Skip to main content
SpringerLink
Log in

Influence of Stefan blowing on nanofluid flow submerged in microorganisms with leading edge accretion or ablation

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

The forced convective boundary layer flow of viscous incompressible time-dependent fluid containing water-based nanofluids and gyrotactic microorganisms simultaneously, from a flat surface with leading edge accretion (or ablation), is theoretically investigated in the present study. In doing so, the governing conservation equations are rendered into a nonlinear system of ordinary differential equations by means of utilizing appropriate coordinates transformations. MAPLE symbolic software is employed to solve these equations, which are subjected to impose boundary conditions using the Runge–Kutta–Fehlberg fourth-fifth order numerical method. It is noteworthy that the results of the present study are in an excellent agreement with previous solutions available in literature. The effect of selected parameters on velocity, temperature, nanoparticle volume fraction and motile microorganism density function is then investigated. Tabular solutions are included for the skin friction, heat transfer rate, nano-particle mass transfer rate and microorganism transfer rate. Applications of the study arise in advanced micro-flow devices to bio-modified nanomaterials processing.

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

Similar content being viewed by others

Abbreviations

\(\tilde{b}\) :

Chemotaxis constant \(({\text{m}})\)

\(C\) :

Nano-particles volume fraction \(( - )\)

\(C_{\text{w}}\) :

Wall nano-particle volume fraction \(( - )\)

\(C_{\infty }\) :

Ambient nano-particle volume fraction \(( - )\)

\(C_{{{\text{f}}_{{\bar{x}}} }}\) :

Local skin friction coefficient along the \(\bar{x}\) \(( - )\)

\(c_{\text{p}}\) :

Specific heat at constant pressure \(\left( {\frac{\text{J}}{\text{kgK}}} \right)\)

\(D_{\text{B}}\) :

Brownian diffusion coefficient \(\left( {\frac{{{\text{m}}^{2} }}{\text{s}}} \right)\)

\(D_{n}\) :

Microorganism diffusion coefficient \(\left( {\frac{{{\text{m}}^{2} }}{\text{s}}} \right)\)

\(D_{\text{T}}\) :

Thermophoresis diffusion coefficient \(\left( {\frac{{{\text{m}}^{2} }}{\text{s}}} \right)\)

\(f(\eta )\) :

Dimensionless stream function \(( - )\)

\(\vec{j}\) :

Vector flux of microorganisms \(\left( {\frac{\text{kg}}{{{\text{m}}^{2} {\text{s}}}}} \right)\)

\(k\) :

Thermal conductivity \(\left( {\frac{\text{W}}{\text{mK}}} \right)\)

\(Lb\) :

Bioconvection Lewis number \(\left( {Lb = \frac{{\alpha_{{}} }}{{D_{n} }}} \right)\,\,\,( - )\)

\(Le\) :

Lewis number \(\left( {Le = \frac{\alpha }{{D_{\text{B}} }}} \right)\,\,\,( - )\)

\(Nb\) :

Brownian motion parameter \(\left( {Nb = \frac{{\tau D_{\text{B}} \left( {C_{\text{w}} - C_{\infty } } \right)}}{{\alpha_{{}} }}} \right)\,\,\,( - )\)

\(Nn_{{\bar{x}}}\) :

Local density number of motile microorganisms \(\,\,( - )\)

\(Nt\) :

Thermophoresis parameter \(\left( {Nt = \frac{{\tau D_{\text{T}} \left( {T_{\text{w}} - T_{\infty } } \right)}}{\alpha T_{\infty }}} \right)\,\,\,( - )\)

\(Nu_{{\bar{x}}}\) :

Local Nusselt number \(( - )\)

\(n\) :

Number of motile microorganisms \(( - )\)

\(n_{\text{w}}\) :

Wall motile microorganisms \(( - )\)

\(Pe\) :

Bioconvection Péclet number \(\left( {Pe = \frac{{\tilde{b}W_{\text{c}} }}{{D_{n} }}} \right)\,\,\,( - )\)

\(Pr\) :

Prandtl number \(\left( {Pr = \frac{\upsilon }{\alpha }} \right)\,\,\,( - )\)

\(Re\) :

Reynolds number \(\left( {\frac{{\bar{U}_{\infty } \bar{x}}}{\upsilon }} \right)\) \(\left( - \right)\)

\(Sh_{{\bar{x}}}\) :

Local Sherwood number \(( - )\)

\(s\) :

Wall mass flux (Stefan blowing)\(\left( {\frac{{C_{\text{w}} - C_{\infty } }}{{1 - C_{\text{w}} }}} \right)\) \(( - )\)

\(\bar{t}\) :

Dimensional time \(({\text{s}})\)

\(T\) :

Nanofluid temperature \(({\text{K}})\)

\(T_{\text{w}}\) :

Wall temperature \((K)\)

\(T_{\infty }\) :

Ambient temperature \(({\text{K}})\)

\(\bar{U}_{\infty }\) :

Dimensional ambient velocity \(\left( {\frac{\text{m}}{\text{s}}} \right)\)

\(\bar{u}\) :

Velocity components along the \(\bar{x}\)-axis \(\left( {\frac{\text{m}}{\text{s}}} \right)\)

\(\overrightarrow {{\bar{v}}}\) :

Velocity vector \(\left( {\frac{\text{m}}{\text{s}}} \right)\)

\(\tilde{\bar{v}}\,\) :

Average swimming velocity vector of microorganism \(\left( {\frac{{{\text{m}}^{2} }}{\text{s}}} \right)\)

\(\bar{v}\) :

Velocity components along the \(\bar{y}\)-axis \(\left( {\frac{\text{m}}{\text{s}}} \right)\)

\(W_{\text{c}}\) :

Maximum cell swimming speed \(\left( {\frac{\text{m}}{\text{s}}} \right)\)

\(\bar{x}\) :

Dimensional coordinate along the surface \(({\text{m}})\)

\(\bar{y}\) :

Coordinate normal to the surface \((m)\)

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

Effective thermal diffusivity \(\left( {\frac{{{\text{m}}^{2} }}{\text{s}}} \right)\)

\(\gamma {}_{{}}\) :

Leading edge accretion/ablation \(( - )\)

\(\eta\) :

Independent similarity variable \(( - )\)

\(\theta (\eta )\) :

Dimensionless temperature \(( - )\)

\(\mu\) :

Dynamic viscosity \(\left( {\frac{\text{kg}}{\text{ms}}} \right)\)

\(\upsilon\) :

Kinematic viscosity \(\left( {\frac{{{\text{m}}^{2} }}{\text{s}}} \right)\)

\(\rho\) :

Fluid density \(\left( {\frac{\text{kg}}{{{\text{m}}^{3} }}} \right)\)

π :

Pi \(( - )\)

\((\rho c)_{\text{f}}\) :

Volumetric heat capacity of the fluid \(\left( {\frac{\text{J}}{{{\text{m}}^{3} {\text{K}}}}} \right)\)

\((\rho c)_{\text{p}}\) :

Volumetric heat capacity of the nanoparticle material \(\left( {\frac{\text{J}}{{{\text{m}}^{3} {\text{K}}}}} \right)\)

\(\sigma\) :

Dimensionless time variable \(\left( {\bar{U}_{\infty }^{{}} \,\bar{t}/\bar{x}} \right)\) \(( - )\)

\(\tau\) :

Ratio of the effective heat capacity of the nanoparticle material to the fluid heat capacity \(\left( {\frac{{(\rho c)_{\text{p}} }}{{(\rho c)_{\text{f}} }}} \right)\left( - \right)\)

\(\phi (\eta )\) :

Dimensionless nanoparticles volume fraction \(( - )\)

\(\chi (\eta )\) :

Dimensionless number of motile microorganisms \(( - )\)

\(\psi\) :

Streamline function \(( - )\)

\((\,\,\,)'\) :

Ordinary differentiation with respect to \(\eta\)

\((\,\,\,)_{\text{w}}\) :

Condition at wall

\((\,\,\,)_{\infty }\) :

Condition in free stream

References

  1. Choi SUS, Eastman JA (1995) Enhancing thermal conductivity of fluids with nanoparticles. ASME Int Mech Eng Congr Expo 66:99–105

    Google Scholar 

  2. Loganathan P, Vimala C (2014) Unsteady flow of nanofluids past a vertical flat plate with leading edge accretion or ablation. Indian J Phys 88:855–859

    Article  Google Scholar 

  3. Hayat T, Imtiaz M, Alsaedi A (2016) Unsteady flow of nanofluid with double stratification and magnetohydrodynamics. Int J Heat Mass Transf 92:100–109

    Article  Google Scholar 

  4. Shehzad N, Zeeshan A, Ellahi R, Vafai K (2016) Convective heat transfer of nanofluid in a wavy channel: Buongiorno’s mathematical model. J Mol Liq 222:446–455

    Article  Google Scholar 

  5. Malvandi A, Ghasemi A, Ganji DD (2016) Thermal performance analysis of hydromagnetic Al2O3-water nanofluid flows inside a concentric microannulus considering nanoparticle migration and asymmetric heating. Int J Therm Sci 109:10–22

    Article  Google Scholar 

  6. Rahman MM, Alam MS, Eltayeb IA (2016) Hydromagnetic natural convective heat transfer flow in an isosceles triangular cavity filled with nanofluid using two-component nonhomogeneous model. Int J Therm Sci 107:272–288

    Article  Google Scholar 

  7. Malvandi A, Moshizi SA, Ganji DD (2016) Effects of temperature-dependent thermophysical properties on nanoparticle migration at mixed convection of nanofluids in vertical microchannels. Powder Technol 303:7–19

    Article  Google Scholar 

  8. Sheremet MA, Pop I, Roşca NC (2016) Magnetic field effect on the unsteady natural convection in a wavy-walled cavity filled with a nanofluid: Buongiorno’s mathematical model. J Taiwan Inst Chem Eng 61:211–222

    Article  Google Scholar 

  9. Akilu S, Sharma KV, Baheta AT, Mamat R (2016) A review of thermophysical properties of water based composite nanofluids. Renew Sustain Energy Rev 66:654–678

    Article  Google Scholar 

  10. Ilhan B, Kurt M, Ertürk H (2016) Experimental investigation of heat transfer enhancement and viscosity change of hBN nanofluids. Exp Therm Fluid Sci 77:272–283

    Article  Google Scholar 

  11. Agarwal R, Verma K, Kumar N, Singh R (2017) Sensitivity of thermal conductivity for Al2O3 nanofluids. Exp Therm Fluid Sci 80:19–26

    Article  Google Scholar 

  12. Loganathan P, Chand PN, Ganesan P (2013) Radiation effects on an unsteady natural convective flow of a nanofluid past an infinite verticle plate. Nano 8:1–10

    Article  Google Scholar 

  13. Mejri I, Mahmoudi A, Abbassi MA, Omri A (2014) Magnetic field effect on entropy generation in a nanofluid-filled enclosure with sinusoidal heating on both side walls. Powder Technol 266:340–353

    Article  Google Scholar 

  14. Uddin MJ, Khan WA, Ismail AIM (2012) Lie group analysis of natural convective flow from a convectively heated upward facing radiating permeable horizontal plate in porous media filled with nanofluid. J Appl Math 2012:18. doi:10.1155/2012/648675

    Article  MATH  MathSciNet  Google Scholar 

  15. Hatami M, Sheikholeslami M, Ganji DD (2014) Laminar flow and heat transfer of nanofluid between contracting and rotating disks by least square method. Powder Technol 253:769–779

    Article  Google Scholar 

  16. Freidoonimehr N, Rashidi MM, Mahmud S (2015) Unsteady MHD free convective flow past a permeable stretching vertical surface in a nano-fluid. Int J Therm Sci 87:136–145

    Article  Google Scholar 

  17. Beg OA, Rashidi MM, Akbari M, Hosseini A (2014) Comparative numerical study of single-phase and two-phase models for bio-nanofluid transport phenomena. J Mech Med Biol 14:1450011

    Article  Google Scholar 

  18. Mutuku WN, Makinde OD (2014) Hydromagnetic bioconvection of nanofluid over a permeable vertical plate due to gyrotactic microorganisms. Comput Fluids 95:88–97

    Article  MathSciNet  Google Scholar 

  19. Siddiqa S, Gul-e-Hina Begum N (2016) Numerical solutions of nanofluid bioconvection due to gyrotactic microorganisms along a vertical wavy cone. Int J Heat Mass Transf 101:608–613

    Article  Google Scholar 

  20. Acharya N, Das K, Kumar P (2016) Framing the effects of solar radiation on magneto-hydrodynamics bioconvection nanofluid flow in presence of gyrotactic microorganisms. J Mol Liq 222:28–37

    Article  Google Scholar 

  21. Kuznetsov AV (2006) The onset of thermo-bioconvection in a shallow fluid saturated porous layer heated from below in a suspension of oxytactic microorganisms. Eur J Mech B Fluids 25:223–233

    Article  MATH  MathSciNet  Google Scholar 

  22. Geng P, Kuznetsov AV (2004) Effect of small solid particles on the development of bioconvection plumes. Int Commun Heat Mass Transf 31:629–638

    Article  Google Scholar 

  23. Kuznetsov AV, Nield DA (2014) Natural convective boundary-layer flow of a nanofluid past a vertical plate: a revised model. Int J Therm Sci 77:126–129

    Article  Google Scholar 

  24. Kuznetsov AV (2012) Nanofluid bioconvection in a horizontal fluid-saturated porous layer. J Porous Media 15:11–27

    Article  Google Scholar 

  25. Kuznetsov AV (2011) Bio-thermal convection induced by two different species of microorganisms. Int Commun Heat Mass Transf 38:548–553

    Article  Google Scholar 

  26. Kuznetsov AV (2005) Thermo-bioconvection in a suspension of oxytactic bacteria. Int Commun Heat Mass Transf 32:991–999

    Article  Google Scholar 

  27. Kuznetsov AV, Avramenko AA, Geng P (2004) Analytical investigation of a falling plume caused by bioconvection of oxytactic bacteria in a fluid saturated porous medium. Int J Eng Sci 42:557–569

    Article  Google Scholar 

  28. Makinde OD, Animasaun IL (2016) Thermophoresis and Brownian motion effects on MHD bioconvection of nano fluid with nonlinear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution. J Mol Liq 221:733–743

    Article  Google Scholar 

  29. Akbar NS, Khan ZH (2016) Magnetic field analysis in a suspension of gyrotactic microorganisms and nanoparticles over a stretching surface. J Magn Magn Mater 410:72–80

    Article  Google Scholar 

  30. Amirsom NA, Uddin MJ, Ismail AI (2016) Three dimensional stagnation point flow of bionanofluid with variable transport properties. Alex Eng J 55:1983–1993

    Article  Google Scholar 

  31. Jayachandra Babu M, Sandeep N (2016) Effect of nonlinear thermal radiation on non-aligned bio-convective stagnation point flow of a magnetic-nanofluid over a stretching sheet. Alex Eng J 55:1931–1939

    Article  Google Scholar 

  32. Raees A, Raees-ul-Haq M, Xu H, Sun Q (2016) Three-dimensional stagnation flow of a nanofluid containing both nanoparticles and microorganisms on a moving surface with anisotropic slip. Appl Math Model 40:4136–4150

    Article  MathSciNet  Google Scholar 

  33. Fang T, Jing W (2014) Flow, heat, and species transfer over a stretching plate considering coupled Stefan blowing effects from species transfer. Commun Nonlinear Sci Numer Simul 19:3086–3097

    Article  MathSciNet  Google Scholar 

  34. Nellis G, Klein S (2009) Heat transfer. Cambridge University Press, New York, USA, p E23–5

    MATH  Google Scholar 

  35. Lienhard JH IV, Lienhard JH V (2005) A heat transfer text book, 3rd edn. Phlogiston Press, Cambridge, pp 662–663

    MATH  Google Scholar 

  36. Uddin J, Kabir MN, Bég OA (2016) Computational investigation of Stefan blowing and multiple-slip effects on buoyancy-driven bioconvection nanofluid flow with microorganisms. Int J Heat Mass Transf 95:116–130

    Article  Google Scholar 

  37. Buongiorno J (2006) Convective transport in nanofluids. J Heat Transf 128:240

    Article  Google Scholar 

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

    Article  Google Scholar 

  39. Garoosi F, Jahanshaloo L, Garoosi S (2015) Numerical simulation of mixed convection of the nano fluid in heat exchangers using a Buongiorno model. Powder Technol 269:296–311

    Article  Google Scholar 

  40. Malvandi A, Moshizi SA, Ghadam E, Ganji DD (2014) Modified Buongiorno’ s model for fully developed mixed convection flow of nanofluids in a vertical annular pipe. Comput Fluids 89:124–132

    Article  Google Scholar 

  41. Garoosi F, Jahanshaloo L, Rashidi MM, Badakhsh A, Ali ME (2015) Numerical simulation of natural convection of the nanofluid in heat exchangers using a Buongiorno model. Appl Math Comput 254:183–203

    MathSciNet  Google Scholar 

  42. Sheremet MA, Pop I (2015) Mixed convection in a lid-driven square cavity filled by a nanofluid: Buongiorno’ s mathematical model. Appl Math Comput 266:792–808

    MathSciNet  Google Scholar 

  43. Sheremet MA, Pop I, Shenoy A (2015) Unsteady free convection in a porous open wavy cavity filled with a nanofluid using Buongiorno’s mathematical model. Int Commun Heat Mass Transf 67:66–72

    Article  Google Scholar 

  44. Tham L, Nazar R, Pop I (2014) Mixed convection flow from a horizontal circular cylinder embedded in a porous medium filled by a nanofluid: Buongiorno–Darcy model. Int J Therm Sci 84:21–33

    Article  Google Scholar 

  45. Sheremet MA, Pop I (2014) Conjugate natural convection in a square porous cavity filled by a nanofluid using Buongiorno’s mathematical model. Int J Heat Mass Transf 79:137–145

    Article  Google Scholar 

  46. Sheremet MA, Pop I (2015) Free convection in a triangular cavity filled with a porous medium saturated by a nanofluid. Int J Numer Methods Heat Fluid Flow 25:1138–1161

    Article  MATH  Google Scholar 

  47. Noghrehabadi A, Behseresht A, Ghalambaz M, Behseresht J (2013) Natural-convection flow of nanofluids over vertical cone embedded in non-Darcy porous media. J Thermophys Heat Transf 27:334–341

    Article  Google Scholar 

  48. Rashidi MM, Momoniat E, Ferdows M, Basiriparsa A (2014) Lie group solution for free convective flow of a nanofluid past a chemically reacting horizontal plate in a porous media. Math Probl Eng 2014:1–21

    Article  MathSciNet  Google Scholar 

  49. Todd L (1997) A family of laminar boundary layers along a semi-infinite flat plate. Fluid Dyn Res 19:235–249

    Article  MATH  MathSciNet  Google Scholar 

  50. Fang T (2008) A note on the unsteady boundary layers over a flat plate. Int J Non Linear Mech 43:1007–1011

    Article  Google Scholar 

  51. Rosca NC, Pop I (2014) Unsteady boundary layer flow of a nanofluid past a moving surface in an external uniform free stream using Buongiorno’s model. Comput Fluids 95:49–55

    Article  MathSciNet  Google Scholar 

  52. Kuznetsov AV (2010) The onset of nanofluid bioconvection in a suspension containing both nanoparticles and gyrotactic microorganisms. Int Commun Heat Mass Transf 37:1421–1425

    Article  Google Scholar 

  53. Stewartson K (2008) On the motion of a flat plate at high speed in a viscous compressible fluid. I. Impulsive motion. Math Proc Camb Philos Soc 51:202

    Article  MATH  MathSciNet  Google Scholar 

  54. Aziz A, Khan WA, Pop I (2012) Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms. Int J Therm Sci 56:48–57

    Article  Google Scholar 

  55. Anoop KB, Sundararajan T, Das SK (2009) Effect of particle size on the convective heat transfer in nanofluid in the developing region. Int J Heat Mass Transf 52:2189–2195

    Article  MATH  Google Scholar 

  56. Javed T, Mehmood Z, Abbas Z (2017) Natural convection in square cavity filled with ferrofluid saturated porous medium in the presence of uniform magnetic field. Phys B Condens Matter 506:122–132

    Article  Google Scholar 

  57. Javed T, Mehmood Z, Siddiqui MA, Pop I (2017) Effects of uniform magnetic field on the natural convection of Cu–water nanofluid in a triangular cavity. Int J Numer Methods Heat Fluid Flow 27:334–357

    Article  Google Scholar 

  58. Javed T, Siddiqui MA, Mehmood Z, Pop I (2015) MHD natural convective flow in an isosceles triangular cavity filled with porous medium due to uniform/non-uniform heated side walls. Zeitschrift fur Naturforsch Sect A J Phys Sci 70:919–928

    Google Scholar 

Download references

Acknowledgements

The authors acknowledge financial support from Universiti Sains Malaysia, RU Grant 1001/PMATHS/8011013.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University Sains Malaysia, 11800, Penang, Malaysia

    Md. Faisal Md. Basir & A. I. Md. Ismail

  2. American International University-Bangladesh, Banani, Dhaka, 1213, Bangladesh

    M. J. Uddin

  3. Aeronautical/Mechanical Engineering, University of Salford, Manchester, M54WT, UK

    O. Anwar Bég

Authors
  1. Md. Faisal Md. Basir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. J. Uddin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. Anwar Bég
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. I. Md. Ismail
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Md. Faisal Md. Basir.

Additional information

Technical Editor: Jader Barbosa Jr.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Basir, M.F.M., Uddin, M.J., Bég, O.A. et al. Influence of Stefan blowing on nanofluid flow submerged in microorganisms with leading edge accretion or ablation. J Braz. Soc. Mech. Sci. Eng. 39, 4519–4532 (2017). https://doi.org/10.1007/s40430-017-0877-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40430-017-0877-7

Keywords

Search

Navigation