# Impact of nanoparticles migration on mixed convection and entropy generation of a \(\hbox {Al}_{2}\hbox {O}_{3}\)–water nanofluid inside an inclined enclosure with wavy side wall

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## Abstract

A numerical study based on the non-homogeneous model of the mixed convection of \(\hbox {Al}_{2}\hbox {O}_{3}\)–water nanofluid in an inclined enclosure is made. The heated side wall of the enclosure is considered to be wavy while the top wall is made to translate horizontally. The wavy physical domain is transformed to a square computational domain through a suitable coordinate transformation. The transformed governing equations are integrated based on the control volume approach in a staggered grid arrangement. A third-order accurate upwind scheme QUICK is used to discretize the convective terms while a central difference type scheme is used to approximate the diffusive terms. The discretized equations along with the specified boundary conditions are solved through a pressure correction-based SIMPLE algorithm. The impact of the inclination angle of the enclosure and surface waviness on the nanofluid mixed convection is elucidated. The parameters governing the Brownian motion, thermophoresis, Lewis number and buoyancy ratio parameter are determined based on the nanofluid thermophysical properties. The effect of the nanoparticle bulk volume fraction and nanoparticle diameter on the mixed convection is analyzed for different choices of nanofluid-to-base fluid thermal buoyancy ratio in the buoyancy-dominated regime as well as shear-dominated regime. A comparison of the non-homogeneous model for the nanofluid with the homogeneous model is also made in this study. The inclination angle of the enclosure is found to have an impact on the mixed convection when buoyancy force is dominant. Heat transfer augmentation occurs as the wave number, and/or wave amplitude of the wavy side wall is increased. The thermodynamic optimization is studied by analyzing the average Nusselt number and the total entropy generation.

## Keywords

\(\hbox {Al}_{2}\hbox {O}_{3}\)–water nanofluid Mixed convection Non-homogeneous model Entropy generation Wavy wall Inclination## List of symbols

- \(D_\mathrm{B}\)
Brownian diffusion coefficient

- \(D_\mathrm{T}\)
Thermophoretic diffusion coefficient

*g*Gravitational acceleration (m s\(^{-2}\))

*Gr*Grashof number, \((1-\phi _\mathrm{b})g\beta _\mathrm{f}\Delta T H^{3}/\nu _\mathrm{f}^{2}\)

*H*Enclosure height (m)

*k*Thermal conductivity (W mK\(^{-1}\))

*Nu*Local Nusselt number

*Nr*Buoyancy ratio, \(\{(\rho _\mathrm{p}-\rho _\mathrm{f})\phi _\mathrm{b}\}/\{\rho _\mathrm{f}\beta _\mathrm{f}\Delta T(1-\phi _\mathrm{b})\}\)

- \(p^{*}\)
Pressure (N m\(^{-2}\))

*Pr*Prandtl number, \(\nu _\mathrm{f}/\alpha _\mathrm{f}\)

*Re*Reynolds number, \(\rho _\mathrm{f}U_{0}H / \mu _\mathrm{f}\)

*Ri*Richardson number, Gr/Re\(^{2}\)

- \(S_\mathrm{f}\)
Dimensionless local entropy generation due to fluid friction irreversibility

- \(S_\mathrm{gen}\)
Dimensionless total entropy generation

- \(S_\mathrm{h}\)
Dimensionless local entropy generation due to heat transfer irreversibility

- \(t^{*}\)
Time (s)

*t*Dimensionless time

*T*Temperature (K)

- (
*u*,*v*) Dimensionless velocity components in

*x*,*y*-direction, respectively- \(U_{0}\)
Reference velocity (m s\(^{-1}\))

## Greek symbols

- \(\alpha\)
Thermal diffusivity (m\(^{2}\) s\(^{-1}\))

- \(\alpha _\mathrm{s}\)
Dimensionless wave amplitude of wavy wall

- \(\beta _\mathrm{f}\)
Coefficient of thermal expansion (\(\mathrm{K}^{-1}\))

- \(\mu\)
Dynamic viscosity (kg m\(^{-1}\) s)

- \(\xi\)
Transformed coordinate in

*x*-direction- \(\eta\)
Transformed coordinate in

*y*-direction- \(\nu\)
Kinematic viscosity (m\(^{2}\) s\(^{-1}\))

- \(\theta\)
Dimensionless temperature

- \(\Pi\)
Dimensionless heat function

- \(\rho\)
Density (kg m\(^{-3}\))

- \(\phi\)
Nanoparticle volume fraction

- \(\omega _\mathrm{n}\)
Wave number of wavy wall

## Subscripts

- av
Average

- b
Bulk

- c
Cold

- f
Clear fluid

- h
Hot

- nf
Nanofluid

- p
Solid particle

- s
Solid wall

- tot
Total

## Superscripts

- \(*\)
Dimensional quantity

- 0
Clear fluid (\(\phi =0\))

## Notes

### Acknowledgements

One of the authors (SB) wishes to acknowledge the financial support received from SERB, Govt. of India through the project grant EMR/2016/000185. The work of I. Pop has been supported by the grant PN-III-P4-ID-PCE-2016-0036, UEFISCDI, Romania. We thank the anonymous reviewers for the constructive comments, which clearly enhanced the quality of the manuscript.

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