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Mixed convection in gravity-driven nano-liquid film containing both nanoparticles and gyrotactic microorganisms

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Abstract

Analysis of a gravity-induced film flow of a fluid containing both nanoparticles and gyrotactic microorganisms along a convectively heated vertical surface is presented. The Buongiorno model is applied. Two kinds of boundary conditions, the passive and the active boundary conditions, are considered to investigate this film flow phenomenon. Through a set of similarity variables, the ordinary differential equations that describe the conservation of the momentum, the thermal energy, the nanoparticles, and the microorganisms are derived and then solved numerically by an efficient finite difference technique. The effects of various physical parameters on the profiles of momentum, thermal energy, nanoparticles, microorganisms, local skin friction, local Nusselt number, local wall mass flux, and local wall motile microorganisms flux are investigated. It is expected that the passively controlled nanofluid model can be much more easily achieved and applied in real circumstances than the actively controlled model.

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Abbreviations

a :

constant (i.e, a = g)

b :

chemotaxis constant

C fx :

skin friction coefficient

C :

nanoparticle volume fraction

C w :

wall nanofluid volume fraction

C ∞ :

ambient nanofluid volume fraction

D B :

Brownian diffusion coefficient

D T :

thermophoretic diffusion coefficient

D n :

diffusivity of microorganism

f(η):

dimensionless stream function

g :

gravitational acceleration vector

Gr :

Grashofnumber

h f(x):

heat transfer coefficient

j :

flux of microorganism

k :

thermal conductivity

Le :

Lewis number

N :

number density of motile microorganisms

N b :

Brownian motion parameter

N r :

bouncy-ratio parameter

N t :

thermophoresis parameter

Nu x :

local Nusselt number

N w :

wall concentration of microorganisms

N ∞ :

ambient concentration of microorganisms

p :

pressure

Pe :

bioconvection Peclet number

Pr :

Prandtl number

R b :

bioconvection Rayleigh number

q w :

wall heat flux

q m :

wall mass flux

q n :

wall motile microorganisms flux

Q mx :

local wall mass flux

Q nx :

local wall motile microorganisms flux

Re x :

local Reynolds number

Sc :

Schmidth number

T :

temperature inside boundary layer

T f :

temperature of fluid on left side of plate

T w :

wall temperature

T ∞ :

ambient temperature

U(x):

free stream velocity

u, v :

velocity components along x- and y-axes, respectively

\(\tilde v\) :

average swimming velocity vector of oxytactic microorganisms

v :

velocity vector

W c :

maximum cell swimming speed

w(η):

dimensionless density of motile microorganisms

x, y :

Cartesian coordinates along surface and normal to it, respectively

α :

thermal diffusivity of nanofluid

▿C :

characteristic nanoparticle volume fraction

▿N :

characteristic motile microorganisms density difference

η :

similarity variable

θ(η):

dimensionless temperature

ϕ(η):

dimensionless nanoparticle volume fraction

γ :

reduced heat transfer parameter

σ :

average volume of a microorganism

β :

volumetric volume expansion coefficient

λ :

constant

μ :

dynamic viscosity

ν :

kinematic viscosity

ρ f :

fluid density

ρ f∞ :

base fluid density at far field

ρ p :

nanoparticle mass density

(ρc)f :

heat capacity of nanofluid

(ρc)p :

effective heat capacity of nanoparticle material

τ :

ratio of effective heat capacity of nanoparticle to that of fluid

ψ :

stream function

τ w :

skin friction/surface shear stress

▿2 :

Laplacian operator

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Author information

Authors and Affiliations

  1. State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    A. Raees & Hang Xu

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

    Qiang Sun

  3. Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, R-400084, Romania

    I. Pop

Authors
  1. A. Raees
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  2. Hang Xu
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  3. Qiang Sun
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  4. I. Pop
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Corresponding author

Correspondence to A. Raees.

Additional information

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

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Raees, A., Xu, H., Sun, Q. et al. Mixed convection in gravity-driven nano-liquid film containing both nanoparticles and gyrotactic microorganisms. Appl. Math. Mech.-Engl. Ed. 36, 163–178 (2015). https://doi.org/10.1007/s10483-015-1901-7

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  • DOI: https://doi.org/10.1007/s10483-015-1901-7

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