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采用Laplace数值反演的Stehfest算法研究了分数阶Oldroyd-B粘弹性流体在两平板间非定常的Poiseuille流动问题.首先,通过数值解与近似解析解的比较验证了Stehfest算法的有效性.其次,运用Stehfest算法对平板Poiseuille流动进行了研究,揭示了分数阶黏弹性平板流的速度过冲和应力过冲现象,指出这些现象对分数导数的阶数存在明显的依赖性.同时,数值结果表明,整数阶本构方程仅仅是分数阶本构方程的特例,分数阶本构方程较整数阶本构方程具有更广泛的适用性.
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关键词:
- 分数阶导数 /
- Oldroyd-B模型 /
- Laplace变换 /
- 黏弹性
In this paper the unsteady Poiseuille flow of fractional Oldroyd-B viscoelastics fluid between two parallel plates is studied, which sheds light on the investigation on fractional differential equations. Stehfest algorithm for numerical inversion of Laplace transform is used for obtaining the numerical solutions, and its validity is verified by comparing the results with approximate analytic solutions. Then the laminar Poiseuille flow of fractional Oldroyd-B viscoelastic fluid is investigated by the Stehfest algorithm. Phenomena of velocity and stress overshootings are found, which are proved to be dependent on the order of fractional derivative. Simultaneously, compared with the integer constitutive equations, the fractional constitutive equations have wider scope of application. This conclusion was drawn based on the obvious fact that the integer constitutive equations are only special cases of the fractional constitutive equations.-
Keywords:
- fractional derivative /
- Oldroyd-B model /
- Laplace transform /
- viscoelastic
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[2] Song D Y, Jiang T Q 1998 Rheol. Acta 37 512
[3] Zhu K Q, Hu K X, Yang F 2009 Chinese Conference of Theoretical and Applied Mechanics August 24—26, 2009 Zhengzhou p204 (in Chinese) [朱克勤、胡开鑫、杨 帆 2009 中国力学学会学术大会 2009.8.24—26 郑州,第204页]
[4] Tan W C, Xu M Y 2002 Mech. Res. Commun. 29 3
[5] Tan W C, Xu M Y 2004 Acta Mech. Sin. 20 471
[6] Qi H T, Jin H 2006 Acta Mech. Sin. 22 301
[7] Khan M, Maqbool K, Hayat T 2006 Acta Mech. 184 1
[8] Khan M, Hayat T, Asghar S 2006 Int. J. Engng. Sci. 44 333
[9] Khan M, Nadeem S, Hayat T, Siddiqui A M 2005 Math. Comput. Model. 41 629
[10] Khan M, Hyder A S, Qi H T 2009 Nonlinear. Anal-Real. 10 980
[11] Huang J Q, He G Y, Liu C Q 1996 Sci. China. Ser. A 26 912 (in Chinese) [黄军旗、何光渝、刘慈群 1996 中国科学(A辑) 26 912]
[12] Stehfest H 1970 Commun. ACM 13 624
[13] Tong D K, Chen Q L 2001 Acta Petrol. Sin. 22 91 (in Chinese) [同登科、陈钦雷 2001 石油学报 22 91]
[14] Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press) p62—108
[15] Chang F X, Chen J, Huang W 2005 Acta Phys. Sin. 54 1113 (in Chinese) [常福宣、陈 进、黄 薇 2005 54 1113]
[16] Wei P J, Zhang S Y, Wu Y L 1999 Adv. Mech. 29 317 (in Chinese) [魏培君、张双寅、吴永礼 1999 力学进展 29 317]
[17] Tong D K, Wang R H 2004 Sci. China. Ser. G 34 87 (in Chinese) [同登科、王瑞和 2004 中国科学(G辑) 34 87]
[18] Erdo Dgˇ an M E, mrak C E 2005 Int. J. Nonlin. Mech. 40 1238
[19] Hayat T, Khan M, Ayub M 2004 Appl. Math. Comput. 151 105
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[1] Bagley R L 1983 J. Rheol. 27 201
[2] Song D Y, Jiang T Q 1998 Rheol. Acta 37 512
[3] Zhu K Q, Hu K X, Yang F 2009 Chinese Conference of Theoretical and Applied Mechanics August 24—26, 2009 Zhengzhou p204 (in Chinese) [朱克勤、胡开鑫、杨 帆 2009 中国力学学会学术大会 2009.8.24—26 郑州,第204页]
[4] Tan W C, Xu M Y 2002 Mech. Res. Commun. 29 3
[5] Tan W C, Xu M Y 2004 Acta Mech. Sin. 20 471
[6] Qi H T, Jin H 2006 Acta Mech. Sin. 22 301
[7] Khan M, Maqbool K, Hayat T 2006 Acta Mech. 184 1
[8] Khan M, Hayat T, Asghar S 2006 Int. J. Engng. Sci. 44 333
[9] Khan M, Nadeem S, Hayat T, Siddiqui A M 2005 Math. Comput. Model. 41 629
[10] Khan M, Hyder A S, Qi H T 2009 Nonlinear. Anal-Real. 10 980
[11] Huang J Q, He G Y, Liu C Q 1996 Sci. China. Ser. A 26 912 (in Chinese) [黄军旗、何光渝、刘慈群 1996 中国科学(A辑) 26 912]
[12] Stehfest H 1970 Commun. ACM 13 624
[13] Tong D K, Chen Q L 2001 Acta Petrol. Sin. 22 91 (in Chinese) [同登科、陈钦雷 2001 石油学报 22 91]
[14] Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press) p62—108
[15] Chang F X, Chen J, Huang W 2005 Acta Phys. Sin. 54 1113 (in Chinese) [常福宣、陈 进、黄 薇 2005 54 1113]
[16] Wei P J, Zhang S Y, Wu Y L 1999 Adv. Mech. 29 317 (in Chinese) [魏培君、张双寅、吴永礼 1999 力学进展 29 317]
[17] Tong D K, Wang R H 2004 Sci. China. Ser. G 34 87 (in Chinese) [同登科、王瑞和 2004 中国科学(G辑) 34 87]
[18] Erdo Dgˇ an M E, mrak C E 2005 Int. J. Nonlin. Mech. 40 1238
[19] Hayat T, Khan M, Ayub M 2004 Appl. Math. Comput. 151 105
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