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Free Convection of Non-Newtonian Nanofluids in Porous media with Gyrotactic Microorganisms

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Abstract

The free convection of non-Newtonian nanofluids along a vertical plate in porous medium is investigated numerically. It is assumed that the medium contains gyrotactic microorganisms along with nanoparticles and the plate is subjected to prescribed temperature, concentration of nanoparticles and density of motile microorganisms. It is further assumed that the plate is impermeable. The governing partial differential equations are reduced to nonlinear ordinary differential equations using similarity transformations. The nonlinear ordinary differential equations are then solved by a finite difference numerical method. The effects of controlling parameters on several dimensionless quantities and numbers of our interest are investigated. The numerical results are compared with the published data and an excellent agreement has been found. It is found that nanofluid and bioconvection parameters have strong effects on local Nusselt, Sherwood and density numbers.

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Abbreviations

\(b\) :

Chemotaxis constant

\(\bar{{C}}\) :

Nanoparticle volume fraction (\(\text{ kg}\, \text{ m}^{-3})\)

\(\bar{{C}}_\mathrm{w}\) :

Wall nanoparticle volume fraction (\(\text{ kg}\, \text{ m}^{-3})\)

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

Ambient nanoparticle volume fraction (\(\text{ kg}\, \text{ m}^{-3})\)

\(D_\mathrm{B}\) :

Brownian diffusion coefficient (\(\text{ m}^{2}\,\text{ s}^{-1})\)

\(D_n \) :

Diffusivity of microorganisms (\(\text{ m}^{2}\,\text{ s}^{-1})\)

\(D_\mathrm{T} \) :

Thermophoretic diffusion coefficient (\(\text{ m}^{2}\,\text{ s}^{-1})\)

\(Ec\) :

Eckert number

\(f(\eta )\) :

Dimensionless stream function

\(g\) :

Acceleration due to gravity (\(\text{ m}\,{\text{ s}}^{-2})\)

\(k\) :

Thermal conductivity (\(\text{ m}^{2}\,\text{ s}^{-1})\)

\(K\) :

Consistency coefficient (\(\text{ kg}\,{\text{ m}}^{-1}\text{ s}^{-1})\)

\(K_0\) :

Permeability of the porous medium (\(\text{ m}^{2})\)

\(L\) :

Characteristic length (\(\text{ m}^{2})\)

\(Lb\) :

Bioconvection Lewis number

\(Le\) :

Lewis number

\(m\) :

Power law index

\(\bar{{n}}_n \) :

Volume fraction of motile microorganisms (\(\text{ kg}\, \text{ m}^{-3})\)

\(Nb\) :

Brownian motion parameter

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

Local density number of the motile microorganisms

\(Nt\) :

Thermophoresis parameter

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

Local Nusselt number

\(Pe\) :

Bioconvection Péclet number

\(Pr \) :

Prandtl number

\(\bar{p}\) :

Pressure (Pa)

\(\bar{q}_\mathrm{m} \) :

Wall mass flux (\(\text{ kg}\,\text{ m}^{-2}\, \text{ s}^{-1})\)

\(\bar{{q}}_n \) :

Wall motile microorganisms flux (\(\text{ kg}\,{\text{ m}}^{-2} \,\text{ s}^{-1})\)

\(\bar{q}_\mathrm{w} \) :

Wall heat flux (\(\text{ J}\,{\text{ m}}^{-2}\, \text{ s}^{-1})\)

\(Ra\) :

Rayleigh number for the porous medium

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

Local Rayleigh number for the porous medium

\(Rb\) :

Bioconvection Rayliegh number

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

Local Sherwood number

\(\bar{{T}}\) :

Nanofluid temperature (K)

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

Wall temperature (K)

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

Ambient temperature (K)

\(\bar{u},\;\bar{v}\) :

Velocity components along \(\bar{x}\)- and \(y\)-axes (\(\text{ m}\,{\text{ s}}^{-1})\)

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

Average directional swimming velocity of microorganisms along \(\bar{x}\)- and \(y\)-axes (m s\(^{-1})\)

\(W_\mathrm{c}\) :

Constant maximum cell swimming speed (\(\text{ m}\, \text{ s}^{-1})\)

\(\bar{x}, \bar{y}\) :

Cartesian coordinates (\(\bar{x}\)-axis is aligned along the stretching surface and \(y\)-axis is normal to it) (m)

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

Effective thermal diffusivity of the porous medium (\(\text{ m}^{2}\,\text{ s}^{-1})\)

\(\phi (\eta )\) :

Rescaled nanoparticle volume fraction

\(\eta \) :

similarity variable

\(\gamma \) :

Average volume of a microorganism (\(\text{ kg}\,\text{ m}^{-3})\)

\(\theta (\eta )\) :

Dimensionless temperature

\(\nu \) :

Kinematic viscosity of the fluid (\(\text{ m}^{2}\, \text{ s}^{-1})\)

\(\rho _\mathrm{f} \) :

Fluid density (\(\text{ kg}\,\text{ m}^{-3})\)

\(\rho _\mathrm{p}\) :

Nanoparticle mass density (\(\text{ kg}\, \text{ m}^{-3}\))

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

Heat capacity of the fluid (\(\text{ J}\, \text{ kg}^{-3}\,\text{ K}^{-1}\))

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

Heat capacity of the nanoparticle material (\(\text{ J}\,\text{ kg}^{-3}\,\text{ K}^{-1})\)

\(\tau \) :

Ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid

\(\sigma (\eta )\) :

Rescaled density of motile microorganisms

\(\psi \) :

Stream function

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Authors and Affiliations

  1. Department of Engineering Sciences, PN Enginnering College, National University of Sciences and Technology, Karachi, 75350, Pakistan

    W. A. Khan

  2. School of Mathematical sciences, University Sains Malaysia, Penang, 11800, Malaysia

    Md. Jashim Uddin & A. I. Md. Ismail

  3. Department of Mathematics, American International University-Bangladesh, Banani-Dhaka, 1213, Bangladesh

    Md. Jashim Uddin

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  1. W. A. Khan
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Khan, W.A., Uddin, M.J. & Ismail, A.I.M. Free Convection of Non-Newtonian Nanofluids in Porous media with Gyrotactic Microorganisms. Transp Porous Med 97, 241–252 (2013). https://doi.org/10.1007/s11242-012-0120-z

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