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Peristaltic Blood Transport in Non-Newtonian Fluid Confined by Porous Soaked Tube: A Numerical Study Through Galerkin Finite Element Technique

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Abstract

This novel investigation suggests the implementation of famous numerical technique namely Galerkin finite element technique for peristaltic study of non-Newtonian fluid confined by a porous tube. The rheological consequences for non-Newtonian materials are executed by using micropolar fluid. The problem is modeled in form of Navier–Stokes expressions. The flow simulations are carried by utilizing the impact of the magnetic field and uniform porous medium. Additionally, the role of inertial forces is also observed as a novelty for current analysis. Unlike typical investigations, the presumptions of lubrication theory are not implemented in the modeling which allows the participation of inertial forces in the governing equations and provided the results independent of wavelength. The solution of the modeled set of coupled non-linear partial differential equations is obtained by Galerkin finite element method with quadratic triangular elements. The verification of attained numerical results with available literature for low Reynolds number approximation is also presented and found in excellent agreement. It is observed that the peristaltic mixing enhances with increasing the inertial and Lorentz force while reverse observations are noticed with for the dense porous medium. An increase in pressure rise per wavelength is observed for micropolar fluid as compared to that of viscous fluid. It is further claimed that the peristaltic mixing is improved with increasing the Reynolds number and permeability of porous medium.

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Abbreviations

\({\varvec{A}}\) :

Unknown nodal vector

\(a\) :

Half width of the channel

\(b\) :

Wave amplitude

\({B}_{0}\) :

Magnetic field parameter

\(c\) :

Velocity of wave train lab frame

\(D\) :

Modified Laplacian

\({E}_{0}\) :

Electric field

\({\varvec{F}}\) :

Global forcing vector

\({{\varvec{g}}}^{\boldsymbol{*}}\) :

Microrotation vector

\(H\) :

Wave in lab frame

\(h\) :

Wave in wave frame

\({J}^{*}\) :

Microgyration parameter in dimensional form

\(J\) :

Microgyration parameter in non-dimensional form

\(k\) :

Permeability of porous medium in dimensional form

\(K\) :

Permeability of porous medium in non-dimensional form

\({\varvec{K}}\) :

Global stiffness matrix

\(M\) :

Hartmann Number

\(m\) :

Micropolar parameter

\(N\) :

Coupling parameter

\({N}_{k}\) :

Interpolation shape function

\(n\) :

Behavior index

\(P\) :

Pressure in lab frame

\({p}_{y}\) :

Yield stress

\(p\) :

Dimensionless pressure in wave frame

\({p}^{*}\) :

Dimensional pressure in wave frame

\({\Delta P}_{\lambda }\) :

Pressure rise per wavelength

\(Q\) :

Time mean flow rate in non-dimensional form

\(q\) :

Flow rate in non-dimensional form

\({Q}^{*}\) :

Time mean flow rate in dimensional form

\({q}^{*}\) :

Flow rate in dimensional form

\(Re\) :

Reynolds number

\({R}_{m}\) :

Magnetic Reynolds number

\(r\) :

Non-dimensional radial component in wave frame

\(R\) :

Axial component in fixed frame

\({r}^{*}\) :

Dimensional radial coordinate in wave frame

\(t\) :

Time dependent variable

\({u}_{i}\) :

Nodal variable

\(\tilde{u }\) :

Approximate nodal variable

\({u}^{*},{v}^{*}\) :

Velocity components in x* & y* direction in wave frame

\(U, V\) :

Velocity components in X & Y direction in lab frame

V :

Velocity vector

\(u, v\) :

Dimensionless velocity components

\({W}_{i}\) :

Weight function

\({w}_{1},{w}_{2}\) , \({w}_{3}\) :

Weight functions

\(z\) :

Non-dimensional axial component in wave frame

\(Z\) :

Axial component in fixed frame

\({z}^{*}\) :

Dimensional axial coordinate in wave frame

\(\alpha \) :

Wave Number

\(\overline{\alpha }, \overline{\beta },\overline{\gamma }\) :

Coefficients of spin gradient viscosity for micropolar fluids

\(\epsilon \) :

Residual error

\({\varepsilon }_{\psi }\) :

Error in approximate stream function

\({\varepsilon }_{\omega }\) :

Error in approximate vorticity function

\({\varepsilon }_{g}\) :

Error in approximate microgyration function

\(\Omega \) :

Domain of the area integral

\({\kappa }^{*}\) :

Thermal conductivity

\(\overline{\kappa }\) :

Coefficient of vortex viscosity

\(\kappa \) :

Consistency index

\(\mu \) :

Dynamic viscosity

\(\rho \) :

Density

\(\Gamma \) :

Domain of the line integral

\(\eta \) :

Dimensionless wave in wave frame

\({\eta }^{*}\) :

Dimensional wave in wave frame

\(\lambda \) :

Wave length in lab frame

\(\phi \) :

Amplitude ratio

\(\psi \) :

Stream function in non-dimensional form

\({\psi }^{*}\) :

Stream function in dimensional form

\({\psi }_{k}\) :

Element nodal approximation of stream function

\(\omega \) :

Non-dimensional vorticity

\({\omega }_{k}\) :

Element nodal approximation of vorticity function

\(\sigma \) :

Electrical conductivity

\(\nu \) :

Kinematic viscosity

\((ele)\) :

Restriction to the relevant variable/function to the element

\(k\) :

Index of the node

References

  1. Latham, T.W., Fluid motions in a peristaltic pump, S.M. Thesis, MIT, (1966)

  2. Imran, M.A.; Shaheen, A.; Sherif, E.M.; Rahimi-Gorji, M.; Seikh, A.H.: Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube. Chin. J. Phys. (2019). https://doi.org/10.1016/j.cjph.2019.11.029

    Article  Google Scholar 

  3. Farooq, S.; Khan, M.I.; Riahi, A.; Chammam, W.; Khan, W.A.: Modeling and interpretation of peristaltic transport in single wall carbon nanotube flow with entropy optimization and Newtonian heating. Comput. Methods Progr. Biomed. 192, 105435 (2020)

    Article  Google Scholar 

  4. Asha, S.K.; Sunitha, G.: Thermal radiation and Hall effects on peristaltic blood flow with double diffusion in the presence of nanoparticles. Case Stud. Therm. Eng. 17, 100560 (2020)

    Article  Google Scholar 

  5. Ahmed, B.; Javed, T.: A study of full Navier-Stokes equations of peristaltic flow in a porous-saturated tube under the inducement of magnetic field: finite element analysis. Chaos, Solitons Fractals 125, 79–87 (2019)

    Article  MathSciNet  Google Scholar 

  6. Shaheen, A.; Asjad, M.I.: Peristaltic flow of a Sisko fluid over a convectively heated surface with viscous dissipation. J. Phys. Chem. Solids 122, 210–217 (2018)

    Article  Google Scholar 

  7. Noreen, S.; Tripathi, Q.D.: Heat transfer analysis on electroosmotic flow via peristaltic pumping in non-Darcy porous medium. Therm. Sci. Eng. Prog. 11, 254–262 (2019)

    Article  Google Scholar 

  8. Hayat, T.; Bibi, S.F.; Farooq, S.; Khan, A.A.: Nonlinear radiative peristaltic flow of Jeffrey nanofluid with activation energy and modified Darcy’s law. J. Braz. Soc. Mech. Sci. Eng. 41, 296 (2019)

    Article  Google Scholar 

  9. Javid, K.; Ali, N.; Asghar, Z.: Rheological and magnetic effects on a fluid flow in a curved channel with different peristaltic wave profiles. J. Braz. Soc. Mech. Sci. Eng. 41, 483 (2019)

    Article  Google Scholar 

  10. Saleem, A.; Akhtar, S.; Nadeem, S.; Ghalambaz, M.: Microphysical analysis for peristaltic flow of SWCNT and MWCNT carbon nanotubes inside a catheterised artery having thrombus: irreversibility effects with entropy. Int. J. Exergy 34(3), 301–314 (2021)

    Article  Google Scholar 

  11. Nadeem, S.; Akhtar, S.; Saleem, A.: Peristaltic flow of a heated Jeffrey fluid inside an elliptic duct: streamline analysis. Appl. Math. Mech. 42(4), 583–592 (2021)

    Article  MathSciNet  Google Scholar 

  12. Saleem, S.; Akhtar, S.; Nadeem, S.; Saleem, A.; Ghalambaz, M.; Issakhov, A.: Mathematical study of electroosmotically driven peristaltic flow of Casson fluid inside a tube having systematically contracting and relaxing sinusoidal heated walls. Chin. J. Phys. 71, 300–311 (2021)

    Article  MathSciNet  Google Scholar 

  13. Akhtar, S.; McCash, L.B.; Nadeem, S.; Saleem, A.: Scientific breakdown for physiological blood flow inside a tube with multi-thrombosis. Sci. Rep. 11(1), 1–14 (2021)

    Article  Google Scholar 

  14. Khan, S.U.; Waqas, H.; Bhatti, M.M.; Imran, M.: Bioconvection in the rheology of magnetized couple stress nanofluid featuring activation energy and Wu’s slip. J. Non-Equilibrium Thermodyn. 45(1), 81–95 (2020)

    Article  Google Scholar 

  15. Al-Khaled, K.; Khan, S.U.; Khan, I.: Chemically reactive bioconvection flow of tangent hyperbolic nanoliquid with gyrotactic microorganisms and nonlinear thermal radiation. Heliyon 6(1), e03117 (2020)

    Article  Google Scholar 

  16. Gholinia, M.; Hosseinzadeh, Kh.; Mehrzadi, H.; Ganji, D.D.; Ranjbar, A.A.: Investigation of MHD Eyring-Powell fluid flow over a rotating disk under effect of homogeneous–heterogeneous reactions. Case Stud. Therm. Eng. 13, 100356 (2019)

    Article  Google Scholar 

  17. Khan, I.; Fatima, S.; Malik, M.Y.; Salahuddin, T.: Exponentially varying viscosity of magnetohydrodynamic mixed convection Eyring-Powell nanofluid flow over an inclined surface. Results Phys. 8, 1194–1203 (2018)

    Article  Google Scholar 

  18. Salawu, S.O.; Ogunseye, H.A.: Entropy generation of a radiative hydromagnetic PowellEyring chemical reaction nanofluid with variable conductivity and electric field loading. Results Eng. 5, 100072 (2020)

    Article  Google Scholar 

  19. Eringen, A.C.: Theory of micropolar fluid. J. Math. Mech. 16, 1–18 (1966)

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  21. Ariman, T.; Turk, M.A.; Sylvester, N.D.: Review article–applications of microcontinuum fluid mechanics. Int. J. Eng. Sci. 12, 273–293 (1974)

    Article  Google Scholar 

  22. Zaib, A.; Khan, U.; Shah, Z.; Kumam, P.; Thounthong, P.: Optimization of entropy generation in flow of micropolar mixed convective magnetite (Fe3O4) ferroparticle over a vertical plate. Alex. Eng. J. 58(4), 1461–1470 (2019)

    Article  Google Scholar 

  23. Bhattacharjee, B.; Chakraborti, P.; Choudhuri, K.: Evaluation of the performance characteristics of double-layered porous micropolar fluid lubricated journal bearing. Tribol. Int. 138, 415–423 (2019)

    Article  Google Scholar 

  24. Reddy, M.G.; Kumar, K.G.: Cattaneo-Christov heat flux feature on carbon nanotubes filled with micropolar liquid over a melting surface: a stream line study. Int. Commun. Heat Mass Transf. 122, 105142 (2021)

    Article  Google Scholar 

  25. Abbas, N.; Malik, M.Y.; Nadeem, S.: Transportation of magnetized micropolar hybrid nanomaterial fluid flow over a Riga curface surface. Comput. Methods Prog. Biomed. 185, 105136 (2020)

    Article  Google Scholar 

  26. Wang, Y.; Ali, N.; Hayat, T.; Oberlack, M.: Peristaltic motion of a magnetohydrodynamic micropolar fluid in a tube. Appl. Math. Model. 35(8), 3737–3750 (2011)

    Article  MathSciNet  Google Scholar 

  27. Nadeem, S.; Abbas, N.; Elmasry, Y.; Malik, M.Y.: Numerical analysis of water based CNTs flow of micropolar fluid through rotating frame. Comput. Methods Prog. Biomed. 186, 105194 (2020)

    Article  Google Scholar 

  28. Sherief, H.H.; Faltas, M.S.; Ragab, K.E.: Motion of a slip spherical particle near a planar micropolar-viscous interface. Eur. J. Mechanics-B/Fluids 89, 274–288 (2021)

    Article  MathSciNet  Google Scholar 

  29. Hayat, T.; Ali, N.: Effects of an endoscope on peristaltic flow of a micropolar fluid. Math. Comput. Model. 48, 721–733 (2008)

    Article  MathSciNet  Google Scholar 

  30. Samaei, L.; Deylami, H.M.; Amanifard, N.; Moayedi, H.: Numerical evaluation of using micropolar fluid model for EHD-induced natural convection heat transfer through a rectangular enclosure. J. Electrost. 101, 103372 (2019)

    Article  Google Scholar 

  31. Hamid, A.H.; Javed, T.; Ahmed, B.; Ali, N.: Numerical study of two-dimensional non-Newtonian peristaltic flow for long wavelength and moderate Reynolds number. J. Braz. Soc. Mech. Sci. Eng. 39(11), 4421–4430 (2017)

    Article  Google Scholar 

  32. Ahmed, B.; Javed, T.; Ali, N.: Numerical study at moderate Reynolds number of peristaltic flow of micropolar fluid through a porous-saturated channel in magnetic field. AIP Adv. 8(1), 015319 (2018)

    Article  Google Scholar 

  33. Ullah, N.; Nadeem, S.; McCash, L.; Saleem, A.; Issakhov, A.: Simulations of micropolar nanofluid-equipped natural convective-driven flow in a cavity. Int. J. Numer. Methods Heat Fluid Flow. (2021). https://doi.org/10.1108/HFF-08-2020-0504

  34. Ahmad, S.; Nadeem, S.; Khan, M.N.: Mixed convection hybridized micropolar nanofluid with triple stratification and Cattaneo-Christov heat flux model. Phys. Scripta 96(7), 075205 (2021)

    Article  Google Scholar 

  35. Kwon, Y.W.; Bang, H.: The finite element method using MATLAB. CRC Press, London (1991)

    MATH  Google Scholar 

  36. Ferreira, A.J.M.: MATLAB codes for finite element analysis solids and structures. Springer, New York (2009)

    MATH  Google Scholar 

  37. Abd-Alla, A.M.; Abo-Dahab, S.M.; Al-Simery, R.D.: Effect of rotation on peristaltic flow of a micropolar fluid through a porous medium with an external magnetic field. J. Magn. Magn. Mater. 348, 33–43 (2013)

    Article  Google Scholar 

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Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia for funding this work through research groups program under grant number R.G.P-1/234/42.

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Authors and Affiliations

  1. Department of Infectious Diseases, Huzhou Central Hospital, Huzhou, 313000, Zhejiang, People’s Republic of China

    Zhao-Wei Tong

  2. Department of Mathematics and Statistics, The University of Lahore, Sargodha Campus, Sargodha, 40100, Pakistan

    Bilal Ahmed & Sohail Ahmad

  3. Department of Mathematics & Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid, 22110, Jordan

    Kamel Al-Khaled

  4. Department of Mathematics, COMSATS University Islamabad, Sahiwal, 57000, Pakistan

    Sami Ullah Khan

  5. Department of Mathematics and Statistics, Riphah International University I-14, Islamabad, 44000, Pakistan

    M. Ijaz Khan

  6. Department of Mathematics, College of Sciences, King Khalid University, Abha, 61413, Kingdom of Saudi Arabia

    M. Y. Malik

  7. School of Engineering, Huzhou University, Huzhou, 313000, People’s Republic of China

    Wei-Feng Xia

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  1. Zhao-Wei Tong
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  2. Bilal Ahmed
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Correspondence to Wei-Feng Xia.

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Tong, ZW., Ahmed, B., Al-Khaled, K. et al. Peristaltic Blood Transport in Non-Newtonian Fluid Confined by Porous Soaked Tube: A Numerical Study Through Galerkin Finite Element Technique. Arab J Sci Eng 47, 1019–1031 (2022). https://doi.org/10.1007/s13369-021-05981-1

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