Advertisement

Journal of Zhejiang University-SCIENCE A

, Volume 20, Issue 4, pp 290–299

# Stagnation point flow and heat transfer past a permeable stretching/shrinking Riga plate with velocity slip and radiation effects

• Nor Ain Azeany Mohd Nasir
• Anuar Ishak
• Ioan Pop
Article
• 16 Downloads

## Abstract

This paper concerns the stagnation point flow and heat transfer of a viscous and incompressible fluid passing through a flat Riga plate with the effects of velocity slip and radiation. An appropriate similarity transformation is chosen to reduce the governing partial differential equations to a system of ordinary differential equations. The numerical results are verified by comparison with existing results from the literature for a special case of the present study. The computed results are analyzed and given in the form of tables and graphs. The behaviors of the skin friction coefficient and the heat transfer rate for various physical parameters are analyzed and discussed. Dual solutions exist for both stretching and shrinking cases. Stability analysis reveals that the solution with lower boundary layer thickness is stable while the other solution is unstable. It is also observed that for the stable solution, the skin friction coefficient and the local Nusselt number increase as the suction effect is increased. For the shrinking case, a solution exists only for a certain range of the shrinking strength and this range increases with increasing value of the suction effect.

## Key words

Riga plate Stagnation-point flow Heat transfer Shrinking sheet Dual solutions

# 通过具有速度滑移和辐射效应的可渗透拉伸/收缩Riga 板的驻点流动和热传导

## 摘要

### 目的

1. 通过分析Riga 板的抽吸效应来控制流体运动和减少摩擦力和压力阻力;2. 利用磁场的速度滑移效应来控制流体的流速;3. 基于辐射原理控制热传导并减小阻力。

### 创新点

1. 本研究可应用于核电厂、飞机、潜艇以及卫星等设施中推进装置的设计;2. 本研究可用于防止边界层分离以减少湍流的产生。

### 方法

1. 构建基于偏微分方程的数理模型;2. 利用相似变换法将偏微分方程简化为常微分方程;3. 利用Matlab 内置求解器bvp4c 对常微分方程组进行数值求解;4. 基于求解结果讨论稳定性。

### 结论

1. 对于拉伸/收缩两种情形的Riga 板问题都存在对偶解;2. 数值求解结果显示表面摩擦系数和表面传热率均会随着吸力的增大而增大,而随拉伸/收缩参数λ的增大而减小;3. 上支解的努塞尔数增大而下支解减小;4. 辐射会提高边界层内的温度,而增强滑移效应则会提高流速同时降低边界层温度;5. 只有上支解是长期稳定的。

## 关键词

Riga 板 驻点流动 热传导 收缩薄片 对偶解

O35

## Preview

Unable to display preview. Download preview PDF.

## References

1. Abd El-Aziz M, 2016. Dual solutions in hydromagnetic stagnation point flow and heat transfer towards a stretching/shrinking sheet with non-uniform heat source/sink and variable surface heat flux. Journal of the Egyptian Mathematical Society, 24(3):479–486. https://doi.org/10.1016/jjoems.2015.09.004
2. Ahmad A, Asghar S, Afzal S, 2016. Flow of nanofluid past a Riga plate. Journal of Magnetism and Magnetic Materials, 402:44–48. https://doi.org/10.1016/j.jmmm.2015.11.043
3. Ahmed N, Khan U, Mohyud-Din ST, 2017a. Influence of nonlinear thermal radiation on the viscous flow through a deformable asymmetric porous channel: a numerical study. Journal of Molecular Liquids, 225:167–173.
4. Ahmed N, Khan U, Mohyud-Din ST, 2017b. Influence of thermal radiation and viscous dissipation on squeezed flow of water between Riga plates saturated with carbon nanotubes. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 522:389–398. https://doi.org/10.1016/j.colsurfa.2017.02.083
5. Ahmed N, Khan U, Mohyud-Din ST, et al., 2017c. Shape effects of nanoparticles on the Squeezed flow between two Riga Plates in the presence of thermal radiation. The European Physical Journal Plus, 132(7):321. https://doi.org/10.1140/epjp/i2017-11576-7 https://doi.org/10.1016/j.molliq.2016.11.021
6. Ahmed N, Khan U, Mohyud-Din ST, et al., 2018. A finite element investigation of the flow of a Newtonian fluid in dilating and squeezing porous channel under the influence of nonlinear thermal radiation. Neural Computing and Applications, 29(2):501–508. https://doi.org/10.1007/s00521-016-2463-9
7. Asadullah M, Khan U, Ahmed N, et al., 2016. Analytical and numerical investigation of thermal radiation effects on flow of viscous incompressible fluid with stretchable convergent/divergent channels. Journal of Molecular Liquids, 224:768–775. https://doi.org/10.1016/j.molliq.2016.10.073
8. Awaludin IS, Weidman PD, Ishak A, 2016. Stability analysis of stagnation-point flow over a stretching/shrinking sheet. AIP Advances, 6(4):045308. https://doi.org/10.1063/L4947130
9. Ayub M, Abbas T, Bhatti MM, 2016. Inspiration of slip effects on electromagnetohydrodynamics (EMHD) nanofluid flow through a horizontal Riga plate. The European Physical Journal Plus, 131(6):193. https://doi.org/10.1140/epjp/i2016-16193-4
10. Aziz A, Niedbalski N, 2011. Thermally developing microtube gas flow with axial conduction and viscous dissipation. International Journal of Thermal Sciences, 50(3):332–340. https://doi.org/10.1016/j.ijthermalsci.2010.08.003
11. Bai Y, Liu XL, Zhang Y, et al., 2016. Stagnation-point heat and mass transfer of MHD Maxwell nanofluids over a stretching surface in the presence of thermophoresis. Journal of Molecular Liquids, 224:1172–1180. https://doi.org/10.1016/j.molliq.2016.10.082
12. Chen S, Tian ZW, 2010. Simulation of thermal micro-flow using lattice Boltzmann method with Langmuir slip model. International Journal of Heat and Fluid Flow, 31(2):227–235. https://doi.org/10.1016/j.ijheatfluidflow.2009.12.006
13. Chiam TC, 1994. Stagnation-point flow towards a stretching plate. Journal of the Physical Society of Japan, 63(6): 2443–2444. https://doi.org/10.1143/JPSJ.63.2443
14. Cortel R, 2008. Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Physics Letters A, 372(5):631–636. https://doi.org/10.1016/j.physleta.2007.08.005
15. Farooq M, Khan MI, Waqas M, et al., 2016a. MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effects. Journal of Molecular Liquids, 221:1097–1103. https://doi.org/10.1016/j.molliq.2016.06.077
16. Farooq M, Anjum A, Hayat T, et al., 2016b. Melting heat transfer in the flow over a variable thicked Riga plate with homogeneous-heterogeneous reactions. Journal of Molecular Liquids, 224:1341–1347. https://doi.org/10.1016/j.molliq.2016.10.123
17. Hady FM, Ibrahim FS, Abdel-Gaied SM, et al., 2012. Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet. Nanoscale Research Letters, 7(1):229. https://doi.org/10.1186/1556-276X-7-229
18. Hayat T, Abbas T, Ayub M, et al., 2016. Flow of nanofluid due to convectively heated Riga plate with variable thickness. Journal of Molecular Liquids, 222:854–862. https://doi.org/10.1016/j.molliq.2016.07.111
19. Hayat T, Khan M, Imtiaz M, et al., 2017. Squeezing flow past a Riga plate with chemical reaction and convective conditions. Journal of Molecular Liquids, 225:569–576. https://doi.org/10.1016/j.molliq.2016.11.089
20. Hiemenz K, 1911. Die Grenzschicht an einem in den gleichformingen Flussigkeitsstrom einge-tauchten graden Kreiszylinder. Dingler’s Polytechnisches Journal, 326: 321–324 (in German).Google Scholar
21. Howarth L, 1951. The boundary layer in three dimensional flow. Part II. The flow near a stagnation point. Philosophical Magazine, 42(7):1433–1440.
22. Hsiao KL, 2016. Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet. Applied Thermal Engineering, 98:850–861. https://doi.org/10.1016/j.applthermaleng.2015.12.138
23. Hsiao KL, 2017a. Combined electrical MHD heat transfer thermal extrusion system using Maxwell fluid with radiative and viscous dissipation effects. Applied Thermal Engineering, 112:1281–1288. https://doi.org/10.1016/j.applthermaleng.2016.08.208
24. Hsiao KL, 2017b. Micropolar nanofluid flow with MHD and viscous dissipation effects towards a stretching sheet with multimedia feature. International Journal of Heat and Mass Transfer, 112:983–990. https://doi.org/10.1016/j.ijheatmasstransfer.2017.05.042
25. Hsiao KL, 2017c. To promote radiation electrical MHD activation energy thermal extrusion manufacturing system efficiency by using Carreau-nanofluid with parameters control method. Energy, 130:486–499. https://doi.org/10.1016/j.energy.2017.05.004
26. Iqbal Z, Mehmood Z, Azhar E, et al., 2017. Numerical investigation of nanofluidic transport of gyrotactic microorganisms submerged in water towards Riga plate. Journal of Molecular Liquids, 234:296–308. https://doi.org/10.1016/j.molliq.2017.03.074
27. Jha BK, Aina B, 2015. Mathematical modelling and exact solution of steady fully developed mixed convection flow in a vertical micro-porous-annulus. Afrika Matematika, 26(7–8):1199–1213. https://doi.org/10.1007/s13370-014-0277-4
28. Khadrawi AF, Al-Shyyab A, 2010. Slip flow and heat transfer in axially moving micro-concentric cylinders. International Communications in Heat and Mass Transfer, 37(8):1149–1152. https://doi.org/10.1016/j.icheatmasstransfer.2010.06.006
29. Khan U, Ahmed N, Bin-Mohsen B, et al., 2017a. Nonlinear radiation effects on flow of nanofluid over a porous wedge in the presence of magnetic field. International Journal of Numerical Methods for Heat & Fluid Flow, 27(1):48–63. https://doi.org/10.1108/HFF-10-2015-0433
30. Khan U, Ahmed N, Mohyud-Din ST, et al., 2017b. Nonlinear radiation effects on MHD flow of nanofluid over a non-linearly stretching/shrinking wedge. Neural Computing and Applications, 28(8):2041–2050. https://doi.org/10.1007/s00521-016-2187-x
31. Khan U, Abbasi A, Ahmed N, et al., 2017c. Particle shape, thermal radiations, viscous dissipation and joule heating effects on flow of magneto-nanofluid in a rotating system. Engineering Computations, 34(8):2479–2498. https://doi.org/10.1108/EC-04-2017-0149
32. Kumari M, Nath G, 1999. Development of flow and heat transfer of a viscous fluid in the stagnation-point region of a three-dimensional body with a magnetic field. Acta Mechanica, 135(1–2):1–12. https://doi.org/10.1007/BF01179042
33. Kuznetsov AV, Nield DA, 2010. Natural convective boundary-layer flow of a nanofluid past a vertical plate. International Journal of Thermal Sciences, 49(2):243–247. https://doi.org/10.1016/j.ijthermalsci.2009.07.015
34. Magyari E, Pantokratoras A, 2011a. Aiding and opposing mixed convection flows over the Riga-plate. Communications in Nonlinear Science and Numerical Simulation, 16(8):3158–3167. https://doi.org/10.1016/j.cnsns.2010.12.003
35. Magyari E, Pantokratoras A, 2011b. Note on the effect of thermal radiation in the linearized Rosseland approximation on the heat transfer characteristics of various boundary layer flows. International Communications in Heat and Mass Transfer, 38(5):554–556. https://doi.org/10.1016/j.icheatmasstransfer.2011.03.006
36. Mahapatra TR, Gupta AS, 2001. Magnetohydrodynamic stagnation-point flow towards a stretching sheet. Acta Mechanica, 152(1–4):191–196. https://doi.org/10.1007/BF01176953
37. Malvandi A, Ganji DD, 2014. Mixed convective heat transfer of water/alumina nanofluid inside a vertical microchannel. Powder Technology, 263:37–44. https://doi.org/10.1016/j.powtec.2014.04.084
38. Nandy SK, Pop I, 2014. Effects of magnetic field and thermal radiation on stagnation flow and heat transfer of nanofluid over a shrinking surface. International Communications in Heat and Mass Transfer, 53:50–55. https://doi.org/10.1016/j.icheatmasstransfer.2014.02.010
39. Nasir NAAM, Ishak A, Pop I, 2017. Stagnation-point flow and heat transfer past a permeable quadratically stretching/shrinking sheet. Chinese Journal of Physics, 55(5): 2081–2091. https://doi.org/10.1016/j.cjph.2017.08.023
40. Nazar R, Amin N, Filip D, et al., 2004. Unsteady boundary layer flow in the region of the stagnation point on a stretching sheet. International Journal of Engineering Science, 42(11–12):1241–1253. https://doi.org/10.1016/j.ijengsci.2003.12.002
41. Noghrehabadi A, Pourrajab R, Ghalambaz M, 2012. Effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet prescribed constant wall temperature. International Journal of Thermal Sciences, 54:253–261. https://doi.org/10.1016/j.ijthermalsci.2011.11.017
42. Oyelakin IS, Mondal S, Sibanda P, 2016. Unsteady Casson nanofluid flow over a stretching sheet with thermal radiation, convective and slip boundary conditions. Alexandria Engineering Journal, 55(2):1025–1035. https://doi.org/10.1016/j.aej.2016.03.003
43. Pantokratoras A, Magyari E, 2009. EMHD free-convection boundary-layer flow from a Riga-plate. Journal of Engineering Mathematics, 64(3):303–315. https://doi.org/10.1007/s10665-008-9259-6
44. Rosca AV, Rosca NC, Pop I, 2016. Numerical simulation of the stagnation point flow past a permeable stretching/shrinking sheet with convective boundary condition and heat generation. International Journal of Numerical Methods for Heat & Fluid Flow, 26(1):348–364. https://doi.org/10.1108/HFF-12-2014-0361
45. Rosca NC, Pop I, 2013. Mixed convection stagnation point flow past a vertical flat plate with a second order slip: heat flux case. International Journal of Heat and Mass Transfer, 65:102–109. https://doi.org/10.1016/j.ijheatmasstransfer.2013.05.061
46. Sharma R, Ishak A, Pop I, 2014. Stability analysis of magneto-hydrodynamic stagnation-point flow toward a stretching/shrinking sheet. Computers & Fluids, 102:94–98. https://doi.org/10.1016/j.compfluid.2014.06.022
47. Torabi M, Peterson GP, 2016. Effects of velocity slip and temperature jump on the heat transfer and entropy generation in micro porous channels under magnetic field. International Journal of Heat and Mass Transfer, 102: 585–595. https://doi.org/10.1016/j.ijheatmasstransfer.2016.06.080
48. Weidman PD, Kubitschek DG, Davis AMJ, 2006. The effect of transpiration on self-similar boundary layer flow over moving surfaces. International Journal of Engineering Science, 44(11–12):730–737. https://doi.org/10.1016/j.ijengsci.2006.04.005
49. Yacob NA, Ishak A, 2011. MHD flow of a micropolar fluid towards a vertical permeable plate with prescribed surface heat flux. Chemical Engineering Research and Design, 89(11):2291–2297. https://doi.org/10.1016/j.cherd.2011.03.011

## Copyright information

© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

## Authors and Affiliations

• Nor Ain Azeany Mohd Nasir
• 1
• 2
• Anuar Ishak
• 2
• Ioan Pop
• 3
1. 1.Department of Mathematics, Centre for Defence Foundation StudiesUniversiti Pertahanan Nasional MalaysiaKuala LumpurMalaysia
2. 2.School of Mathematical Sciences, Faculty of Science and TechnologyUniversiti Kebangsaan MalaysiaBangiMalaysia
3. 3.Department of MathematicsBabeş-Bolyai UniversityCluj-NapocaRomania