A Chebyshev Spectral Method for the Unsteady Maxwell Oblique Stationary Point Flow on an Axially Cosine Oscillating Cylinder
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摘要: 研究了非稳态Maxwell流体斜撞击轴向余弦振荡圆柱的斜驻点流动. 首先，基于斜驻点流动特性，在柱面坐标系下求得关于压力的二阶常微分方程，对压强进行修正，建立了非稳态Maxwell流体在振荡圆柱上斜驻点流动的边界层模型. 接着，合理的相似变换将模型转化，使用Chebyshev谱方法求得模型的数值解. 结果表明，在贴近圆柱表面的流体随着圆柱体做周期性运动；圆柱的曲率越大越会使在同一时刻同一位置处的流体质点的速度越大；相反，非稳态参数及流体的记忆特性也会在更靠近圆柱壁面处阻碍流体流动.
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关键词:
- 非稳态斜驻点流动 /
- Maxwell流体 /
- 振荡圆柱 /
- 修正压强场 /
- Chebyshev谱方法
Abstract: The oblique stationary point flow of the Maxwell fluid impacting an axially cosine oscillating cylinder was studied. Firstly, based on the oblique stationary point flow characteristics, the pressure was corrected with the 2nd-order ordinary differential equation of pressure obtained in the cylindrical coordinate system. Later, the boundary layer model for the unsteady Maxwell fluid on an oscillating cylinder was established. The model was converted through the reasonable similarity transform, and the numerical solutions were obtained with the Chebyshev spectral method. The results show that, the fluid near the surface of the cylinder moves periodically with the cylinder. The larger the curvature of the cylinder is, the higher the velocity of the fluid particle will be in the same position at the same time. In contrast, the unsteady state parameter and the memory properties of the fluid hinder the flow closer to the cylinder wall. -
图 2 不同振荡参数ω下的速度三维图
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 2. 3D plots of velocities u for different values of oscillation parameter ω
图 4 不同圆柱曲率参数β下的速度u
Figure 4. Velocities u for different values of curvature parameter β
图 5 不同速度比参数s1下的速度u
Figure 5. Velocities u for different values of velocity ratio parameter s1
图 6 不同Deborah拉数De下的速度u
Figure 6. Velocities u for different values of Deborah number De
表 1 现有f″(0)的数据与文献[33-34]中f″(0)的数据对比结果
Table 1. Comparison results between the present data and the data in ref. [33-34]
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表 2 选取不同长度L时f″(0)的数据对比结果
Table 2. Data comparison results of f″(0) with different values of length L
s1 β=0，De=0，N=120 L=6 L=7 L=8 L=9 0.15(a/c=0.3) -0.849 420 808 3 -0.849 420 047 2 -0.849 420 014 1 -0.849 420 022 6 0.25(a/c=0.5) -0.667 263 677 5 -0.667 263 660 4 -0.667 263 666 5 -0.667 263 681 3 0.4(a/c=0.8) -0.299 388 804 7 -0.299 388 802 1 -0.299 388 810 2 -0.299 388 830 4 1(a/c=2) 2.017 502 833 5 2.017 502 837 70 2.017 502 821 3 2.017 502 787 82 1.5(a/c=3) 4.729 282 401 84 4.729 282 403 29 4.729 282 384 3 4.729 282 346 45 2(a/c=4) 8.000 429 507 31 8.000 429 504 18 8.000 429 481 5 8.000 429 443 28
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[1] ABBASI A, FAROOQ W, MABOOD F, et al. Finite difference simulation for oblique stagnation point flow of viscous nanofluid towards a stretching cylinder[J]. Physica Scripta, 2020, 96(1): 015212. doi: 10.1088/1402-4896/abc927 [2] KOLSI L, ABBASI A, ALQSAIR U F, et al. Thermal enhancement of ethylene glycol base material with hybrid nanofluid for oblique stagnation point slip flow[J]. Case Studies in Thermal Engineering, 2021, 28: 101468. doi: 10.1016/j.csite.2021.101468 [3] DRAZIN P G, RILEY N. The Navier-Stokes Equations: a Classification of Flows and Exact Solutions[M]. Cambridge: Cambridge University Press, 2006. [4] POZRIKIDIS C. Introduction to Theoretical and Computational Fluid Dynamics[M]. Oxford: Oxford University Press, 2011. [5] WANG C Y. Axisymmetric stagnation flow on a cylinder[J]. Quarterly of Applied Mathematics, 1974, 32(2): 207-213. doi: 10.1090/qam/99683 [6] WEIDMAN P D, PUTKARADZE V. Axisymmetric stagnation flow obliquely impinging on a circular cylinder[J]. European Journal of Mechanics B: Fluids, 2003, 22(2): 123-131. doi: 10.1016/S0997-7546(03)00019-0 [7] RAHIMI A B, ESMAEILPOUR M. Axisymmetric stagnation flow obliquely impinging on a moving circular cylinder with uniform transpiration[J]. International Journal for Numerical Methods in Fluids, 2011, 65(9): 1084-1095. doi: 10.1002/fld.2230 [8] ABBASI A, MABOOD F, FAROOQ W, et al. Non-orthogonal stagnation point flow of Maxwell nano-material over a stretching cylinder[J]. International Communications in Heat and Mass Transfer, 2021, 120: 105043. doi: 10.1016/j.icheatmasstransfer.2020.105043 [9] RAHIMI A B, BAYAT R. Effect of the angle of oblique stagnation-point flow impinging axisymmetrically on a vertical circular cylinder with mixed convection heat transfer[J]. International Journal of Sustainable Energy, 2019, 38(9): 849-865. doi: 10.1080/14786451.2019.1601628 [10] MABOOD F, ABBASI A, FAROOQ W, et al. Effects of non-linear radiation and chemical reaction on Oldroyd-B nanofluid near oblique stagnation point flow[J]. Chinese Journal of Physics, 2022, 77: 1197-1208. doi: 10.1016/j.cjph.2022.03.049 [11] GHAFFARI A, JAVED T, HSIAO K L. Heat transfer analysis of unsteady oblique stagnation point flow of elastico-viscous fluid due to sinusoidal wall temperature over an oscillating-stretching surface: a numerical approach[J]. Journal of Molecular Liquids, 2016, 219: 748-755. doi: 10.1016/j.molliq.2016.04.014 [12] STUART J T. The viscous flow near a stagnation point when the external flow has uniform vorticity[J]. Journal of Aerosol Science, 1959, 26(2): 124-125. [13] DORREPAA J M. An exact solution of the Navier-Stokes equation which describes non-orthogonal stagnation-point flow in two dimensions[J]. Journal of Fluid Mechanics, 1986, 163: 141-147. doi: 10.1017/S0022112086002240 [14] TAMADA K J. Two-dimensional stagnation-point flow impinging obliquely on an oscillating flat plate[J]. Journal of the Physical Society of Japan, 1979, 46(1): 310-311. doi: 10.1143/JPSJ.46.310 [15] REZA M, GUPTA A S. Steady two-dimensional oblique stagnation point flow towards a stretching surface[J]. Fluid Dynamics Research, 2005, 37(5): 334-340. doi: 10.1016/j.fluiddyn.2005.07.001 [16] REZA M, GUPTA A S. Some aspects of non-orthogonal stagnation-point flow towards a stretching surface[J]. Engineering, 2010, 2(9): 705-709. doi: 10.4236/eng.2010.29091 [17] LOK Y Y, AMIN N, POP I. Non-orthogonal stagnation point flow towards a stretching sheet[J]. International Journal of Non-Linear Mechanics, 2006, 41(4): 622-627. doi: 10.1016/j.ijnonlinmec.2006.03.002 [18] HAYAT T, SAIF R S, ELLAHI R, et al. Simultaneous effects of melting heat and internal heat generation in stagnation point flow of Jeffrey fluid towards a nonlinear stretching surface with variable thickness[J]. International Journal of Thermal Sciences, 2018, 132: 334-354. [19] NADEEM S, MEHMOOD R, AKBAR N S. Non-orthogonal stagnation point flow of a nano non-Newtonian fluid towards a stretching surface with heat transfer[J]. International Journal of Heat and Mass Transfer, 2013, 57(2): 679-689. doi: 10.1016/j.ijheatmasstransfer.2012.10.019 [20] TOOKE R M, BLYT M G. A note on oblique stagnation-point flow[J]. Physics of Fluids, 2008, 20(3): 33101. doi: 10.1063/1.2876070 [21] 阿斯拉夫M, 阿斯拉夫M M. 微极流体向受热面的MHD驻点流动[J]. 应用数学和力学, 2011, 32(1): 44-52. doi: 10.3879/j.issn.1000-0887.2011.01.005ASHRAF M, ASHRAF M M. MHD stagnation point flow of a micropolar fluid towards a heated surface[J]. Applied Mathematics and Mechanics, 2011, 32(1): 44-52. (in Chinese) doi: 10.3879/j.issn.1000-0887.2011.01.005 [22] BAI Y, TANG Q L, ZHANG Y. Unsteady inclined stagnation point flow and thermal transmission of Maxwell fluid on a stretched/contracted plate with modified pressure field[J]. International Journal of Numerical Methods for Heat & Fluid Flow, 2022, 32(12): 3824-3847. [23] BAI Y, TANG Q L, ZHANG Y. Unsteady MHD oblique stagnation slip flow of Oldroyd-B nanofluids by coupling Cattaneo-Christov double diffusion and Buongiorno model[J]. Chinese Journal of Physics, 2022, 79: 451-470. doi: 10.1016/j.cjph.2022.09.013 [24] HIEMENZ K. Die grenzschicht an einem in den gleichformingen flussigkeits-strom einge-tauchten graden kreiszylinder[J]. Dingler's Polytechnic Journal, 1911, 326: 321-410. [25] HOWARTH L. The boundary layer in three-dimensional flow, part Ⅱ: the flow near a stagnation point[J]. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1951, 42(335): 1433-1440. doi: 10.1080/14786445108560962 [26] DAVEY A. Boundary-layer flow at a saddle point of attachment[J]. Journal of Fluid Mechanics, 1961, 10(4): 593-610. doi: 10.1017/S0022112061000391 [27] 朱婧, 郑连存, 张欣欣. 具有延伸表面的驻点流动和传热问题的级数解[J]. 应用数学和力学, 2009, 30(4): 432-442. http://www.applmathmech.cn/article/id/1220ZHU Jing, ZHENG Liancun, ZHANG Xinxin. Analytic solution of stagnation-point flow and heat transfer over a stretching sheet by means of homotopy analysis method[J]. Applied Mathematics and Mechanics, 2009, 30(4): 432-442. (in Chinese) http://www.applmathmech.cn/article/id/1220 [28] 陈亚飞, 郑云英. 不可压缩黏性流体的二维Navier-Stokes方程的间断有限元模拟[J]. 应用数学和力学, 2020, 41(8): 844-852. doi: 10.21656/1000-0887.400379CHEN Yafei, ZHENG Yunying. A discontinuous Galerkin FEM for 2D Navier-Stokes equations of incompressible viscous fluids[J]. Applied Mathematics and Mechanics, 2020, 41(8): 844-852. (in Chinese) doi: 10.21656/1000-0887.400379 [29] 王尕平, 刘竟慧. 端部旋转的圆柱形容器内的Stokes流[J]. 应用数学和力学, 2023, 44(1): 52-60. doi: 10.21656/1000-0887.430197WANG Gaping, LIU Jinghui. Stokes flow in cylindrical containers with rotating ends[J]. Applied Mathematics and Mechanics, 2023, 44(1): 52-60. (in Chinese) doi: 10.21656/1000-0887.430197 [30] MOTSA S S. A new spectral local linearization method for nonlinear boundary layer flow problems[J]. Journal of Applied Mathematics, 2013, 2013(6): 423628. [31] MAJEE A, JAVED T, GHAFFARI A, et al. Analysis of heat transfer due to stretching cylinder with partial slip and prescribed heat flux: a Chebyshev spectral Newton iterative scheme[J]. Alexandria Engineering Journal, 2015, 54(4): 1029-1036. doi: 10.1016/j.aej.2015.09.015 [32] TREFETHEN L N. Spectral Methods in MATLAB[M]. Philadelphia: SIAM, 2000. [33] LABROPULU F, LI D, POP I. Non-orthogonal stagnation-point flow towards a stretching surface in a non-Newtonian fluid with heat transfer[J]. International Journal of Thermal Sciences, 2010, 49(6): 1042-1050. doi: 10.1016/j.ijthermalsci.2009.12.005 [34] MAHAPATRA T, GUPTA A S. Heat transfer in stagnation-point flow towards a stretching sheet[J]. Heat and Mass transfer, 2002, 38(6): 517-521. -
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