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Three-dimensional free bio-convection of nanofluid near stagnation point on general curved isothermal surface

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Abstract

In this paper, the three-dimensional nanofluid bio-convection near a stagnation attachment is studied. With a set of similarity variables, the governing equations embodying the conservation of total mass, momentum, thermal energy, nanoparticles and microorganisms are reduced to a set of fully coupled nonlinear differential equations. The homotopy analysis method (HAM)-finite difference method (FDM) technique is used to obtain exact solutions. The effect of various physical parameters on distribution of the motile microorganisms and the important physical quantities of practical interests are presented and discussed.

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Abbreviations

a :

semi-major radii of curvature tangent to stagnation point surface

b :

semi-minor radii of curvature tangent to stagnation point surface

c :

ratio of two principal radii of curvature, c = b/a

C :

nanoparticle volume fraction

C w :

nanoparticle volume fraction at body surface

C∞:

ambient nanofluid volume fraction

C fx :

C fy, local skin friction coefficients along x-and y-directions, respectively

C fx :

C fy, local skin friction coefficients along xand y-directions, respectively

D B :

Brownian diffusion coefficient

D T :

thermophoretic diffusion coefficient

D m :

diffusivity of microorganisms

g :

gravitational acceleration

g :

vector of gravitational acceleration

Gr :

Grashof number

j :

vector of flux of microorganisms

Le :

Lewis number

N :

number density of motile microorganisms

N w :

density of microorganisms at body surface

N∞:

ambient density of microorganisms

N B :

Brownian motion parameter

N r :

bouncy-ratio parameter

N T :

thermophoresis parameter

Nu x :

local Nusselt number

p :

pressure

Pr :

Prandtl number

Pe :

bioconvection Péclet number

qwC :

wall mass flux

qwN :

wall motile microorganism flux

qwT :

wall heat flux

Qmx,Qmy :

local wall flux of nanoparticles along x- and y-directions, respectively

Qnx,Qny :

local wall flux of motile microorganisms along x- and y-directions, respectively

Ra b :

bioconvection Rayleigh number

Re x ,Re y :

local Reynolds numbers along x- and y-directions, respectively

Sc :

Schmidt number

T :

temperature

T w :

temperature at body surface

T∞:

ambient temperature

u, v, w :

velocity components along x-, y- and z-axes, respectively

v :

velocity vector

ṽ :

average swimming velocity vector of gyrotactic microorganisms

W c :

maximum cell swimming speed

x, y, z :

Cartesian coordinates

ṽ x , ṽ y , ṽ z :

average swimming velocities of gyrotactic microorganisms

α :

thermal diffusivity of nanofluid

β :

volumetric volume expansion coefficient

ν :

kinematic viscosity, ν = µ/ρf

μ:

dynamic viscosity

ρf:

nanofluid density

ρf∞:

ambient base fluid density

ρp:

nanoparticle density

Δρ :

density difference between cell and base fluid, Δρ = ρcell - ρf

τ:

ratio of effective heat capacity of nanoparticle to that of fluid

τwx, τwy :

local wall skin frictions along x- and y-directions, respectively

γ :

average volume of microorganism

ε :

maximum relative error

η :

similarity variable

f(η):

reduced stream function in x-direction

s(η):

reduced stream function in y-direction

θ(η):

reduced temperature

ϕ(η):

reduced nanoparticle volume fraction

ξ(η):

reduced density of motile microorganisms

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Authors and Affiliations

  1. State Key Laboratory of Ocean Engineering, Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Qingkai Zhao, Hang Xu & A. Raees

  2. School of Marine Science and Technology, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

    Longbin Tao

  3. Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, 119260, Singapore

    Qiang Sun

Authors
  1. Qingkai Zhao
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  2. Hang Xu
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  3. Longbin Tao
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  4. A. Raees
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  5. Qiang Sun
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Corresponding author

Correspondence to Qingkai Zhao.

Additional information

Project supported by the Program for New Century Excellent Talents in University of China (No.NCET-12-0347)

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Zhao, Q., Xu, H., Tao, L. et al. Three-dimensional free bio-convection of nanofluid near stagnation point on general curved isothermal surface. Appl. Math. Mech.-Engl. Ed. 37, 417–432 (2016). https://doi.org/10.1007/s10483-016-2046-9

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  • DOI: https://doi.org/10.1007/s10483-016-2046-9

Keywords

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2010 Mathematics Subject Classification

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