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The Bingham flow in periodic domains Mardare, Sorin


We consider a Bingham flow in a domain which is periodic in one direction and we are interested in the asymptotic behaviour of the solution to the stationary Bingham problem as the length 2l of the domain (in the periodic direction) goes to infinity. In order to do that we follow the methods used in [CM], where the same study has been done for the Stokes system. However, the techniques in [CM] need some important adaptations in order to treat the difficulties related to the nonlinearity of the problem. Our main result states that the velocity of the fluid converges strongly in the H1-norm to the solution of a Bingham problem in the infinite periodic domain. Nevertheless, the speed of the convergence is much lower than the one obtained for the (linear) Stokes problem. More specifically, for a Bingham fluid, the rate of convergence is of the order of l^−a with 0 < a < 1/2, while being exponential for the Stokes problem, 2 i.e. of the order of e^−αl for some positive α. This is a joint work with Patrizia Donato and Bogdan Vernescu.

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