Abstract
The present work concerns the effect of hematocrit-dependent viscosity on pulsatile flow of blood through narrow tube with porous walls. Two-fluid model of blood is assumed to be consisting of a core region (Casson fluid) and a plasma region (Newtonian fluid). No slip condition is assumed on wall and pressure gradient has been considered as periodic function of time. The wall of the blood vessel composed of a thin porous (Brinkman) layer. The stress jump condition has been employed at the fluid–porous interface in the plasma region. Up to first order, approximate solutions of governing equations are obtained using perturbation approach. A comparative analysis for relative change in resistance offered against the flow between our model and previously studied single and two-fluid models without porous walls has also been done. Mathematical expressions for velocity, rate of flow and resistance offered against the flow have been obtained analytically for different regions and influence of plasma layer thickness, varying viscosity, stress jump parameter, permeability and viscosity ratio parameter on above quantities are pictorially discussed. It is perceived that the values of flow rate for two-fluid model with porous region near walls are higher in comparison with two-fluid model without porous region near walls. Dependency of hematocrit (Ht) on the porosity parameters is graphically discussed. The study reveals a significant impact of various parameters on hematocrit (Ht). A novel observation is that a slight increase in pressure wave amplitude leads to significant fluctuation in hematocrit (Ht) which also indicates how systole and diastole (which controls the pressure gradient amplitudes) leads to changes on blood hematocrit (Ht).
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Abbreviations
- ‘\({-}\)’:
-
Represents dimensional quantities
- r, z :
-
Radial and axial distance, respectively
- t :
-
Time
- h :
-
Plasma layer thickness
- \(R_{\mathrm{p}}\) :
-
Plug core radius
- \(v_1, v_2, v_3\) :
-
Axial velocities in core, plasma and Brinkman region, respectively
- \(v_{\mathrm{p}}\) :
-
Plug core velocity
- K :
-
Constant in varying viscosity relation (viscosity parameter)
- n :
-
Viscosity index
- \(p_1,p_2,p_3\) :
-
Pressures in core, plasma and Brinkman region, respectively
- q(z):
-
Pressure gradient in steady flow state
- Q :
-
Volumetric flow rate
- \(R_1, R_2, R_3\) :
-
Radius of artery in core, plasma and Brinkman region, respectively
- k :
-
Permeability in porous region
- \(R_0\) :
-
Radius of blood vessel without porous walls
- L :
-
Length of blood vessel
- SFM, TFM:
-
Single-fluid model and two-fluid model, respectively
- CF, NF:
-
Casson and Newtonian fluid, respectively
- PW:
-
Porous region near walls
- Ht:
-
Hematocrit (i.e., volume concentration of all RBCs in whole blood)
- \(C_{\mathrm{v}}(r)\) :
-
Volume concentration of all RBCs
- \(c_{\mathrm{v}}\) :
-
Constant in concentration relation
- H(r):
-
Heaviside unit function
- \(\phi \) :
-
Azimuthal angle
- \(\alpha \) :
-
Womersley parameter
- \(\alpha _{\mathrm{p}}=1/\lambda _1^2\) :
-
Porosity parameter (Srivastava and Srivastava 2005)
- \(\tau \) :
-
Shear stress of Casson fluid
- \(\tau _{\mathrm{y}}\) :
-
Yield stress
- \(\theta \) :
-
Dimensionless yield stress
- \(\tau _{\mathrm{w}}\) :
-
Wall shear stress
- \(\mu _1(r)\) :
-
Variable viscosity of Casson fluid in core region
- \(\mu _2\) :
-
Constant viscosity coefficient of peripheral region
- \(\mu _{\mathrm{e}}\) :
-
Effective viscosity coefficient of Brinkman region (Srivastava and Srivastava 2005)
- \(\lambda _1\) :
-
Viscosity ratio parameter in Brinkman region
- \(\beta \) :
-
Stress jump parameter
- \(\lambda \) :
-
Flow resistance
- 1, 2, 3:
-
Denote for core, plasma and Brinkman region, respectively (for \(p_i, v_i, R_i\))
- p:
-
Represent plug flow value (for \(v_{\mathrm{p}}, R_{\mathrm{p}}\))
- e:
-
Represent effective viscosity coefficient in Brinkman region (for \(\mu _{\mathrm{e}}\))
- w:
-
Value at wall (for \(\tau _{\mathrm{w}}\))
- y:
-
Value of yield stress (for \(\tau _{\mathrm{y}}\))
References
Akbar, N.S., Rahman, S.U., Ellahi, R., Nadeem, S.: Blood flow study of Williamson fluid through stenosed arteries with permeable walls. Eur. Phys. J. Plus 129, 256 (2014)
Aroesty, J., Gross, J.F.: The mathematics of pulsatile flow in small blood vessels I. Casson theory. Microvasc. Res. 4, 1–12 (1972a)
Aroesty, J., Gross, J.F.: Pulsatile flow in small blood vessels I. Casson theory. Biorheology 9, 33–43 (1972b)
Bali, R., Awasthi, U.: Effect of a magnetic field on the resistance to blood flow through stenotic artery. Appl. Math. Comput. 188, 1635–1641 (2007)
Bhattacharyya, A., Raja Sekhar, G.P.: Stokes flow inside a porous spherical shell: stress jump boundary condition. ZAMP 56, 475–496 (2005)
Boodoo, C., Bhatt, B., Comissiong, D.: Two-phase fluid flow in a porous tube: a model for blood flow in capillaries. Rheol. Acta 52, 579–588 (2013)
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 (1947a)
Brinkman, H.C.: On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. A 1, 81–86 (1947b)
Bugliarello, G., Sevilla, J.: Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes. Biorheology 7, 85–107 (1970)
Casson, N.: A flow equation for pigment-oil suspensions of the printing ink type. In: Rheology of Disperse System, pp. 84–102. Pergamon (1959)
Chakravarty, S., Mandal, P.K.: Numerical simulation of Casson fluid flow through differently shaped arterial stenoses. ZAMP 65, 767–782 (2014)
Chaturani, P., Samy, R.P.: Pulsatile flow of a Casson’s fluid through stenosed arteries with application to blood flow. Biorheology 23, 499–511 (1986)
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)
Deo, S., Filippov, A.N., Tiwari, A., Vasin, S.I., Starov, V.: Hydrodynamic permeability of aggregates of porous particles with an impermeable core. Adv. Colloid Interface Sci. 164(1), 21–37 (2011)
Ellahi, R., Rahman, S.U., Nadeem, S., Akbar, N.S.: Blood flow of nanofluid through an artery with composite stenosis and permeable walls. Appl. Nanosci. 4, 919–926 (2014)
Lih, M.M.: Transport Phenomena in Medicine and Biology, 1st edn. Wiley, New York (1975)
Mehmood, O.U., Mustapha, N., Shafie, S.: Unsteady two-dimensional blood flow in porous artery with multi-irregular stenosis. Transp. Porous Media 92, 259–275 (2012)
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)
Misra, J.C., Ghosh, S.K.: Flow of Casson fluid in a narrow tube with a side branch. Int. J. Eng. Sci. 38, 2045–2077 (2000)
Misra, J.C., Adhikary, S.D., Shit, G.C.: Mathematical analysis of blood flow through an arterial segment with time-dependent stenosis. Math. Model. Anal. 13, 401–412 (2008)
Nadeem, S., Akbar, N.S., Hameed, M.: Peristaltic transport and heat transfer of a MHD Newtonian fluid with variable viscosity. Int. J. Numer. Mathods Fluids 63, 1375–1393 (2010)
Nagarani, P., Sarojamma, G.: Effect of body acceleration on pulsatile flow of Casson fluid through a mild stenosed artery. Korea Aust. Rheol. J. 20, 189–196 (2008)
Nayfeh, A.H.: Introduction to Perturbation Techniques, 1st edn. 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 Transf. 38, 2635–2646 (1995a)
Ochoa-Tapia, J.A., Whitaker, S.: 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 (1995b)
Ponalagusamy, R., Selvi, R.T.: A study on two-layered model (Casson–Newtonian) for blood flow through an arterial stenosis: axially variable slip velocity at the wall. J. Frankl. Inst. 348, 2308–2321 (2011)
Ponalagusamy, R., Selvi, R.T.: Blood flow in stenosed arteries with radially variable viscosity, peripheral plasma layer thickness and magnetic field. Meccanica 48, 2427–2438 (2013)
Rohlf, K., Tenti, G.: The role of the Womersley number in pulsatile blood flow a theoretical study of the Casson model. J. Biomech. 34, 141–148 (2001)
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. Theor. Phys. 2(5), 243–260 (2008)
Sankar, D.S.: Two-fluid nonlinear mathematical model for pulsatile blood flow through stenosed arteries. Bull. Malays. Math. Sci. Soc. 35, 487–498 (2012)
Sankar, D.S., Lee, U.: Two-fluid Casson model for pulsatile blood flow through stenosed arteries: a theoretical model. Commun. Nonlinear Sci. Numer. Simul. 15, 2086–2097 (2010)
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)
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)
Shit, G.C., Roy, M., Sinha, A.: Mathematical modelling of blood flow through a tapered overlapping stenosed artery with variable viscosity. Appl. Bionics Biomech. 11, 185–195 (2014)
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 (1980a)
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 (1980b)
Siddiqui, S.U., Verma, N.K., Mishra, S., Gupta, R.S.: Mathematical modelling of pulsatile flow of Casson’s fluid in arterial stenosis. Appl. Math. Comput. 210, 1–10 (2009)
Srivastava, V.P., Saxena, M.: Two-layered model of Casson fluid flow through stenotic blood vessels: applications to the cardiovascular system. J. Biomech. 27, 921–928 (1994)
Srivastava, A.C., Srivastava, N.: Flow past a porous sphere at small Reynolds number. ZAMP 56, 821–835 (2005)
Straughan, B.: Stability and Wave Motion in Porous Media, vol. 165, 165th edn. Springer, New York (2008)
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 (2019a)
Tiwari, A., Chauhan, S.S.: Effect of varying viscosity on a two-layer model of the blood flow through porous blood vessels. Eur. Phys. J. Plus 134, 41 (2019b)
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)
Vafai, K.: Porous Media: Applications in Biological Systems and Biotechnology. CRC Press, Boca Raton (2010)
Venkatesan, J., Sankar, D.S., Hemalatha, K., Yatim, Y.: Mathematical analysis of Casson fluid model for blood rheology in stenosed narrow arteries. J. Appl. Math. 2013(583809), 1–11 (2013)
Acknowledgements
Authors are thankful to reviewers for their valuable suggestions which motivated us for some new observations. Authors also 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)).
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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 Med 129, 721–741 (2019). https://doi.org/10.1007/s11242-019-01302-1
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DOI: https://doi.org/10.1007/s11242-019-01302-1