# 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.

show all references

##### 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 $S = 2.1$
 [1] Jacinto Marabel Romo. A closed-form solution for outperformance options with stochastic correlation and stochastic volatility. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1185-1209. doi: 10.3934/jimo.2015.11.1185 [2] Sigve Hovda. Closed-form expression for the inverse of a class of tridiagonal matrices. Numerical Algebra, Control and Optimization, 2016, 6 (4) : 437-445. doi: 10.3934/naco.2016019 [3] Nora Merabet. Global convergence of a memory gradient method with closed-form step size formula. Conference Publications, 2007, 2007 (Special) : 721-730. doi: 10.3934/proc.2007.2007.721 [4] Azam Chaudhry, Rehana Naz. Closed-form solutions for the Lucas-Uzawa growth model with logarithmic utility preferences via the partial Hamiltonian approach. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 643-654. doi: 10.3934/dcdss.2018039 [5] Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569 [6] Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure and Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 [7] Eemeli Blåsten, Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Problems and Imaging, 2015, 9 (3) : 709-723. doi: 10.3934/ipi.2015.9.709 [8] J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177 [9] Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917 [10] Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015 [11] Antonio Iannizzotto, Nikolaos S. Papageorgiou. Existence and multiplicity results for resonant fractional boundary value problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 511-532. doi: 10.3934/dcdss.2018028 [12] Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061 [13] Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234 [14] Juan Dávila, Louis Dupaigne, Marcelo Montenegro. The extremal solution of a boundary reaction problem. Communications on Pure and Applied Analysis, 2008, 7 (4) : 795-817. doi: 10.3934/cpaa.2008.7.795 [15] Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431 [16] Mauro Garavello. Boundary value problem for a phase transition model. Networks and Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89 [17] Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084 [18] Marta García-Huidobro, Raul Manásevich. A three point boundary value problem containing the operator. Conference Publications, 2003, 2003 (Special) : 313-319. doi: 10.3934/proc.2003.2003.313 [19] Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 [20] Mohan Mallick, Sarath Sasi, R. Shivaji, S. Sundar. Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2022, 21 (2) : 705-726. doi: 10.3934/cpaa.2021195

2021 Impact Factor: 1.497