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Laminar MHD natural convection of nanofluid containing gyrotactic microorganisms over vertical wavy surface saturated non-Darcian porous media

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Abstract

Magnetohydrodynamic (MHD) bioconvection of an incompressible electrically conducting nanofluid near a vertical wavy surface saturated porous medium containing both nanoparticle and gyrotactic microorganisms is investigated. The nanofluid is represented by a model that includes both Brownian motion and thermophoresis effects. A suitable set of non-dimensional variables are used to transform the governing boundary layer equations into a dimensionless form. The resulting nonlinear system is mapped to the vertical flat plate domain, and a non-similar solution is used to the obtained equations. The obtained non-similar system is then solved numerically using the fourth-order Runge-Kutta method. The influence of various physical parameters on the local Nusselt number, the local Sherwood number, the local density number of the motile microorganisms, the dimensionless velocity, the dimensionless temperature, and the rescaled density of motile microorganisms is studied. It is found that the local Nusselt number, the local Sherwood number, and the local density number of the motile microorganisms decrease by increasing either the Grashof number or the magnetic field parameter.

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Abbreviations

ã:

amplitude of wavy surface

A :

microorganisms concentration difference parameter

C :

nanoparticle volume fraction

D :

Brownian diffusion coefficient

D t :

thermophoretic diffusion coefficient

D n :

diffusivity of microorganisms

f :

dimensionless stream function

g :

acceleration due to gravity

Gr:

Grashofnumber

k :

thermal conductivity

K :

permeability of porous medium

\(\bar K\) :

material parameter

L :

characteristic length of wavy surface

Le b :

bioconvectionLewisnumber

Le :

Lewis number

M n :

magnetic field parameter

N :

rescaled density of motile microorganisms

N b :

Brownian motion parameter

N r :

buoyancy ratio

N t :

thermophoresis parameter

N nx :

local density number of motile microorganisms

N ux :

local Nusselt number

P :

pressure

Pe b :

bioconvection Péclet number

qm :

surface mass flux

qn:

surface motile microorganisms flux

qw:

surface heat flux

Ra :

Rayleigh number

Ra b :

bioconvection Rayleigh number

Sh x :

local Sherwood number

T :

temperature

W :

constant maximum cell swimming speed

(u, v):

velocity components of fluid

(x, y):

coordinate axes

α :

thermal diffusivity of porous media

φ :

dimensionless nanoparticle volume fraction

β :

volumetric expansion coefficient

θ :

dimensionless temperature

(ξ, η):

non-similarity variables

(ρc)p:

effective heat capacity of nanoparticle material

(ρc)f:

heat capacity of fluid

ρf:

density of fluid

ρp:

nanoparticle mass density

j :

ratio between effective heat capacity of nanoparticle material and heat capacity of fluid

μ :

dynamic viscosity

γ :

average volume of microorganisms

σ :

wavy surface

ψ :

stream function

w :

condition at surface

f :

fluid

∞:

condition in free stream

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, South Valley University, Qena, 83523, Egypt

    S. E. Ahmed & A. Mahdy

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  1. S. E. Ahmed
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  2. A. Mahdy
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Correspondence to S. E. Ahmed.

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Ahmed, S.E., Mahdy, A. Laminar MHD natural convection of nanofluid containing gyrotactic microorganisms over vertical wavy surface saturated non-Darcian porous media. Appl. Math. Mech.-Engl. Ed. 37, 471–484 (2016). https://doi.org/10.1007/s10483-016-2044-9

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  • DOI: https://doi.org/10.1007/s10483-016-2044-9

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

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