# Boundary layer flow of a nanofluid past a permeable exponentially shrinking surface with convective boundary condition using Buongiorno’s model

M.M. Rahman (Department of Mathematics and Statistics, Sultan Qaboos University, Muscat, Sultanate of Oman)
Alin V. Rosca (Department of Statistics, Forecasts and Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania)
I. Pop (Department of Mathematics, Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania)

ISSN: 0961-5539

Article publication date: 2 March 2015

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## Abstract

### Purpose

The purpose of this paper is to numerically solve the problem of steady boundary layer flow of a nanofluid past a permeable exponentially shrinking surface with convective surface condition. The Buongiorno’s mathematical nanofluid model has been used.

### Design/methodology/approach

Using appropriate similarity transformations, the basic partial differential equations are transformed into ordinary differential equations. These equations have been solved numerically for different values of the governing parameters, stretching/shrinking parameter λ, suction parameter s, Prandtl number Pr, Lewis number Le, Biot number, the Brownian motion parameter Nb and the thermophoresis parameter Nt, using the bvp4c function from Matlab. The effects of these parameters on the reduced skin friction coefficient, heat transfer from the surface of the sheet, Sherwood number, dimensionless velocity, and temperature and nanoparticles volume fraction distributions are presented in tables and graphs, and are in details discussed.

### Findings

Numerical results are obtained for the reduced skin-friction, heat transfer and for the velocity and temperature profiles. The results indicate that dual solutions exist for the shrinking case (λ<0). A stability analysis has been performed to show that the upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and, therefore, not physically possible. In addition, it is shown that for a regular fluid (Nb=Nt=0) a very good agreement exists between the present numerical results and those reported in the open literature.

### Research limitations/implications

The problem is formulated for an incompressible nanofluid with no chemical reactions, dilute mixture, negligible viscous dissipation, negligible radiative heat transfer and a new boundary condition is imposed on nanoparticles and base fluid locally in thermal equilibrium. The analysis reveals that the boundary layer separates from the plate. Beyond the turning point it is not possible to get the solution based on the boundary-layer approximations. To obtain further solutions, the full basic partial differential equations have to be solved.

### Originality/value

The present results are original and new for the boundary-layer flow and heat transfer past a shrinking sheet in a nanofluid. Therefore, this study would be important for the researchers working in the relatively new area of nanofluids in order to become familiar with the flow behavior and properties of such nanofluids. The results show that in the presence of suction the dual solutions may exist for the flow of a nanofluid over an exponentially shrinking as well as stretching surface.

## Acknowledgements

M. M. Rahman is thankful to College of Science, Sultan Qaboos University for funding through the research grant IG/SCI/DOMS/13/05 to support his work. The work of Alin V. Roşca was supported by a research grant of the Babeş-Bolyai University, Cluj-Napoca, Romania, project number GTC_34073/2013. The authors wish also to express their thanks to the very competent Reviewers for the valuable comments and suggestions.

## Citation

Rahman, M.M., Rosca, A.V. and Pop, I. (2015), "Boundary layer flow of a nanofluid past a permeable exponentially shrinking surface with convective boundary condition using Buongiorno’s model", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 25 No. 2, pp. 299-319. https://doi.org/10.1108/HFF-12-2013-0361

## Publisher

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Emerald Group Publishing Limited