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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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
- nanofluid
- stagnation point
- bioconvection
- gyrotactic microorganisms, homotopy analysis method (HAM)-finite difference method (FDM)