Advertisement

Neural Computing and Applications

, Volume 30, Issue 4, pp 1237–1249 | Cite as

A mathematical model of MHD nanofluid flow having gyrotactic microorganisms with thermal radiation and chemical reaction effects

  • M. M. Bhatti
  • S. R. Mishra
  • T. Abbas
  • M. M. Rashidi
Original Article

Abstract

In this article, we have examined three-dimensional unsteady MHD boundary layer flow of viscous nanofluid having gyrotactic microorganisms through a stretching porous cylinder. Simultaneous effects of nonlinear thermal radiation and chemical reaction are taken into account. Moreover, the effects of velocity slip and thermal slip are also considered. The governing flow problem is modelled by means of similarity transformation variables with their relevant boundary conditions. The obtained reduced highly nonlinear coupled ordinary differential equations are solved numerically by means of nonlinear shooting technique. The effects of all the governing parameters are discussed for velocity profile, temperature profile, nanoparticle concentration profile and motile microorganisms’ density function presented with the help of tables and graphs. The numerical comparison is also presented with the existing published results as a special case of our study. It is found that velocity of the fluid diminishes for large values of magnetic parameter and porosity parameter. Radiation effects show an increment in the temperature profile, whereas thermal slip parameter shows converse effect. Furthermore, it is also observed that chemical reaction parameter significantly enhances the nanoparticle concentration profile. The present study is also applicable in bio-nano-polymer process and in different industrial process.

Keywords

Magnetic field Chemical reaction Nanofluid Numerical solution Gyrotactic microorganisms 

List of symbols

\(\bar{u},\bar{v},\bar{w}\)

Velocity components

\(\bar{r},\bar{z}\)

Cylindrical coordinate

Re

Reynolds number

\(\tilde{t}\)

Time

\(\bar{C}\)

Nanoparticle volume fraction

\(\bar{P}\)

Pressure

Fw

Suction/injection parameter

\(\bar{T}_{\infty }\)

Atmosphere temperature

\(\bar{C}_{\infty }\)

Atmosphere concentration

\(\bar{T}_{w}\)

Surface temperature

Nt

Thermophoresis parameter

Nb

Brownian motion parameter

N1

Velocity slip parameter

\(\bar{n}_{w}\)

Surface density of motile-organism

\(\bar{k}_{\text{p}}\)

Permeability of porous medium

\(\bar{t}\)

Time

J

Heat flux of microorganisms

a0 (>0)

Constant

cF

Forchheimer coefficient

Q0

Heat source/sink

\(\bar{D}_{1}\)

Thermal slip factor

\(D_{{\bar{T}}}\)

Thermophoretic diffusion coefficient

\(\bar{n}\)

Density of motile microorganisms

\(\bar{b}\)

Chemotaxis constant

Wc

Maximum cell swimming speed

S

Unsteady parameter

DB

Brownian diffusion coefficient

\(D_{{\bar{n}}}\)

A micro-organism diffusivity

Pr

Prandtl number

Sc

Schmidt number

Sb

Bio-convection Schmidt number

Pe

Peclet number

kf

Forchheimer number

M

Magnetic parameter

Rd

Thermal radiation parameter

HS

Heat source/sink parameter

qw

Surface heat flux

qM

Surface mass flux

qN

Motile surface microorganism flux

k

Mean absorption coefficient

\(C_{{F\bar{x}}}\)

Skin friction coefficient

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

Nusselt number

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

Sherwood number

\(N_{{n\bar{x}}}\)

Density number of motile microorganisms

Greek symbols

\(\bar{\beta }\)

Contraction expansion strength

\(\bar{\alpha }_{m}\)

Thermal conductivity

(ρc)f

Heat capacity of the fluid

(ρc)p

Heat capacity of nanoparticle

σ

Electrical conductivity

μ

Viscosity of nanofluid

θ

Temperature profile

ϕ

Nanoparticle concentration profile

Φ

Motile microorganism density profile

\(\bar{\sigma }\)

Stefan–Boltzmann constant

β

Velocity slip

τw

Shear stress

ρp

Density of nanoparticles

σ

Electrical conductivity

βT

Thermal slip

ρ

Density

ν

Kinematic viscosity

γ

Chemical reaction parameter

Notes

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.

References

  1. 1.
    Thomas S, Yang W (eds) (2009) Advances in polymer processing: from macro-to nano-scales. Woodhead Publishing, OxfordGoogle Scholar
  2. 2.
    Bachok N, Ishak A (2010) Flow and heat transfer over a stretching cylinder with prescribed surface heat flux. Malays J Math Sci 4:159–169zbMATHGoogle Scholar
  3. 3.
    Stasiak J, Squires AM, Castelletto V, Hamley IW, Moggridge GD (2009) Effect of stretching on the structure of cylinder and sphere-forming styrene–isoprene–styrene block copolymers. Macromolecules 42:5256–5265CrossRefGoogle Scholar
  4. 4.
    Sakiadis BC (1961) Boundary layer behaviour on continuous solid surfaces: I. Boundary layer equations for two-dimensional and axisymmetric flow. AIChE J 7:26–28CrossRefGoogle Scholar
  5. 5.
    Crane LJ (1970) Flow past a stretching plate. Z Angew Math Phys 21(4):645–647. doi: 10.1007/BF01587695 CrossRefGoogle Scholar
  6. 6.
    Datta BK, Roy P, Gupta AS (1985) Temperature field over a stretching sheet with uniform heat flux. Int J Heat Mass Trans 12:89–94CrossRefGoogle Scholar
  7. 7.
    Chen CK, Char MI (1988) Heat transfer of a continuous stretching surface with suction and blowing. J Math Anal Appl 135:568–580MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bhatti MM, Shahid A, Rashidi MM (2016) Numerical simulation of fluid flow over a shrinking porous sheet by successive linearization method. Alex Eng J 55:51–56CrossRefGoogle Scholar
  9. 9.
    Bhatti MM, Abbas T, Rashidi MM (2016) A New numerical simulation of MHD stagnation-point flow over a permeable stretching/shrinking sheet in porous media with heat transfer. Iran J Sci Technol Trans A Sci. doi: 10.1007/s40995-016-0027-6 Google Scholar
  10. 10.
    Saidur R, Leong KY, Mohammad HA (2011) A review on applications and challenges of nanofluids. Renew sustain Energy Rev 15:1646–1668CrossRefGoogle Scholar
  11. 11.
    Wong KV, De Leon O (2010) Applications of nanofluids: current and future. Adv Mech Eng 2:519659CrossRefGoogle Scholar
  12. 12.
    Yu W, Xie H (2012) A review on nanofluids: preparation, stability mechanisms, and applications. J Nanomater 2012:1Google Scholar
  13. 13.
    Raees A, Xu H, Liao SJ (2015) Unsteady mixed nano-bioconvection flow in a horizontal channel with its upper plate expanding or contracting. Int J Heat Mass Transf 86:174–182CrossRefGoogle Scholar
  14. 14.
    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 52:2189–2195CrossRefzbMATHGoogle Scholar
  15. 15.
    Pedley TJ (2010) Instability of uniform micro-organism suspensions revisited. J Fluid Mech 647:335–359MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Xu H, Pop I (2014) Mixed convection flow of a nanofluid over a stretching surface with uniform free stream in the presence of both nanoparticles and gyrotactic microorganisms. Int J Heat Mass Transf 75:610–623CrossRefGoogle Scholar
  17. 17.
    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 56:48–57CrossRefGoogle Scholar
  18. 18.
    Tham L, Nazar R, Pop I (2013) Mixed convection flow over a solid sphere embedded in a porous medium filled by a nanofluid containing gyrotactic microorganisms. Int J Heat Mass Transf 62:647–660CrossRefGoogle Scholar
  19. 19.
    Saranya S, Radha KV (2014) Review of nanobiopolymers for controlled drug delivery. Polym Plast Technol Eng 53:1636–1646CrossRefGoogle Scholar
  20. 20.
    Oh JK, Lee DI, Park JM (2009) Biopolymer-based microgels/nanogels for drug delivery applications. Prog Polym Sci 34:1261–1282CrossRefGoogle Scholar
  21. 21.
    Abo-Eldahab EM, Abd El-Aziz M (2000) Radiation effect on heat transfer in electrically conducting fluid at a stretching surface with uniform free stream. J Phys D Appl Phys 33:3180–3185CrossRefGoogle Scholar
  22. 22.
    Cortell R (2008) Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Phys Lett A 372:631–636CrossRefzbMATHGoogle Scholar
  23. 23.
    Cortell R (2006) Effects of viscous dissipation and work done by deformation on MHD flow and heat transfer of a viscoelastic fluid over a stretching sheet. Phys Lett A 357:298–305CrossRefzbMATHGoogle Scholar
  24. 24.
    Bhatti MM, Abbas T, Rashidi MM, Ali MES (2016) Numerical simulation of entropy generation with thermal radiation on MHD carreau nanofluid towards a shrinking sheet. Entropy 18(6):200MathSciNetCrossRefGoogle Scholar
  25. 25.
    Bhatti MM, Rashidi MM (2016) Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet. J Mol Liq 221:567–573CrossRefGoogle Scholar
  26. 26.
    Bhatti MM, Rashidi MM (2016) Entropy generation with nonlinear thermal radiation in MHD boundary layer flow over a permeable shrinking/stretching sheet: numerical solution. J Nanofluids 5:543–548CrossRefGoogle Scholar
  27. 27.
    Anjalidevi SP, Kandasamy R (1999) Effects of chemical reaction, heat and mass transfer on laminar flow along a semi infinite horizontal plate. Heat Mass Transf 35:465–467CrossRefGoogle Scholar
  28. 28.
    Hayat T, Muhammad T, Shehzad AS, Alsaedi A (2015) Similarity solution to threedimensional boundary layer flow of second grade nanofluid past a stretching surface with thermal radiation and heat source/sink. AIP Adv 5:017107CrossRefGoogle Scholar
  29. 29.
    Abel MS, Siddheshwar PG, Mahesha N (2009) Effects of thermal buoyancy and variable thermal conductivity on the MHD flow and heat transfer in a power-law fluid past a vertical stretching sheet in the presence of a non-uniform heat source. Int J Non Linear Mech 44:1–12CrossRefzbMATHGoogle Scholar
  30. 30.
    Sheikholeslami M, Rashidi MM (2015) Effect of space dependent magnetic field on free convection of Fe3O4–water nanofluid. J Taiwan Inst Chem Eng. doi: 10.1016/jjtice.2015.03.035 Google Scholar
  31. 31.
    Sheikholeslami M (2014) Effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition. Eur Phys J Plus 2014:129–248Google Scholar
  32. 32.
    Sheikholeslami M, Ganji DD, Rashidi MM (2015) Ferrofluid flow and heat transfer in a semi annulus enclosure in the presence of magnetic source considering thermal radiation. J Taiwan Inst Chem Eng 47:6–17CrossRefGoogle Scholar
  33. 33.
    Ishak A, Nazar R, Pop I (2008) Magnetohydrodynamics flow and heat transfer due to a stretching cylinder. Energy Convers Manag 49:3265–3269CrossRefzbMATHGoogle Scholar
  34. 34.
    Mukhopadhyay S (2013) MHD boundary layer slip flow along a stretching cylinder. Ain Shams Eng J 4:317–324CrossRefGoogle Scholar
  35. 35.
    Ishak A (2010) Unsteady MHD flow and heat transfer over a stretching plate. J Appl Sci 10:2127–2131CrossRefGoogle Scholar
  36. 36.
    Pop I, Na TY (1998) A note on MHD flow over a stretching permeable surface. Mech Res Commun 25:263–269MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Khan K, Abel MS, Sonth RM (2003) Visco-elastic MHD flow heat and mass transfer over a porous stretching sheet with dissipation of energy and stress work. Heat Mass Trans 40:47–57CrossRefGoogle Scholar
  38. 38.
    Abed Mahdi R, Mohammed HA, Munisamy KM, Saeid NH (2015) Review of convection heat transfer and fluid flow in porous media with nanofluid. Renew Sust Energy Rev 41:715–734CrossRefGoogle Scholar
  39. 39.
    Abed Mahdi R, Mohammed HA, Munisamy KM (2013) Improvement of convection heat transfer by using porous media and nanofluid: review. Int J Sci Res 2:34–47Google Scholar
  40. 40.
    Das KS, Choi SU, Yu W, Pradep T (2007) Nanofluid: science and technology. Wiley, LondonCrossRefGoogle Scholar
  41. 41.
    James M, Mureithi EW, Kuznetsov D (2014) Natural convection flow past an impermeable vertical plate embedded in nanofluid saturated porous medium with temperature dependent viscosity. Asian J Math Appl 2014:ama0165Google Scholar
  42. 42.
    Ishak A, Nazar R, Pop I (2008) Magnetohydrodynamic (MHD) flow and heat transfer due to a stretching cylinder. Energy Convers Manag 49:3265–3269CrossRefzbMATHGoogle Scholar
  43. 43.
    Wang CY (1988) Fluid flow due to a stretching cylinder. Phys Fluids 31:466–468CrossRefGoogle Scholar
  44. 44.
    Basir MFM, Uddin MJ, Ismail AM, Bég OA (2016) Nanofluid slip flow over a stretching cylinder with Schmidt and Péclet number effects. AIP Adv 6:055316CrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  • M. M. Bhatti
    • 1
  • S. R. Mishra
    • 2
  • T. Abbas
    • 3
  • M. M. Rashidi
    • 4
  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiChina
  2. 2.Department of MathematicsSiksha ‘O’ Anusandhan UniversityKhandagiri, BhubaneswarIndia
  3. 3.Department of MathematicsQuaid-I-Azam UniversityIslamabadPakistan
  4. 4.Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management SystemsTongji UniversityShanghaiChina

Personalised recommendations