwari, A.: Creeping flow of Jeffrey fluid through a swarm of porous cylindrical particles: Brinkman-Forchheimer model. Int. J. Multiphase Flow 145, 103803 103803 (2021)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  • Selimefendigil, F., Ismael, M.A., Chamkha, A.J.: Mixed convection in superposed nanofluid and porous layers in square enclosure with inner rotating cylinder. Int. J. Mech. Sci. 124–125, 95–108 (2017)

    Article  Google Scholar 

  • Shah, P.D., Tiwari, A., Chauhan, S.S.: Solute dispersion in micropolar-Newtonian fluid flowing through porous layered tubes with absorbing walls. Int. Commun. Heat Mass Transfer 119, 104724 (2020)

    Article  Google Scholar 

  • Sharan, M., Singh, B., Kumar, P.: A two-layer model for studying the effect of plasma layer on the delivery of oxygen to tissue using a finite element method. Appl. Math. Model. 21, 419–426 (1997)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Sharma, B.D., Yadav, P.K.: A mathematical model of blood flow in narrow blood vessels in presence of magnetic field. National Acad. Sci. Lett. 42, 239–243 (2019)

    Article  Google Scholar 

  • Sharma, B.D., Yadav, P.K., Filippov, A.: A Jeffrey-fluid model of blood flow in tubes with stenosis. Coll. J. 79, 849–856 (2017)

    Article  Google Scholar 

  • Shukla, J.B., Parihar, R.S., Rao, B.R.P.: Effects of stenosis on non-Newtonian flow of the blood in an artery. Bull. Math. Biol. 42, 283–294 (1980)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Tiwari, A., Chauhan, S.S.: 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)

    Article  Google Scholar 

  • Tiwari, A., Chauhan, S.S.: 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)

    Article  Google Scholar 

  • Tiwari, A., Chauhan, S.S.: 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)

    Article  Google Scholar 

  • Tiwari, A., Deo, S.: Pulsatile flow in a cylindrical tube with porous walls: applications to blood flow. J. Porous Media 16(4), 335–340 (2013)

    Article  Google Scholar 

  • Tiwari, A., Shah, P.D., Chauhan, S.S.: Analytical study of micropolar fluid flow through porous layered microvessels with heat transfer approach. Euro. Phys. J. Plus 135, 209 (2020)

    Article  Google Scholar 

  • Tiwari, A., Shah, P.D., Chauhan, S.S.: Unsteady solute dispersion in two-fluid flowing through narrow tubes: a temperature-dependent viscosity approach. Int. J. Therm. Sci. 161, 106651 (2021)

    Article  Google Scholar 

  • Toghraie, D., Mashayekhi, R., Arasteh, H., Sheykhi, S., Niknejadi, M., Chamkha, A.J.: Two-phase investigation of water-Al2O3 nanofluid in a micro concentric annulus under non-uniform heat flux boundary conditions. Int. J. Num. Method. Heat Fluid Flow 30(4), 1795–1814 (2020)

    Article  Google Scholar 

  • Yadav, P.K., Sharma, B.D., Filippov, A.: An Oscillatory viscoelastic model of blood flow in stenotic artery. Coll. J. 82, 617–625 (2020)

    Article  Google Scholar 

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

    Article  Google Scholar 

Download references

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Chennai, Tamil Nadu, 601103, India

    Satyendra Singh Chauhan

  2. Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani, Rajasthan, 333031, India

    Pallav Dhanendrakumar Shah & Ashish Tiwari

Authors
  1. Satyendra Singh Chauhan
    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. Ashish Tiwari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pallav Dhanendrakumar Shah.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chauhan, S.S., Shah, P.D. & Tiwari, A. Analytical Study of the Effect of Variable Viscosity and Heat Transfer on Two-Fluid Flowing through Porous Layered Tubes. Transp Porous Med 142, 641–668 (2022). https://doi.org/10.1007/s11242-022-01765-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-022-01765-9

Keywords

Mathematics Subject Classification

Search

Navigation