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December  2019, 9(4): 423-431. doi: 10.3934/naco.2019041

## Stability analysis of stagnation point flow in nanofluid over stretching/shrinking sheet with slip effect using buongiorno's model

 1 Institute for Mathematical Research, Universiti Putra Malaysia, 43400, Serdang, Selangor, Malaysia 2 Department of Mathematics, Universiti Putra Malaysia, 43400, Serdang, Selangor, Malaysia

* Corresponding author: najwamohdnajib@ymail.com

The reviewing process of the paper is handled by Gafurjan Ibragimov, Siti Hasana Sapar and Siti Nur Iqmal Ibrahim

Received  January 2018 Revised  September 2018 Published  August 2019

Fund Project: The first author is supported by Putra grant of Universiti Putra Malaysia

The study on stagnation boundary layer flow in nanofluid over stretching/shrinking sheet with the effect of slip at the boundary was considered by applying the Buongiorno's model. The partial differential equations of the governing equations were transformed into ordinary differential equations by using appropriate similarity transformation in order to obtain the similarity equations. The equations then were substituted into bvp4c code in Matlab software to get the numerical results. The results of skin friction coefficient, heat transfer coefficient as well as mass transfer coefficient on the governing parameters such as slip parameter, Brownian motion parameter, and thermophoresis parameter are shown graphically. The presence of slip parameter is significantly affected the skin friction, heat and mass transfer coefficient. The smallest number of Brownian motion is sufficient to increase the heat transfer coefficient while largest number of thermophoresis parameter is required to increase mass transfer coefficient. The stability analysis results expressed that the first solution is stable and physically realizable whereas the second solution is not.

Citation: Najwa Najib, Norfifah Bachok, Norihan Md Arifin, Fadzilah Md Ali. Stability analysis of stagnation point flow in nanofluid over stretching/shrinking sheet with slip effect using buongiorno's model. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 423-431. doi: 10.3934/naco.2019041
##### References:
 [1] N. Bachok, N. Najib, N. M. Arifin and N. Senu, Stability of dual solutions in boundary layer flow and heat transfer on a moving plate in a copper-water nanofluid with slip effect, WSEAS Transactions on Fluid Mechanics, 11 (2016), 151-158.   Google Scholar [2] J. Buongiorno, Convective transport in nanofluids, Journal of Heat Transfer, 128 (2006), 240-250.   Google Scholar [3] S. K. Das, S. U. S. Choi, W. Yu and T. Pradeep, Nanofluids Science and Technology, John Wiley & Sons, New York, 2007. Google Scholar [4] N. F. Dzulkifli, N. Bachok, I. Pop, N. A. Yacob, N. M. Arifin and H. Rosali, Soret and Dufour effects on unsteady boundary layer flow and heat transfer of nanofluid over a stretching/shrinking sheet: A stability analysis, Journal of Chemical Engineering & Process Technology, 8 (2017), 1-9.   Google Scholar [5] R. Hamid, R. Nazar and I. Pop, Non-alignment stagnationpoint flow of a nanofluid past a permeable stretching$/$shrinking sheet: Buongiorno's model, , Scientific Reports, 5 (2015), 14640. Google Scholar [6] S. D. Harris, D. B. Ingham and I. Pop, Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip, Transport Porous Media, 77 (2009), 267-285.   Google Scholar [7] W. Ibrahim, B. Shankar and M. M. Nandeppanavar, MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet, International Journal of Heat and Mass Transfer, 56 (2013), 1-9.   Google Scholar [8] A. Ishak, Flow and heat transfer over a shrinking sheet: A stability analysis, International Journal of Mechanical, Aerospace, Industrial and Mechatronics Engineering, 8 (2014), 905-909.   Google Scholar [9] R. Jusoh and R. Nazar, Stagnation point flow and heat transfer of a nanofluid over a stretching$/$shrinking sheet with convective boundary conditions and suction, AIP Proceeding, 1830 (2017), 020043. Google Scholar [10] M. I. Khan, M. Tamoor, T. Hayat and A. Alsaedi, MHD boundary layer thermal slip flow by nonlinearly stretching cylinder with suction/blowing and radiation, Results in Physics, 7 (2017), 1207-1211.   Google Scholar [11] S. Mansur, A. Ishak and I. Pop, Stagnation-point flow towards a stretching$/$shrinking sheet in a nanofluid using Buongiorno's model, Journal of Process Mechanical Engineering, 4 (2015), 1-9.   Google Scholar [12] S. Mansur and A. Ishak, The magnetohydrodynamic boundary layer flow of a nanofluid past a stretching/shrinking sheet with slip boundary conditions, Journal of Applied Mathematics, 90752 (2014), 1-7.  doi: 10.1155/2013/350647.  Google Scholar [13] S. Mansur, A. Ishak and I. Pop, The magnetohydrodynamic stagnation point flow of a nanofluid over a stretching/shrinking sheet with suction, Plos One, 11 (2015), 1-14.  doi: 10.1155/2013/350647.  Google Scholar [14] A. Mehmood and A. Ali, The effect of slip condition on unsteady MHD oscillatory ow of a viscous uid in a planer channel, Romanian Journal of Physics, 52 (2007), 85-91.   Google Scholar [15] K. Merill, M. Beauchesne, J. Previte, J. Paullet and P. Weidman, Final steady flow near a stagnation point on a vertical surface in a porous medium, International Journal of Heat and Mass Transfer, 49 (2006), 4681-4686.   Google Scholar [16] J. H. Merkin, On dual solutions occurring in mixed convection in a porous medium, Journal of Engineering Mathematics, 20 (1985), 171-179.   Google Scholar [17] M. K. A. Mohamed, N. A. Z. Noar, M. Z. Salleh and A. Ishak, Slip flow on stagnation point over a stretching sheet in a viscoelastic nanofluid, , AIP Proceedings, 1830 (2017), 020015. Google Scholar [18] S. Mukhopadhyay, MHD boundary layer slip flow along a stretching cylinder, Ain Shams Engineering Journal, 4 (2013), 317-324.   Google Scholar [19] N. Najib, N. Bachok and N. M. Arifin, Stability of dual solutions in boundary layer flow and heat transfer over an exponentially shrinking cylinder, Indian Journal of Science and Technology, 9 (2016), 1-6.   Google Scholar [20] N. Najib, N. Bachok, N. M. Arifin and N. Senu, Boundary layer flow and heat transfer of nanofluids over a moving plate with partial slip and thermal convective boundary condition: Stability analysis, International Journal of Mechanics, 11 (2017), 19-24.   Google Scholar [21] A. V. Rosca and I. Pop, Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip, International Journal of Heat and Mass Transfer, 6 (2013), 355-364.   Google Scholar [22] R. Sharma and A. Ishak, Stagnation point flow of a micropolar fluid over a stretching/ shrinking sheet with second-order velocity slip, Journal of Aerospace Engineering, 04016025 (2016), 1-8.   Google Scholar [23] P. D. Weidman, D. G. Kubitschek and A. M. J. Davis, The effect of transpiration on self-similar boundary layer flow over moving surfaces, International Journal of Engineering Science, 44 (2006), 730-737.   Google Scholar [24] K. Zaimi and A. Ishak, Stagnation-point flow towards a stretching vertical sheet with slip effects, Mathematics, 4 (2016), 1-8.   Google Scholar

show all references

##### References:
 [1] N. Bachok, N. Najib, N. M. Arifin and N. Senu, Stability of dual solutions in boundary layer flow and heat transfer on a moving plate in a copper-water nanofluid with slip effect, WSEAS Transactions on Fluid Mechanics, 11 (2016), 151-158.   Google Scholar [2] J. Buongiorno, Convective transport in nanofluids, Journal of Heat Transfer, 128 (2006), 240-250.   Google Scholar [3] S. K. Das, S. U. S. Choi, W. Yu and T. Pradeep, Nanofluids Science and Technology, John Wiley & Sons, New York, 2007. Google Scholar [4] N. F. Dzulkifli, N. Bachok, I. Pop, N. A. Yacob, N. M. Arifin and H. Rosali, Soret and Dufour effects on unsteady boundary layer flow and heat transfer of nanofluid over a stretching/shrinking sheet: A stability analysis, Journal of Chemical Engineering & Process Technology, 8 (2017), 1-9.   Google Scholar [5] R. Hamid, R. Nazar and I. Pop, Non-alignment stagnationpoint flow of a nanofluid past a permeable stretching$/$shrinking sheet: Buongiorno's model, , Scientific Reports, 5 (2015), 14640. Google Scholar [6] S. D. Harris, D. B. Ingham and I. Pop, Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip, Transport Porous Media, 77 (2009), 267-285.   Google Scholar [7] W. Ibrahim, B. Shankar and M. M. Nandeppanavar, MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet, International Journal of Heat and Mass Transfer, 56 (2013), 1-9.   Google Scholar [8] A. Ishak, Flow and heat transfer over a shrinking sheet: A stability analysis, International Journal of Mechanical, Aerospace, Industrial and Mechatronics Engineering, 8 (2014), 905-909.   Google Scholar [9] R. Jusoh and R. Nazar, Stagnation point flow and heat transfer of a nanofluid over a stretching$/$shrinking sheet with convective boundary conditions and suction, AIP Proceeding, 1830 (2017), 020043. Google Scholar [10] M. I. Khan, M. Tamoor, T. Hayat and A. Alsaedi, MHD boundary layer thermal slip flow by nonlinearly stretching cylinder with suction/blowing and radiation, Results in Physics, 7 (2017), 1207-1211.   Google Scholar [11] S. Mansur, A. Ishak and I. Pop, Stagnation-point flow towards a stretching$/$shrinking sheet in a nanofluid using Buongiorno's model, Journal of Process Mechanical Engineering, 4 (2015), 1-9.   Google Scholar [12] S. Mansur and A. Ishak, The magnetohydrodynamic boundary layer flow of a nanofluid past a stretching/shrinking sheet with slip boundary conditions, Journal of Applied Mathematics, 90752 (2014), 1-7.  doi: 10.1155/2013/350647.  Google Scholar [13] S. Mansur, A. Ishak and I. Pop, The magnetohydrodynamic stagnation point flow of a nanofluid over a stretching/shrinking sheet with suction, Plos One, 11 (2015), 1-14.  doi: 10.1155/2013/350647.  Google Scholar [14] A. Mehmood and A. Ali, The effect of slip condition on unsteady MHD oscillatory ow of a viscous uid in a planer channel, Romanian Journal of Physics, 52 (2007), 85-91.   Google Scholar [15] K. Merill, M. Beauchesne, J. Previte, J. Paullet and P. Weidman, Final steady flow near a stagnation point on a vertical surface in a porous medium, International Journal of Heat and Mass Transfer, 49 (2006), 4681-4686.   Google Scholar [16] J. H. Merkin, On dual solutions occurring in mixed convection in a porous medium, Journal of Engineering Mathematics, 20 (1985), 171-179.   Google Scholar [17] M. K. A. Mohamed, N. A. Z. Noar, M. Z. Salleh and A. Ishak, Slip flow on stagnation point over a stretching sheet in a viscoelastic nanofluid, , AIP Proceedings, 1830 (2017), 020015. Google Scholar [18] S. Mukhopadhyay, MHD boundary layer slip flow along a stretching cylinder, Ain Shams Engineering Journal, 4 (2013), 317-324.   Google Scholar [19] N. Najib, N. Bachok and N. M. Arifin, Stability of dual solutions in boundary layer flow and heat transfer over an exponentially shrinking cylinder, Indian Journal of Science and Technology, 9 (2016), 1-6.   Google Scholar [20] N. Najib, N. Bachok, N. M. Arifin and N. Senu, Boundary layer flow and heat transfer of nanofluids over a moving plate with partial slip and thermal convective boundary condition: Stability analysis, International Journal of Mechanics, 11 (2017), 19-24.   Google Scholar [21] A. V. Rosca and I. Pop, Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip, International Journal of Heat and Mass Transfer, 6 (2013), 355-364.   Google Scholar [22] R. Sharma and A. Ishak, Stagnation point flow of a micropolar fluid over a stretching/ shrinking sheet with second-order velocity slip, Journal of Aerospace Engineering, 04016025 (2016), 1-8.   Google Scholar [23] P. D. Weidman, D. G. Kubitschek and A. M. J. Davis, The effect of transpiration on self-similar boundary layer flow over moving surfaces, International Journal of Engineering Science, 44 (2006), 730-737.   Google Scholar [24] K. Zaimi and A. Ishak, Stagnation-point flow towards a stretching vertical sheet with slip effects, Mathematics, 4 (2016), 1-8.   Google Scholar
Skin friction coefficient $f''(0)$, heat transfer coefficient $-\theta'(0)$ and mass transfer coefficient $-\phi'(0)$ vs $\varepsilon$ for different $\sigma$
Heat transfer coefficient $-\theta'(0)$ and mass transfer coefficient $-\phi'(0)$ vs $\varepsilon$ for different $Nb$
Heat transfer coefficient $-\theta'(0)$ and mass transfer coefficient $-\phi'(0)$ vs $\varepsilon$ for different $Nt$
Velocity profile $f'(\eta)$, temperature profile $\theta(\eta)$ and concentration profile $\phi(\eta)$ for different $\varepsilon$
Smallest eigenvalues $\gamma$ for selected values of $\varepsilon$ with different $\sigma$
 $\sigma$ $\varepsilon$ First solution Second solution 0 -1.246 0.0622 -0.0614 -1.24 0.2121 -0.2036 -1.2 0.5780 -0.5172 0.2 -1.388 0.0802 -0.0791 -1.38 0.2390 -0.2300 -1.3 0.7707 -0.6783 0.4 -1.582 0.0719 -0.0712 -1.58 0.1273 -0.1251 -1.5 0.6936 -0.6301
 $\sigma$ $\varepsilon$ First solution Second solution 0 -1.246 0.0622 -0.0614 -1.24 0.2121 -0.2036 -1.2 0.5780 -0.5172 0.2 -1.388 0.0802 -0.0791 -1.38 0.2390 -0.2300 -1.3 0.7707 -0.6783 0.4 -1.582 0.0719 -0.0712 -1.58 0.1273 -0.1251 -1.5 0.6936 -0.6301
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