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Natural convection boundary layer flow due to gyrotactic microorganisms about a vertical cone in porous media saturated by a nanofluid

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Abstract

The similarity solution for Darcy free convection about an isothermal vertical cone with fixed apex half angle, pointing downward in a nanofluid saturated porous medium, has been made. It is assumed that the medium contains gyrotactic microorganisms along with nanoparticles and the cone is subjected to concentration of nanoparticles and density of motile microorganisms. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. The governing partial differential equations are converted into nonlinear ordinary differential equations using unique similarity transformations. The effects of the governing parameters on the dimensionless quantities such as velocity, temperature, nanoparticle concentration, density of motile microorganisms, local Nusselt, local Sherwood and local density numbers for both nanoparticles, and motile microorganism density are explored. A comprehensive numerical computation is carried out for various values of the parameters that describe the flow characteristics.

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Abbreviations

C :

Nanoparticle volume fraction

D :

Brownian diffusion coefficient

\( \bar{D} \) :

Thermophoretic diffusion coefficient

\( \tilde{D} \) :

Diffusivity of microorganisms

f :

Dimensionless stream function

g :

Acceleration due to gravity

k :

Thermal conductivity

K :

Permeability of porous medium

L b :

Bioconvection Lewis number

L e :

Lewis number

N :

Concentration of microorganisms

N b :

Brownian motion parameter

N r :

Buoyancy ratio

N t :

Thermophoresis parameter

Nn :

Density number of motile microorganisms

Nu :

Nusselt number

P :

Pressure

P e :

Bioconvection Péclet number

q m :

Surface mass flux

q n :

Surface motile microorganisms flux

q w :

Surface heat flux

R a :

Rayleigh number

R b :

Bioconvection Rayleigh number

S :

Dimensionless density of motile microorganisms

Sh :

Sherwood number

T :

Temperature

\( \hat{W} \) :

Constant maximum cell swimming speed

(u, v):

Velocity components of the fluid

(x, y):

Coordinate axes

α :

Thermal diffusivity of porous media

ϕ :

Dimensionless nanoparticle volume fraction

β :

Volumetric expansion coefficient

θ :

Dimensionless temperature

η :

Similarity variable

(ρc)p :

Effective heat capacity of nanoparticle material

(ρc) f :

Heat capacity of the fluid

ρ f :

Density of the fluid

ρ p :

Nanoparticle mass density

τ :

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

μ :

Dynamic viscosity

γ:

Average volume of a microorganisms

ω :

Half angle of the cone

ϵ :

Porosity

σ :

Microorganisms concentration difference parameter

ψ:

Stream function

\( w \) :

Conditions at the surface

f :

Fluid

∞:

Conditions in the free stream

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Authors and Affiliations

  1. Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt

    A. Mahdy

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  1. A. Mahdy
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Correspondence to A. Mahdy.

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Technical Editor: Francisco Ricardo Cunha.

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Mahdy, A. Natural convection boundary layer flow due to gyrotactic microorganisms about a vertical cone in porous media saturated by a nanofluid. J Braz. Soc. Mech. Sci. Eng. 38, 67–76 (2016). https://doi.org/10.1007/s40430-015-0313-9

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  • DOI: https://doi.org/10.1007/s40430-015-0313-9

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