# American Institute of Mathematical Sciences

November  2020, 25(11): 4119-4126. doi: 10.3934/dcdsb.2020090

## Analysis of nanofluid flow past a permeable stretching/shrinking sheet

 School of Science, Penn State Behrend, Erie, Pennsylvania 16563-0203, USA

* Corresponding author: Joseph E. Paullet

Received  May 2019 Revised  December 2019 Published  November 2020 Early access  April 2020

In this article we analyze a recently proposed model for boundary layer flow of a nanofluid past a permeable stretching/shrinking sheet. The boundary value problem (BVP) resulting from this model is governed by two physical parameters; ${{\lambda}}$, which controls the stretching (${{\lambda}} >0$) or shrinking (${{\lambda}} < 0$) of the sheet, and $S$, which controls the suction ($S>0$) or injection ($S<0$) of fluid through the sheet. For ${{\lambda}} \ge 0$ and $S\in \mathbb{R}$, we present a closed-form solution to the BVP and prove that this solution is unique. For ${{\lambda}} < 0$ and $S< 2\sqrt{-{{\lambda}}}$ we prove no solution exists. For ${{\lambda}} < 0$ and $S = 2\sqrt{-{{\lambda}}}$ we present a closed-form solution to the BVP and prove that it is unique. For ${{\lambda}} < 0$ and $S> 2\sqrt{-{{\lambda}}}$ we present two closed-form solutions to the BVP and prove the existence of an infinite number of solutions in this parameter range. The analytical results proved here differ from the numerical results reported in the literature. We discuss the mathematical aspects of the problem that lead to the difficulty in obtaining accurate numerical approximations to the solutions.

Citation: Joseph E. Paullet, Joseph P. Previte. Analysis of nanofluid flow past a permeable stretching/shrinking sheet. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4119-4126. doi: 10.3934/dcdsb.2020090
##### References:
 [1] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer., 128 (2006), 240-250.  doi: 10.1115/1.2150834. [2] S. K. Das, S. U. Choi, W. Yu and T. Pradeep, Nanofluids: Science and Technology, John Wiley and Sons, New York, 2007. [3] D. B. Ingham and S. N. Brown, Flow past a suddenly heated vertical plate in a porous medium, Proc. R. Soc. Lond. A, 403 (1986), 51-80. [4] S. Jahan, H. Sakidin, R. Nazar and I. Pop, Analysis of heat transfer in nanofluid past a convectively heated permeable stretching/shrinking sheet with regression and stability analyses, Results in Physics, 10 (2018), 395-405.  doi: 10.1016/j.rinp.2018.06.021. [5] J. B. McLeod and K. R. Rajagopal, On the uniqueness of flow of a Navier-Stokes fluid due to a stretching boundary, Arch. Rational Mech. Anal., 98 (1987), 385-393.  doi: 10.1007/BF00276915. [6] E. E. S. Michaelides, Nanofluidics. Thermodynamic and Transport Properties, Springer International Publishing, Switzerland, 2014. [7] K. R. Rajagopal, T. Y. Na and A. S. Gupta, Flow of a viscoelastic fluid over a stretching sheet, Rheol. Acta, 23 (1984), 213-215.  doi: 10.1007/BF01332078. [8] W. C. Troy, E. A. Overman, G. B. Ermentrout and J. P. Keener, Uniqueness of flow of a second-order fluid past a stretching sheet, Quart. Appl. Math., 44 (1987), 753-755.  doi: 10.1090/qam/872826.

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##### References:
 [1] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer., 128 (2006), 240-250.  doi: 10.1115/1.2150834. [2] S. K. Das, S. U. Choi, W. Yu and T. Pradeep, Nanofluids: Science and Technology, John Wiley and Sons, New York, 2007. [3] D. B. Ingham and S. N. Brown, Flow past a suddenly heated vertical plate in a porous medium, Proc. R. Soc. Lond. A, 403 (1986), 51-80. [4] S. Jahan, H. Sakidin, R. Nazar and I. Pop, Analysis of heat transfer in nanofluid past a convectively heated permeable stretching/shrinking sheet with regression and stability analyses, Results in Physics, 10 (2018), 395-405.  doi: 10.1016/j.rinp.2018.06.021. [5] J. B. McLeod and K. R. Rajagopal, On the uniqueness of flow of a Navier-Stokes fluid due to a stretching boundary, Arch. Rational Mech. Anal., 98 (1987), 385-393.  doi: 10.1007/BF00276915. [6] E. E. S. Michaelides, Nanofluidics. Thermodynamic and Transport Properties, Springer International Publishing, Switzerland, 2014. [7] K. R. Rajagopal, T. Y. Na and A. S. Gupta, Flow of a viscoelastic fluid over a stretching sheet, Rheol. Acta, 23 (1984), 213-215.  doi: 10.1007/BF01332078. [8] W. C. Troy, E. A. Overman, G. B. Ermentrout and J. P. Keener, Uniqueness of flow of a second-order fluid past a stretching sheet, Quart. Appl. Math., 44 (1987), 753-755.  doi: 10.1090/qam/872826.
The value of $f''(0)$ as a function of ${{\lambda}}$ for various values of $S$, from far left, $S = 2.5$, $S = 2.3$ and $S = 2.1$
The values of $a_1$ (solid curve) and $a_2$ (dashed curve) as a function of ${{\lambda}}$ for various values of $S$, from far left, $S = 2.5$, $S = 2.3$ and $S = 2.1$
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