Chin. Ann. Math. Ser. B 39(2), 2018, 183–200 DOI: 10.1007/s11401-018-1059-3 Chinese Annals of Mathematics, Series B c  The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2018 Bingham Flows in Periodic Domains of Infinite Length Patrizia DONATO 1 Sorin MARDARE 1 Bogdan VERNESCU 2 (Dedicated to Philippe G. Ciarlet on the occasion of his 80th birthday) Abstract The Bingham fluid model has been successfully used in modeling a large class of non-Newtonian fluids. In this paper, the authors extend to the case of Bingham fluids the results previously obtained by Chipot and Mardare, who studied the asymptotics of the Stokes flow in a cylindrical domain that becomes unbounded in one direction, and prove the convergence of the solution to the Bingham problem in a finite periodic domain, to the solution of the Bingham problem in the infinite periodic domain, as the length of the finite domain goes to infinity. As a consequence of this convergence, the existence of a solution to a Bingham problem in the infinite periodic domain is obtained, and the uniqueness of the velocity field for this problem is also shown. Finally, they show that the error in approximating the velocity field in the infinite domain with the velocity in a periodic domain of length 2ℓ has a polynomial decay in ℓ, unlike in the Stokes case (see [Chipot, M. and Mardare, S., Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction, Journal de Math´ematiques Pures et Appliqu´ees, 90(2), 2008, 133–159]) where it has an exponential decay. This is in itself an important result for the numerical simulations of non-Newtonian flows in long tubes. Keywords Bingham fluids, Variational inequalities 2000 MR Subject Classification 35B40, 35B27, 35J87, 76D07, 74C10 1 Introduction The Bingham fluid model has been proposed by Bingham [2] in 1916 to model plastic flows. Bingham fluids behave at high stresses like a Newtonian fluid, however at low stresses they do not deform. In other words, the shear rate depends linearly on the shear stress only past a certain value of the shear stress, called yield stress; below the yield stress there is no shear. More precisely, the stress tensor σ is given by σ = −pI + τ , with τ =2μD(u)+ √ 2g D(u) |D(u)| , if |D(u)|=0, |τ |≤ √ 2g, if |D(u)| =0, where u is the velocity, g is the yield stress and the shear rate tensor D is defined by D ij = 1 2 ( ∂ui ∂xj + ∂uj ∂xi ) . Thus the flow region of a Bingham fluid can have zones under stress but with no Manuscript received December 14, 2016. Revised May 11, 2017. 1 Universit´e de Rouen, Laboratoire de Math´ematiques Rapha¨el Salem, UMR CNRS 6085, Avenue de l’Universit´e, BP 12, 76801 Saint- ´ Etienne-du-Rouvray, France. E-mail: Patrizia.Donato@univ-rouen.fr Sorin.Mardare@univ-rouen.fr 2 Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Rd., Worcester, MA01609, USA. E-mail: vernescu@wpi.edu