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Effect of local thermal non-equilibrium on unsteady heat transfer by natural convection of a nanofluid over a vertical wavy surface

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Abstract

This paper uses thermal non-equilibrium model to study transient heat transfer by natural convection of a nanofluid over a vertical wavy surface. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. Three-temperature model is applied to represent the local thermal non-equilibrium among the particle, fluid, and solid-matrix phases. Finite difference method is used to solve the dimensionless governing equations of the problem. The obtained results are displayed in 2D graphs to illustrate the influences of the different physical parameters on local skin-friction coefficient, local Nusselt numbers for fluid, particle and solid phases and local Sherwood number. The results for velocity component, nanoparticle volume fraction, fluid temperature, particle temperature and solid-matrix temperature are presented in 3D graphs as a function of the axial and transverse coordinates. All the obtained results are discussed.

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Abbreviations

\(\bar{a}\) :

dimensional amplitude of the wavy surface

A :

amplitude-wave length ratio, \(\bar{a}/L\)

C f :

local skin-friction coefficient, defined by Eq. (43)

Da :

Darcy number, defined by Eq. (16)

D B :

Brownian diffusion coefficient

D T :

thermophoretic diffusion coefficient

g :

gravitational acceleration

h fp :

heat transfer coefficient for fluid/particle interface

h fs :

heat transfer coefficient for fluid/solid interface

k :

thermal conductivity

K :

permeability of the porous medium

L :

wavelength of the wavy surface

Le :

Lewis number, defined by Eq. (18)

N b :

Brownian motion parameter, defined by Eq. (20)

N r :

buoyancy ratio, defined by Eq. (19)

N t :

thermophoresis parameter, defined by Eq. (21)

N hp :

Nield number for the fluid/particle interface, defined by Eq. (22)

N hs :

Nield number for the fluid/solid interface, defined by Eq. (23)

Nu x :

local Nusselt number, defined by Eqs. (37)–(39)

P :

pressure

Pr :

Prandtl number, defined by Eq. (17)

Ra :

Rayleigh number, defined by Eq. (15)

Sh :

Sherwood number, defined by Eq. (40)

t :

dimensional time

T :

dimensional temperature

V :

velocity vector

u, v :

dimensional velocity components

\(\bar{u}\), \(\bar{v}\) :

dimensionless velocity components, defined by Eq. (7)

\(\tilde{U}\) :

characteristic velocity

x, y :

dimensional Cartesian coordinates

\(\bar{x}\), \(\bar{y}\) :

dimensionless Cartesian coordinates, defined by Eq. (7)

α :

thermal diffusivity

β :

volumetric expansion coefficient for the fluid

γ p :

modified thermal capacity ratio, defined by Eq. (24)

γ s :

modified thermal capacity ratio, defined by Eq. (25)

ε :

porosity

ε p :

modified thermal diffusivity ratio, defined by Eq. (26)

ε s :

modified thermal diffusivity ratio, defined by Eq. (27)

μ :

viscosity of the fluid

ρ :

density

(ρc):

heat capacity

\(\bar{\sigma}\) :

dimensional coordinate of the wavy surface

σ :

dimensionless coordinate of the wavy surface, defined by Eq. (7)

τ :

dimensionless time parameter, defined by Eq. (7)

τ w :

wall shear stress

θ :

dimensionless temperature, defined by Eq. (7)

ϕ :

dimensional nanoparticle volume fraction

Φ :

dimensionless nanoparticle volume fraction, defined by Eq. (7)

f :

fluid phase

p :

particle phase

s :

solid-matrix phase

w :

surface condition

∞:

condition far away from the surface

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Ahmed, S.E., Abd El-Aziz, M.M. Effect of local thermal non-equilibrium on unsteady heat transfer by natural convection of a nanofluid over a vertical wavy surface. Meccanica 48, 33–43 (2013). https://doi.org/10.1007/s11012-012-9581-y

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  • DOI: https://doi.org/10.1007/s11012-012-9581-y

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