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Non-uniform magnetic field effect on nanofluid hydrothermal treatment considering Brownian motion and thermophoresis effects

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Abstract

In this paper, heat and mass transfer of nanofluid in presence of variable magnetic field is investigated. The effects of Brownian motion and thermophoresis are taken into account. Control Volume-based Finite Element Method is applied to solve the governing equations in which both effect of ferrohydrodynamic and magnetohydrodynamic are considered. The effects of Rayleigh number, Hartmann number arising from MHD, buoyancy ratio number and Lewis number on the flow and heat transfer characteristics have been examined. Results are presented in the form of streamline, isotherm, isoconcentration and heatline plots. Results show that Nusselt number has direct relationship with Rayleigh number, buoyancy ratio number and Lewis number while it has reverse relationship with Hartmann number.

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Abbreviations

A :

Amplitude

B :

Magnetic induction \(\left( { = \mu_{0} H} \right)\)

C p :

Specific heat at constant pressure

D B :

Brownian diffusion coefficient

D T :

Thermophoretic diffusion coefficient

Ec :

Eckert number \(\left( { = \left( {\alpha \mu } \right)/\left[ {\left( {\rho C_{P} } \right)\Delta TL^{2} \,} \right]} \right)\)

Gr f :

Grashof number

\(\vec{g}\) :

Gravitational acceleration vector

H x , H y :

Components of the magnetic field intensity

H :

The magnetic field strength

Ha :

Hartmann number \(\left( { = LH_{0} \mu_{0} \sqrt {\sigma /\mu } } \right)\)

k :

Thermal conductivity

\(K^{\prime}\) :

Constant parameter

L :

Gap between inner and outer boundary of the enclosure L = r out−r in

Le :

Lewis number \(( = \alpha /D_{B} )\)

\(Mn_{F}\) :

Magnetic number arising from FHD \(\left( { = \mu_{0} H_{0}^{2} K^{\prime}\Delta T\,L^{2} /\left( {\mu \alpha } \right)} \right)\)

M :

Magnetization \(\left( { = K^{\prime}\overline{H} \left( {T_{c}^{\prime } - T} \right)} \right)\)

Nu :

Nusselt number

N :

Number of undulations

Nb :

Brownian motion parameter \(( = (\rho c)_{p} D_{B} (\phi_{h} - \phi_{c} )/(\rho c)_{f} \alpha )\)

Nt :

Thermophoretic parameter \(( = (\rho c)_{p} D_{T} (\Delta T)/[(\rho c)_{f} \alpha T_{c} ])\)

Nr :

Buoyancy ratio number \(\left( { = \left( {\rho_{p} - \rho_{0} } \right)\left( {\phi_{h} - \phi_{c} } \right)/[\left( {1 - \phi_{c} } \right)\rho_{{f_{0} }} \beta L\left( {\Delta T} \right)]} \right)\)

Pr :

Prandtl number\(( = \mu /\rho_{f} \alpha )\)

Ra :

Thermal Rayleigh number \(( = \left( {1 - \phi_{c} } \right)\rho_{{f_{0} }} g\beta L^{3} \left( {\Delta T} \right)/\left( {\mu \alpha } \right))\)

T :

Fluid temperature

\(T_{c}^{\prime }\) :

Curie temperature

u, v :

Velocity components in the x-direction and y-direction

U, V :

Dimensionless velocity components in the X-direction and Y-direction

x, y :

Space coordinates

X, Y :

Dimensionless space coordinates

\(\zeta\) :

Angle measured from the lower right plane

α :

Thermal diffusivity

\(\phi\) :

Volume fraction

γ :

Magnetic field strength at the source

\(\varepsilon_{1}\) :

Temperature number \(\left( { = T_{1} /\Delta T} \right)\)

\(\varepsilon_{2}\) :

Curie temperature number \(\left( { = T_{c}^{\prime } /\Delta T} \right)\)

σ :

Electrical conductivity

μ :

Dynamic viscosity

μ 0 :

Magnetic permeability of vacuum \(\left( { = 4\pi \times 10^{ - 7} {\text{Tm/A}}} \right)\)

\(\upsilon\) :

Kinematic viscosity

\(\varTheta\) :

Dimensionless temperature

ρ :

Fluid density

β :

Thermal expansion coefficient

\(\omega ,\varOmega\) :

Vorticity and dimensionless vorticity

c :

Cold

h :

Hot

ave:

Average

loc:

Local

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Authors and Affiliations

  1. Department of Mechanical Engineering, Babol University of Technology, Babol, Islamic Republic of Iran

    M. Sheikholeslami

  2. Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, 4800 Cao An Rd., Jiading, Shanghai, 201804, China

    M. M. Rashidi

  3. ENN-Tongji Clean Energy Institute of Advanced Studies, Shanghai, China

    M. M. Rashidi

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Correspondence to M. Sheikholeslami.

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Technical Editor: Francisco Ricardo Cunha.

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Sheikholeslami, M., Rashidi, M.M. Non-uniform magnetic field effect on nanofluid hydrothermal treatment considering Brownian motion and thermophoresis effects. J Braz. Soc. Mech. Sci. Eng. 38, 1171–1184 (2016). https://doi.org/10.1007/s40430-015-0459-5

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