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Effects of magnetic Reynolds number on swimming of gyrotactic microorganisms between rotating circular plates filled with nanofluids

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Abstract

The three-dimensional (3D) nanofluid flow among the rotating circular plates filled with nanoparticles and gyrotactic microorganisms is studied. A generalized form of the magnetic Reynolds number is used for the mathematical modeling of the ferro-nanofluid flow. The torque effects on the lower and upper plates are calculated. A differential transform scheme with the Padé approximation is used to solve the coupled highly nonlinear ordinary differential equations. The results show that the squeeze Reynolds number significantly suppresses the temperature, microorganism, and nanoparticle concentration distribution, and agree well with those obtained by the numerical method.

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Abbreviations

H θ :

axial components

H z :

azimuthal components

μ 1 :

magnetic permeability inside the plate

μ 2 :

magnetic permeability outside the plate

μ f :

free space permeability (N·A−2)

B(r, θ, z):

induced magnetic field

T 1 :

constant temperature of the lower plate (K)

T u :

constant temperature of the upper plate (K)

C 1 :

concentration at the lower plate

C u :

concentration at the upper plate

p :

pressure (Pa)

ρ :

fluid density (kg·m−3)

μ :

fluid viscosity (N·s·m−2)

δ :

electrical conductivity (S·m−1)

T :

temperature (K)

C :

concentration

T m :

mean fluid temperature (K)

c p :

specific heat (J·kg−1·K−1)

D B :

Brownian diffusivity

D T :

thermophoretic diffusion coefficient

j :

microorganism flux

R M :

magnetic force strength in the axial direction

M T :

magnetic force in the tangential direction

M R :

magnetic Reynolds number

T b :

Brownian motion parameter

T t :

thermophoresis parameter

Pr :

Prandtl number

Sc :

Schmidt number

B n :

bioconvection number

f :

axial velocity

g :

tangential velocity

ξ :

angular velocity

b :

radius of the disk

T up :

dimensionless torque applied on the upper plate

T low :

dimensionless torque applied on the lower plate

b̄ :

chemotaxis constant

W max :

maximal speed

D mo :

diffusivity of microorganisms

R Q :

squeeze Reynolds number

References

  1. EASTMAN, J. A. and CHOI, S. U. S. Enhancing thermal conductivity of fluids with nanoparticles. ASME International Mechanical Engineering Congress and Exposition, San Fransisco, 99–105 (1995)

    Google Scholar 

  2. Pak, B. C. and CHO, Y. I. Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles. Experimental Heat Transfer: An International Journal, 11, 151–170 (1998)

    Article  Google Scholar 

  3. LIN, J. Z. and YANG, H. L. A review on the flow instability of nanofluids. Applied Mathematics and Mechanics (English Edition), 40(9), 1227–1238 (2019) https://doi.org/10.1007/s10483-019-2521-9

    Article  MathSciNet  Google Scholar 

  4. DAS, S. K., PUTRA, N., THIESEN, P., and ROETZEL, W. Temperature dependence of thermal conductivity enhancement for nanofluids. Journal of Heat Transfer, 125, 567–574 (2003)

    Article  Google Scholar 

  5. XUAN, Y. and ROETZEL, W. Conceptions for heat transfer correlation of nanofluids. International Journal of Heat and Mass Transfer, 43, 3701–3707 (2000)

    Article  MATH  Google Scholar 

  6. GHERASIM, I., ROY, G., NGUYEN, C. T., and VO-NGOC, D. Experimental investigation of nanofluids in confined laminar radial flows. International Journal of Thermal Sciences, 48, 1486–1493 (2009)

    Article  Google Scholar 

  7. ZHANG, X., GU, H., and FUJII, M. Effective thermal conductivity and thermal diffusivity of nanofluids containing spherical and cylindrical nanoparticles. Experimental Thermal and Fluid Science, 31, 593–599 (2007)

    Article  Google Scholar 

  8. HAMILTON, R. L. and CROSSER, O. K. Thermal conductivity of heterogeneous two-component systems. Industrial and Engineering Chemistry Fundamentals, 1, 187–191 (1962)

    Article  Google Scholar 

  9. SHEIKHOLESLAMI, M., HAYAT, T., and ALSAEDI, A. MHD free convection of Al2O3-water nanofluid considering thermal radiation: a numerical study. International Journal of Heat and Mass Transfer, 96, 513–524 (2016)

    Article  Google Scholar 

  10. FAKOUR, M., VAHABZADEH, A., and GANJI, D. D. Study of heat transfer and flow of nanofluid in permeable channel in the presence of magnetic field. Propulsion and Power Research, 4, 50–62 (2015)

    Article  Google Scholar 

  11. ELLAHI, R., ZEESHAN, A., HUSSAIN, F., and ABBAS, T. Thermally charged MHD bi-phase flow coatings with non-Newtonian nanofluid and hafnium particles along slippery walls. Coatings, 9, 300 (2019)

    Article  Google Scholar 

  12. ZHU, J., WANG, S., ZHENG, L., and ZHANG, X. Heat transfer of nanofluids considering nanoparticle migration and second-order slip velocity. Applied Mathematics and Mechanics (English Edition), 38(1), 125–136 (2017) https://doi.org/10.1007/s10483-017-2155-6

    Article  MathSciNet  Google Scholar 

  13. BASKAYA, E., KOMURGOZ, G., and OZKOL, I. Investigation of oriented magnetic field effects on entropy generation in an inclined channel filled with ferrofluids. Entropy, 19, 377 (2017)

    Article  Google Scholar 

  14. HAMID, M., USMAN, M., HAQ, R. U., KHAN, Z. H., and WANG, W. Wavelet analysis of stagnation point flow of non-Newtonian nanofluid. Applied Mathematics and Mechanics (English Edition), 40(8), 1211–1226 (2019) https://doi.org/10.1007/s10483-019-2508-6

    Article  Google Scholar 

  15. ZEESHAN, A., SHEHZAD, N., ABBAS, T., and ELLAHI, R. Effects of radiative electromagnetohydrodynamics diminishing internal energy of pressure-driven flow of titanium dioxide-water nanofluid due to entropy generation. Entropy, 21, 236 (2019)

    Article  Google Scholar 

  16. IMTIAZ, M., SHAHID, F., HAYAT, T., and ALSAEDI, A. Melting heat transfer in Cu-water and Ag-water nanofluids flow with homogeneous-heterogeneous reactions. Applied Mathematics and Mechanics (English Edition), 40(4), 465–480 (2019) https://doi.org/10.1007/s10483-019-2462-8

    Article  MathSciNet  MATH  Google Scholar 

  17. HUSSAIN, F., ELLAHI, R., ZEESHAN, A., and VAFAI, K. Modelling study on heated couple stress fluid peristaltically conveying gold nanoparticles through coaxial tubes: a remedy for gland tumors and arthritis. Journal of Molecular Liquids, 268, 149–155 (2018)

    Article  Google Scholar 

  18. ELLAHI, R., HASSAN, M., and ZEESHAN, A. Shape effects of nanosize particles in Cu-H2O nanofluid on entropy generation. International Journal of Heat and Mass Transfer, 81, 449–456 (2015)

    Article  Google Scholar 

  19. HEIDARY, H., HOSSEINI, R., PIRMOHAMMADI, M., and KERMANI, M. J. Numerical study of magnetic field effect on nano-fluid forced convection in a channel. Journal of Magnetism and Magnetic Materials, 374, 11–17 (2015)

    Article  Google Scholar 

  20. LIN, J., XIA, Y., and KU, X. K. Friction factor and heat transfer of nanofluids containing cylindrical nanoparticles in laminar pipe flow. Journal of Applied Physics, 116, 133513 (2014)

    Article  Google Scholar 

  21. LIN, J. Z., XIA, Y., and KU, X. K. Pressure drop and heat transfer of nanofluid in turbulent pipe flow considering particle coagulation and breakage. Journal of Heat Transfer, 136, 111701 (2014)

    Article  Google Scholar 

  22. LIN, J. Z., XIA, Y., and KU, X. K. Flow and heat transfer characteristics of nanofluids containing rod-like particles in a turbulent pipe flow. International Journal of Heat and Mass Transfer, 93, 57–66 (2016)

    Article  Google Scholar 

  23. SOBAMOWO, M. G., JAYESIMI, L. O., and WAHEED, M. A. Magnetohydrodynamic squeezing flow analysis of nanofluid under the effect of slip boundary conditions using variation of parameter method. Karbala International Journal of Modern Science, 4, 107–118 (2018)

    Article  Google Scholar 

  24. VAJRAVELU, K., PRASAD, K. V., NG, C. O., and VAIDYA, H. MHD squeeze flow and heat transfer of a nanofluid between parallel disks with variable fluid properties and transpiration. International Journal of Mechanical and Materials Engineering, 12, 9 (2017)

    Article  Google Scholar 

  25. HOSSEINZADEH, K., ALIZADEH, M., and GANJI, D. D. Hydrothermal analysis on MHD squeezing nanofluid flow in parallel plates by analytical method. International Journal of Mechanical and Materials Engineering, 13, 4 (2018)

    Article  Google Scholar 

  26. HASHMI, M. M., HAYAT, T., and ALSAEDI, A. On the analytic solutions for squeezing flow of nanofluid between parallel disks. Nonlinear Analysis: Modelling and Control, 17, 418–430 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. SINGH, K., RAWAT, S. K., and KUMAR, M. Heat and mass transfer on squeezing unsteady MHD nanofluid flow between parallel plates with slip velocity effect. Journal of Nanoscience, 1–11 (2016)

    Google Scholar 

  28. BHATTA, D. P., MISHRA, S. R., and DASH, J. K. Unsteady squeezing flow of water-based nanofluid between two parallel disks with slip effects: analytical approach. Heat Transfer-Asian Research, 48, 1–20 (2019)

    Article  Google Scholar 

  29. MADAKI, A. G., ROSLAN, R., MOHAMED, M., and KAMARDAN, M. G. Analytical solutions of squeezing unsteady nanofluid flow in the presence of thermal radiation. Journal of Computer Science and Computational Mathematics, 6, 451–463 (2016)

    Google Scholar 

  30. VIMALA, P. and MANIMEGALAI, K. Natural convection flow of nanofluids in squeeze film with an exponential curvature. Journal of Informatics and Mathematical Sciences, 10, 371–381 (2010)

    Article  Google Scholar 

  31. ACHARYA, N., DAS, K., and KUNDU, P. K. The squeezing flow of Cu-water and Cu-kerosene nanofluids between two parallel plates. Alexandria Engineering Journal, 55, 1177–1186 (2016)

  32. THONGMOON, M. and PUSJUSO, S. The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations. Nonlinear Analysis: Hybrid Systems, 4, 425–431 (2010)

    MathSciNet  MATH  Google Scholar 

  33. FATOOREHCHI, H. and ABOLGHASEMI, H. Differential transform method to investigate mass transfer phenomenon to a falling liquid film system. Australian Journal of Basic and Applied Sciences, 5, 337–345 (2011)

    MATH  Google Scholar 

  34. YOUSIF, M. A., HATAMI, M., and ISMAEL, H. F. Heat transfer analysis of MHD three dimensional Casson fluid flow over a porous stretching sheet by DTM-Padé. International Journal of Applied and Computational Mathematics, 3, 813–828 (2017)

    Article  MathSciNet  Google Scholar 

  35. ÇILINGIR, S., İINCI, A. A., BULUT, H., HAMMOUCH, Z., BASKONUS, H. M., Mekkaoui, T., MUHAMMAD, B., and BIN, F. Numerical investigation on MHD flow and heat transfer over an exponentially stretching sheet with viscous dissipation and radiation effects, ITM Web of Conferences, 13, 01025 (2017)

    Article  Google Scholar 

  36. KHAN, W. A., RASHAD, A. M., ABDOU, M. M. M., and TLILI, I. Natural bioconvection flow of a nanofluid containing gyrotactic microorganisms about a truncated cone. European Journal of Mechanics-B/Fluids, 75, 133–142 (2019)

    Article  MathSciNet  Google Scholar 

  37. Saleem, S., Rafiq, H., Al-Qahtani, A., El-Aziz, M. A., Malik, M. Y., and animasaun, i. l. magneto jeffrey nanofluid bioconvection over a rotating vertical cone due to gyrotactic microorganism}. mathematical problems in engineering, 2019, 3478037 (2

  38. KHAN, W. A., MAKINDE, O. D., and KHAN, Z. H. MHD boundary layer flow of a nanofluid containing gyrotactic microorganisms past a vertical plate with Navier slip. International Journal of Heat and Mass Transfer, 74, 285–291 (2014)

    Article  Google Scholar 

  39. FAROOQ, S., HAYAT, T., ALSAEDI, A., and AHMAD, B. Numerically framing the features of second order velocity slip in mixed convective flow of Sisko nanomaterial considering gyrotactic microorganisms. International Journal of Heat and Mass Transfer, 112, 521–532 (2017)

    Article  Google Scholar 

  40. WAQAS, H., KHAN, S. U., HASSAN, M., BHATTI, M. M., and IMRAN, M. Analysis on the bioconvection flow of modified second-grade nanofluid containing gyrotactic microorganisms and nanoparticles. Journal of Molecular Liquids, 291, 111231 (2019)

    Article  Google Scholar 

  41. WAQAS, H., KHAN, S. U., IMRAN, M., and BHATTI, M. M. Thermally developed Falkner-Skan bioconvection flow of a magnetized nanofluid in the presence of a motile gyrotactic microorganism: Buongiorno’s nanofluid model. Physica Scripta, 94, 115304 (2019)

    Article  Google Scholar 

  42. AZIZ, A., KHAN, W. A., and POP, I. Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms. International Journal of Thermal Sciences, 56, 48–57 (2012)

    Article  Google Scholar 

  43. RAEES, A., XU, H., SUN, Q., and POP, I. Mixed convection in gravity-driven nano-liquid film containing both nanoparticles and gyrotactic microorganisms. Applied Mathematics and Mechanics (English Edition), 36(2), 163–178 (2015) https://doi.org/10.1007/s10483-015-1901-7

    Article  MathSciNet  MATH  Google Scholar 

  44. MEADE, D. B., HARAN, B. S., and WHITE, R. E. The shooting technique for the solution of two-point boundary value problems. Maple Technical Newsletter, 3, 1–8 (1996)

    Google Scholar 

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Author information

Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong Province, China

    Lijun Zhang, M. B. Arain & M. M. Bhatti

  2. Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

    M. B. Arain & A. Zeeshan

  3. Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jaddeh, 21589, Saudi Arabia

    H. Hal-Sulami

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  1. Lijun Zhang
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  2. M. B. Arain
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  3. M. M. Bhatti
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  4. A. Zeeshan
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  5. H. Hal-Sulami
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Correspondence to M. M. Bhatti.

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Zhang, L., Arain, M.B., Bhatti, M.M. et al. Effects of magnetic Reynolds number on swimming of gyrotactic microorganisms between rotating circular plates filled with nanofluids. Appl. Math. Mech.-Engl. Ed. 41, 637–654 (2020). https://doi.org/10.1007/s10483-020-2599-7

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  • DOI: https://doi.org/10.1007/s10483-020-2599-7

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