×

A comparison of macroscopic models describing the collective response of sedimenting rod-like particles in shear flows. (English) Zbl 1376.76071

Summary: We consider a kinetic model, which describes the sedimentation of rod-like particles in dilute suspensions under the influence of gravity, presented in [the authors, Multiscale Model. Simul. 15, No. 1, 500–536 (2017; Zbl 1362.76059)]. Here we restrict our considerations to shear flow and consider a simplified situation, where the particle orientation is restricted to the plane spanned by the direction of shear and the direction of gravity. For this simplified kinetic model we carry out a linear stability analysis and we derive two different nonlinear macroscopic models which describe the formation of clusters of higher particle density. One of these macroscopic models is based on a diffusive scaling, the other one is based on a so-called quasi-dynamic approximation. Numerical computations, which compare the predictions of the macroscopic models with the kinetic model, complete our presentation.

MSC:

76T20 Suspensions

Citations:

Zbl 1362.76059
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Guazzelli, É.; Hinch, J., Fluctuations and instability in sedimentation, Annu. Rev. Fluid Mech., 42, 97-116 (2011) · Zbl 1299.76285
[2] Herzhaft, B.; Guazzelli, É., Experimental study of the sedimentation of a dilute fiber suspension, Phys. Rev. Lett., 77, 290-293 (1996)
[3] Herzhaft, B.; Guazzelli, É., Experimental study of the sedimentation of dilute and semi-dilute suspensions of fibers, J. Fluid Mech., 384, 133-158 (1999) · Zbl 0939.76504
[4] Metzger, B.; Butlerz, J. E.; Guazzelli, É., Experimental investigation of the instability of a sedimenting suspension of fibers, J. Fluid Mech., 575, 307-332 (2007) · Zbl 1108.76316
[5] Koch, D. L.; Shaqfeh, E. S.G., The instability of a dispersion of sedimenting spheroids, J. Fluid Mech., 209, 521-542 (1989) · Zbl 0681.76047
[7] Chalub, F. A.C. C.; Markowich, P. A.; Perthame, B.; Schmeiser, C., Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142, 123-141 (2004) · Zbl 1052.92005
[8] Othmer, H. G.; Hillen, T., The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math., 62, 1222-1250 (2002) · Zbl 1103.35098
[9] Goudon, Th., Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: Analysis of the two-dimensional case, Math. Models Methods Appl. Sci., 15, 737-752 (2005) · Zbl 1074.82021
[10] Doi, M.; Edwards, S. F., The Theory of Polymer Dynamics (1986), Oxford University Press
[11] Otto, F.; Tzavaras, A., Continuity of velocity gradients in suspensions of rod-like molecules, Comm. Math. Phys., 277, 3, 729-758 (2008) · Zbl 1158.76051
[12] Helzel, C.; Otto, F., Multiscale simulations for suspensions of rod-like molecules, J. Comput. Phys., 216, 1, 52-75 (2006) · Zbl 1107.82040
[13] Poupaud, F.; Soler, J., Parabolic limit and stability of the Vlasov-Poisson-Fokker-Planck system, Math. Models Methods Appl. Sci., 10, 10271045 (2000) · Zbl 1018.76048
[14] Bird, R. B.; Curtiss, Ch. F.; Armstrong, R. C.; Hassager, O., Dynamics of Polymeric Liquids, Vol. 2, Kinetic Theory (1987), Wiley Interscience
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.