{"title":"Ginzburg-Landau Model for Curved Two-Phase Shallow Mixing Layers","authors":"Irina Eglite, Andrei A. Kolyshkin","volume":64,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":381,"pagesEnd":386,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/14646","abstract":"Method of multiple scales is used in the paper in order\nto derive an amplitude evolution equation for the most unstable mode\nfrom two-dimensional shallow water equations under the rigid-lid\nassumption. It is assumed that shallow mixing layer is slightly curved\nin the longitudinal direction and contains small particles. Dynamic\ninteraction between carrier fluid and particles is neglected. It is\nshown that the evolution equation is the complex Ginzburg-Landau\nequation. Explicit formulas for the computation of the coefficients of\nthe equation are obtained.","references":"[1] G.H. Jirka, \"Large scale flow structures and mixing processes in shallow\nflows\", J. Hydr. Res., vol. 39, pp. 567-573, 2001.\n[2] V.H. Chu and S. Babarutsi, \"Confinement and bed-friction effects in\nshallow turbulent mixing layers\", J. Hydr. Eng., vol. 114, pp. 1257-\n1274, 1988.\n[3] W.S.J. Uijttewaal and R. Booij, \"Effect of shallowness on the\ndevelopment of free-surface mixing layers\", Phys. Fluids., vol. 12, pp.\n1257-1274, 1988.\n[4] W.S.J. Uijttewaal and J. Tukker, \"Development of quasi twodimensional\nstructures in a shallow free-surface mixing layer\", Exp.\nFluids., vol. 24, pp. 192-200, 1998.\n[5] B.C. Prooijen and W.S.J. Uijttewaal, \"A linear approach for the\nevolution of coherent structures in shallow mixing layers\", Phys.\nFluids., vol. 14, pp. 4105-4114, 2002.\n[6] V.H. Chu, J.H. Wu, and R.E. Khayat, \"Stability of transverse shear\nflows in shallow open channels\", J. Hydr. Eng., vol. 117, pp. 1370-\n1388, 1991.\n[7] D. Chen and G.H. Jirka, \"Linear stability analysis of turbulent mixing\nlayers and jets in shallow water layers\", J. Hydr. Res., vol. 36, pp. 815-\n830, 1998.\n[8] M.S. Ghidaoui and A.A. Kolyshkin, \"Linear stability analysis of lateral\nmotions in compound open channels\", J. Hydr. Eng., vol. 125, pp. 871-\n880, 1999.\n[9] A.A. Kolyshkin and M.S. Ghidaoui, \"Gravitational and shear\ninstabilities in compound and composite channels\", J. Hydr. Eng., vol.\n128, pp. 1076-1086, 2002.\n[10] Y. Yang, J.N. Chung, T.R. Troutt, and C.T. Crowe, \"The influence of\nparticles on the stability of two-phase mixing layers\", Phys. Fluids., vol.\nA2, pp. 1839-1845, 1990.\n[11] Y. Yang, J.N. Chung, T.R. Troutt, and C.T. Crowe, \"The effect of\nparticles on the stability of a two-phase wake flow\", Int. J. Multiphase\nFlow, vol. 19, pp. 137-149, 1993.\n[12] K. Stewartson and J.T. Stuart, \"A non-linear instability theory for a\nwave system in plane Poiseuille flow\", J. Fluid Mech.., vol. 48, pp. 529-\n545, 1971.\n[13] F. Feddersen, \"Weakly nonlinear shear waves\", J. Fluid Mech., vol. 372,\npp. 71-91, 1998.\n[14] P.J. Blennerhassett, \"On the generation of waves by wind\",\nPhilosophical Transactions of the Royal Society of London, Ser. A., vol.\n298, pp. 451-494, 1980.\n[15] A.A. Kolyshkin and M.S. Ghidaoui, \"Stability analysis of shallow wake\nflows\", J. Fluid Mech.., vol. 494, pp. 355-377, 2003.\n[16] A.A. Kolyshkin and S. Nazarovs, \"Linear and weakly nonlinear analysis\nof two-phase shallow wake flows\", WSEAS Transactions on\nMathematics., vol. 6, pp. 1-8, 2007.\n[17] I. Eglite and A.A. Kolyshkin, \"Linear and weakly nonlinear instability\nof slightly curved shallow mixing layers\", WSEAS Transactions on\nFluid Mechanics., vol. 6, pp. 123-132, 2011.\n[18] D. Zwillinger, Handbook of differential equations. New York:\nAcademic Press, 1998.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 64, 2012"}