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Asymptotic states of a Smoluchowski equation. (English) Zbl 1086.76003
From the text: We study the high-concentration asymptotics of steady states of a Smoluchowski equation arising in the modeling of nematic liquid crystalline polymers. We study the Smoluchowski equation on the unit sphere with the Maier-Saupe potential. This choice of the potential allows us to investigate rigorously the asymptotics of the steady-state solutions for large values of the potential intensity, corresponding to large concentrations. We reduce the problem of finding steady state solutions of the Smoluchowski partial differential equation with Maier-Saupe potential to the finite-dimensional problem of finding the eigenvalues of a symmetric, traceless matrix. Linear combinations of these eigenvalues are critical points of a function associated with them. We find multiple steady solutions which are clustered in three distinct groups. As the concentration is increased, these steady solutions converge to uniform, prolate and oblate states. Furthermore, the methods of study allow us to devise asymptotic expansions for the steady states, expansions that are valid at high but finite concentrations.

76A15 Liquid crystals
76T20 Suspensions
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
[1] Brenner, M., Constantin, P., Kadanoff, L., Schenkel, A., Venkataramani, S.: Diffusion, attraction and collapse. Nonlinearity 12, 1071-1098 (1999) · Zbl 0942.35018 · doi:10.1088/0951-7715/12/4/320
[2] Constantin, P., Dupont, T., Goldstein, R., Kadanoff, L., Shelley, M., Zhou, S.: Droplet breakup in a model of the Hele-Shaw cell. Phys. Rev. E 47, 593-596 (1993)
[3] Constantin, P., Kevrekidis, I., Titi, E.S.: Remarks on a Smoluchowski equation. To appear Discrete and Continuous Dynamical Systems, 2004 · Zbl 1138.35336
[4] Doi, M.: Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases. J. Polym. Sci., Polym. Phys. Ed. 19, 229-243 (1981) · doi:10.1002/pol.1981.180190205
[5] Faraoni, V., Grosso, M., Crescitelli, S., Maffetone, P.L.: The rigid rod model for nematic polymers: An analysis of the shear flow problem. J. Rheol. 43, 829-843 (1999) · doi:10.1122/1.551005
[6] Forest, G., Wang, Q.: Monodomain response of finite-aspect-ratio macromolecules in shear and related linear flows. Rheologica Acta, 42, 20-46 (2003) · doi:10.1007/s00397-002-0252-0
[7] Forest, G., Wang, Q., Zhou, R.: The weak shear kinetic phase diagram for nematic polymers. Preprint, May 2003
[8] Forest, G., Zhou, R., Wang, Q.: Scaling behavior of kinetic orientational distributions for dilute nematic polymers in weak shear. Preprint, July 17, 2003 · Zbl 1106.76331
[9] Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, London, 1978 · Zbl 0451.53038
[10] Jendrejack, R.M., de Pablo, J.J., Graham, M.D.: A method for multiscale simulation of flowing complex fluids. J. Non-Newtonian Fluid Mech. 108, 123-142 (2002) · Zbl 1143.76535 · doi:10.1016/S0377-0257(02)00127-1
[11] Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal 29, 1-17 (1998) · Zbl 0915.35120 · doi:10.1137/S0036141096303359
[12] Larson, R.G.: The Structure and Rheology of Complex Fluids. Oxford University Press, London, 1999
[13] Larson, R., Öttinger, H.C.: The effect of molecular elasticity on out-of-plane orientations in shearing flows of liquid crystalline polymers. Macromolecules 24, 6270-6282 (1991) · doi:10.1021/ma00023a033
[14] Laso, M., Öttinger, H.C.: Calculation of viscoelastic flow using molecular models: the CONNFFESSIT approach. J. Non-Newtonian Fluid Mech. 47, 1-20 (1993) · Zbl 0774.76012 · doi:10.1016/0377-0257(93)80042-A
[15] Onsager, L.: The effects of shape on the interaction of colloidal particles. Ann. N. Y. Acad. Sci 51, 627-659 (1949) · doi:10.1111/j.1749-6632.1949.tb27296.x
[16] Öttinger, H.C.: Stochastic Phenomena in Polymeric Fluids, Tools and Examples for Developing Simulation Algorithms. Springer-Verlag, New York, 1996 · Zbl 0995.60098
[17] Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26, 101-174 (2001) · Zbl 0984.35089 · doi:10.1081/PDE-100002243
[18] Siettos, C., Graham, M.D., Kevrekidis, I.G.: Coarse Brownian dynamics computations for nematic liquid crystals. J. Chem. Phys. 118, 10149-10157 (2003) · doi:10.1063/1.1572456
[19] Suen, J.K.C., Joo, Y.L., Armstrong, R.C.: Molecular orientation effects in viscoelasticity. Annu. Rev. Fluid Mech. 34, 417-444 (2002) · Zbl 1047.76502 · doi:10.1146/annurev.fluid.34.083101.134818
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