Arabian Journal for Science and Engineering

, Volume 44, Issue 2, pp 739–752 | Cite as

Analytical Solution for Fully Developed Flows of Nanofluids in Mixed-Convection Zone Within Vertical Channels

  • Fahad G. Al-Amri
Research Article - Mechanical Engineering


In this paper, a closed-form analytical solution is presented for a fully developed mixed-convection laminar flow of nanofluids between two vertical parallel plates. The Buongiorno model, which considers the Brownian motion and thermophoresis force, is employed to investigate the hydrodynamic and heat transfer behavior of the nanofluid flow. The equations for the conservation of mass, momentum, energy, and the nanoparticle concentration field have been analytically solved, and expressions for the velocity, temperature, and nanoparticle concentration profiles as well as for the Nusselt number are given. The results show that in addition to the mixed-convection buoyancy parameter (Gr/Re), the immersed-particle buoyancy parameter additionally enriches the momentum and enhances the heat transfer inside the channel. Moreover, in the mixed-convection regime, in contrast to the case of forced convection, the heat transfer rate decreases sharply and then gradually as the solid/fluid thermal conductivity ratio increases. The present results contradict the prevailing perception that higher thermal conductivities of nanoparticles are always desirable and boost heat transfer. The study findings will be helpful in selecting an appropriate nanoparticle material that would provide a high heat transfer rate based on the application’s thermal conditions.


Nanofluid Vertical plate Mixed convection Brownian motion Thermophoresis 

List of symbols


Channel spacing


Specific heat at constant pressure


Nanoparticle diameter


Brownian diffusivity, \(= {K}_{\mathrm{BO}} {T}/3\pi \mu _{\mathrm{bf}} {d}_{\mathrm{p}}\)


Thermophoresis diffusivity, \(= 0.26{k}_{\mathrm{bf}} /({2k}_{\mathrm{bf}} + {k}_{\mathrm{p}} )*\mu _{\mathrm{bf}} /\rho _{\mathrm{bf}} *\phi _0\)


Grashof number, \(\frac{{g}\beta _{\mathrm{bf}} {q}_1 {b}^{4}}{\upsilon ^{2}{K}_{\mathrm{bf}}}\)


Thermal conductivity


Boltzmann constant


Solid/fluid thermal conductivity ratio, \({K}_{\mathrm{p}} /{K}_{\mathrm{bf}}\)


Ratio of Brownian and thermophoretic diffusivities, \(= {D}_{\mathrm{B}}/{D}_{\mathrm{T}}\)


Nusselt number


Fluid pressure at any cross section

\({p}^{\prime }\)

Pressure defect at any cross section, \({p-p}_{\mathrm{s}}\)


Fluid pressure at channel entrance


Hydrostatic pressure, \(-\rho _{0} \, {gz}\)


Dimensionless pressure at any cross section, \(\frac{{p}^{\prime }-{p}_0}{\rho _0 {u}_0^2}\)


Prandtl number


Heat flux ratio, \(\frac{{q}_2}{{q}_1}\)


Reynolds number, \(=\frac{{u}_{\mathrm{o}} {b}}{\upsilon }\)


Temperature at any point


Inlet temperature


Mean temperature in each cross section \({T}_{\mathrm{m}}(z)\)


Entrance axial velocity


Longitudinal velocity component at any point


Dimensionless longitudinal velocity, \(= {u}/{u}_{\mathrm{o}}\)


Horizontal coordinate


Dimensionless horizontal coordinate, y / b


Vertical coordinate


Dimensionless vertical coordinate, z / (bRe)

Greek symbols

\(\upsilon \)

Kinematic fluid viscosity

\(\rho \)

Fluid density

\(\mu \)

Dynamic fluid viscosity

\(\theta \)

Dimensionless temperature at any point, \([{=\;\frac{{k}_{\mathrm{bf}} ({T-T}_{\mathrm{m}})}{{q}_1 {b}}}]\)

\(\beta \)

Thermal expansion coefficient

\(\phi \)

Particle volume fraction

\(\varPhi \)

Rescaled nanoparticle volume fraction, \([{=\frac{\phi }{\phi _0}}]\)

\(\gamma \)

Immersed-particle buoyancy parameter, \(= 0.78\pi \frac{\mu _{\mathrm{bf}}^2}{\rho _{\mathrm{bf}}}\frac{{\mathrm{d}}_{\mathrm{p}}}{{K}_{\mathrm{BO}} \beta _{\mathrm{bf}} {T}_0^2}(\frac{\rho _{\mathrm{p}}}{\rho _{\mathrm{bf}}}-1\))



Base fluid










Duct wall at \(Y = 0\)


Duct wall at \(Y = 1\)


Condition at the entrance


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Energy Engineering, College of EngineeringImam Abdulrahman Bin Faisal UniversityDammamSaudi Arabia

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