Abstract
A differential constraint method is used to obtain analytical solutions of a second-grade fluid flow. By using the first-order differential constraint condition, exact solutions of Poiseuille flows, jet flows and Couette flows subjected to suction or blowing forces, and planar elongational flows are derived. In addition, two new classes of exact solutions for a second-grade fluid flow are found. The obtained exact solutions show that the non-Newtonian second-grade flow behavior depends not only on the material viscosity but also on the material elasticity. Finally, some boundary value problems are discussed.
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References
Yanenko, N. N. Selected Works (in Russian), Nauka, Moscow (1991)
Sidorov, A. F., Shapeev, V. P., and Yanenko, N. N. Method of Differential Constraints and its Applications in Gas Dynamics (in Russian), Nauka, Novosibirsk (1984)
Andreev, V. K., Kaptsov, O. V., Pukhnachev, V. V., and Rodionov, A. A. Applications of Group-Theoretic Methods in Hydrodynamics, Kluwer, Dordrecht (1998)
Olver, P. and Rosenau, P. The construction of special solutions to partial differential equations. SIAM J. Appl. Math. 47(2), 263–278 (1987)
Olver, P. Direct reduction and differential constraints. Proceedings of the Royal Society of London (Series A), 444(1922), 509–523 (1994)
Levi, D. and Winternitz, P. Nonclassical symmetry reduction: example of the Boussinesq equation. J. Phys. A: Math. Gen. 22(15), 2915–2924 (1989)
Polyanin, A. D. and Zaitsev, V. F. Handbook of Nonlinear Partial Differential Equations, CRC Press, Florida (2004)
Nemenyi, P. F. Recent developments in inverse and semi-inverse methods in the mechanics of continua. Advances in Applied Mechanics 2(11), 123–151 (1951)
Hayat, T., Mohyuddin, M. R., and Asghar, S. Some inverse solutions for unsteanian fluid. Tamsui Oxford Journal of Mathematical Sciences 21(1), 1–20 (2005)
Asghar, S., Mohyuddin, M. R., Hayat, T., and Siddiqui, A. M. On inverse solutions of unsteady riabouchinsky flows of second grade fluid. Tamsui Oxford Journal of Mathematical Sciences 22(2), 221–229 (2006)
Labropulu, F. A few more exact solutions of a second grade fluid via inverse method. Mechanics Research Communications 27(6), 713–720 (2000)
Mohyuddin, M. R. and Ahmad, A. Corrigendum to: inverse solutions for a second-grade fluid for porous medium channel and Hall current effects by Muhammad R. Mohyuddin and Ehsan Ellahi Ashraf. Proc. Indian Acad. Sci. (Math. Sci.) 117(2), 283–285 (2007)
Siddiqui, A. M., Mohyuddin, M. R., Hayat, T., and Asghar, S. Some more inverse solutions for steady flows of a second-grade fluid. Arch. Mech. 55(4), 373–387 (2003)
Siddiqui, A. M., Islam, S., and Ghori, Q. K. Two dimensional viscous incompressible flows in a porous medium. Journal of Porous Media 9(6), 591–596 (2006)
Xie, S. B. Some inverse soluitions for non-Newtonian fluid (in Chinese). Journal of Beijing Normal University (Nature Science) 37(1), 19–21 (2001)
Liu, C. Q. and Huang, J. Q. Analytical solutions for equations of unsteady flow of non-Newtonian fluids in tube (in Chinese). Appl. Math. Mech. -Engl. Ed. 10(11), 989–996 (1989) DOI:10.1007/BF02014545
Zhu, W. H. and Liu, C. Q. Analytical solution of flow of second-order non-Newtonian fluid through annular pipes (in Chinese). Appl. Math. Mech. -Engl. Ed. 14(3), 209–215 (1993) DOI:10.1007/BF02451405
Huang, J. Q. and Liu, C. Q. An analytical solution and analysis of characters for viscoelastic fluid flow in annular pipe (in Chinese). Appl. Math. Mech. -Engl. Ed. 18(6), 535–541 (1997) DOI:10.1007/BF02454112
Yürüsoy, M. Similarity solutions for creeping flow and heat transfer in second grade fluids. Appl. Math. Mech. -Engl. Ed. 25(4), 467–474 (2004) DOI:10.1007/BF02437531
Shen, F., Tan, W. C., Zhao, Y. H., and Masuoka, T. Decay of vortex velocity and diffusion of temperature in a generalized second grade fluid. Appl. Math. Mech. -Engl. Ed. 25(10), 1151–1159 (2004) DOI:10.1007/BF02439867
Tan, W. C., Xian, F., and Wei, L. An exact solution of unsteady Couette flow of generalized second grade fluid. Chinese Science Bulletin 47(21), 1783–1785 (2002)
Xu, M. Y. and Tan, W. C. Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion. Science in China (Series A) 44(11), 1387–1399 (2001)
Coleman, B. D. and Noll, W. An approximation theorem for functionals with applications in continuum mechanics. Arch. Rat. Mech. Anal. 6, 355–370 (1960)
Kaloni, P. N. and Siddqui, A. M. The flow of a second grade fluid. Int. J. Eng. Sci. 21(10), 1157–1169 (1983)
Rivlin, R. S. and Ericksen, J. L. Stress deformation relations for isotropic materials. J. Rat. Mech. Anal. 4(4), 323–425 (1955)
Wang, Z. X. and Guo, D. R. Conspectus of Special Functions (in Chinese), Peking University Press, Beijing (2000)
Thomas, A. H. and Todd, B. D. On the Arnold cat map and periodic boundary conditions for planar elongational flow. Molecular Physics 101(23), 3445–3454 (2003)
d’Avino, G., Maffettone, P. L., Hulsen, M. A., and Peters, G. W. M. Numerical simulation of planar elongational flow of concentrated rigid particle suspensions in a viscoelastic fluid. Journal of Non-Newtonian Fluid Mechanics 150(2–3), 65–79 (2008)
Chen, W. F. Non-Newtonian Fluid Mechanics (in Chinese), Science Press, Beijing (1984)
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(Contributed by Yu-lu LIU)
Project supported by the National Natural Science Foundation of China (No. 10772110)
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Zhang, Dx., Feng, Sx., Lu, Zm. et al. Application of differential constraint method to exact solution of second-grade fluid. Appl. Math. Mech.-Engl. Ed. 30, 403–412 (2009). https://doi.org/10.1007/s10483-009-0401-x
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DOI: https://doi.org/10.1007/s10483-009-0401-x