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Modelling two-layer nanofluid flow in a micro-channel with electro-osmotic effects by means of Buongiorno’s mode

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Abstract

A fully developed steady immiscible flow of nanofluid in a two-layer microchannel is studied in the presence of electro-kinetic effects. Buongiorno’s model is employed for describing the behavior of nanofluids. Different from the previous studies on two-layer channel flow of a nanofluid, the present paper introduces the flux conservation conditions for the nanoparticle volume fraction field, which makes this work new and unique, and it is in coincidence with practical observations. The governing equations are reduced into a group of ordinary differential equations via appropriate similarity transformations. The highly accurate analytical approximations are obtained. Important physical quantities and total entropy generation are analyzed and discussed. A comparison is made to determine the significance of electrical double layer (EDL) effects in the presence of an external electric field. It is found that the Brownian diffusion, the thermophoresis diffusion, and the viscosity have significant effect on altering the flow behaviors.

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Abbreviations

Br1, Br2 :

Brinkman numbers

B 0 :

magnetic field in z-direction

\({\overline{C}_{1}},\;{\overline{C}_{2}}\) :

nano-particle volume fractions

C 0 :

reference nano-particle volume fraction

C w :

nano-particle volume fraction on the micro-channel walls

Cf1, Cf2 :

local skin friction coefficients

(cp)f, (cp)s :

specific heat of fluid and nanoparticles

DB1, DB2 :

Brownian diffusion coefficients

Dt1, Dt2 :

thermophoretic diffusion coefficients

e :

charge of a proton

E(m):

error for homotopy analysis method (HAM) computation order m

E s :

non-dimensional external electric field parameter

Ex, Ey :

electric field in x- and y-directions respectively

F1, F2 :

electrical body forces from uniform electromagnetic field

H :

total distance between the boundaries of the channel

H1, H2 :

distances of two-layer fluid in Regions I and II

h1, h2 :

non-dimensional distances of Regions I and II

Ha1, Ha2 :

Hartman numbers

k1, k2 :

Debye-Hückel parameters

k B :

Boltzmann constant

\(k_{\text{f}_{1}},\;k_{\text{f}_{2}}\) :

thermal conductivities of the fluid

kf:

ratio of thermal conductivities of the fluid

L :

length of the micro-channel

M D :

dimensionless mass diffusion parameter

n 0 :

the bulk ionic concentration

NB1,NB2 :

Brownian motion parameters

Nt1,Nt2 :

thermophoresis parameters

Nu1,Nu2 :

local Nusselt numbers

̄p :

pressure

P1, P2 :

non-dimensional pressure gradient parameters

qw1, qw2 :

wall heat fluxes on the two wall of the channel

R D :

universal gas constant

Re1,Re2 :

Reynolds numbers

s1, s2 :

non-dimensional nano-particle volume fractions

Se1, Se2 :

strengths of lateral direction electric field

\(S_G^1, \; S_G^2\) :

entropy generated in the respective channels

S total :

total entropy generated in the channel

S r :

ratio of entropy generated in Region I and Region II

̄T1, ̄T2 :

temperatures

T 0 :

reference temperature

T w :

temperature on the micro-channel wall surface

T̂ :

absolute temperature

u1, u2 :

non-dimensional velocities of the fluid

Ua1,Ua2 :

average velocities of the fluid

̄u1, ̄u2 :

x-component of the fluid velocities

W :

width of the micro-channel

̄x, ̄y, ̄z :

Cartesian coordinates

ẑ :

the valences of ions

α1, β2 :

thermal diffusivities of the nanofluid

ε :

ratio of dielectric constants of the medium

ε1, ε2 :

dielectric constants of the medium

ε 0 :

permittivity of vacuum

η :

non-dimensional spatial variable

Г1, Г2 :

non-dimensional pressure gradient parameters

k1, k2 :

electro-osmotic parameters

Ʌ 0 :

dimensionless reference nanoparticle parameter

λ N :

ratio of the respective quantity N such that N ∈ ε, kf, μ, σ, DB,Dt, α, τ, ρ

μ1, μ2 :

dynamic viscosities of the fluid

θ 0 :

dimensionless reference temperature parameter

θ1, θ2 :

non-dimensional temperature distributions

(ρ1)f, (ρ2)s:

densities of the fluids or nanoparticles

ρ ̄e1, ρ ̄e2 :

charge densities

τw1, τw2 :

shear stresses on the wall of the micro-channel

ϕ̄1, ϕ̄2 :

electrostatic potentials

ϕ1, ϕ2 :

non-dimensional electrostatic potentials

ζ1, ζ2 :

ratios of Joule heating to the applied temperature differences between wall and ambient fluid

ζ1, ζ2 :

non-dimensional zeta potentials

ζ̄1, ζ̄2 :

zeta potentials

1, 2:

refer to quantities for Regions I and II

f, s:

refer to the fluid and solid particles

w:

physical quantities on the micro-channel wall

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Authors and Affiliations

  1. State Key Lab 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

    M. D. K. Niazi & Hang Xu

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  1. M. D. K. Niazi
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  2. Hang Xu
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Correspondence to Hang Xu.

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Project supported by the National Natural Science Foundation of China (No. 11872241)

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Niazi, M.D.K., Xu, H. Modelling two-layer nanofluid flow in a micro-channel with electro-osmotic effects by means of Buongiorno’s mode. Appl. Math. Mech.-Engl. Ed. 41, 83–104 (2020). https://doi.org/10.1007/s10483-020-2558-7

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  • DOI: https://doi.org/10.1007/s10483-020-2558-7

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