Barrett, John W. and Suli, Endre (2011) Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers. Technical Report. EDP Sciences, SMAI. (Submitted)

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Abstract
We construct a Galerkin finite element method for the numerical approximation of weak solutions to a general class of coupled FENEtype finitely extensible nonlinear elastic dumbbell models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible NavierStokes equations in a bounded domain , d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extrastress tensor appearing on the righthand side in the momentum equation. The extrastress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a FokkerPlanck type parabolic equation, a crucial feature of which is the presence of a centreofmass diffusion term. We require no structural assumptions on the drag term in the FokkerPlanck equation; in particular, the drag term need not be corotational. We perform a rigorous passage to the limit as first the spatial discretization parameter, and then the temporal discretization parameter tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled NavierStokesFokkerPlanck system. The passage to the limit is performed under minimal regularity assumptions on the data: a squareintegrable and divergencefree initial velocity datum for the NavierStokes equation and a nonnegative initial probability density function for the FokkerPlanck equation, which has finite relative entropy with respect to the Maxwellian M.
Item Type:  Technical Report (Technical Report) 

Subjects:  O  Z > Partial differential equations O  Z > Statistical mechanics, structure of matter D  G > Fluid mechanics H  N > Numerical analysis 
Research Groups:  Numerical Analysis Group 
ID Code:  1056 
Deposited By:  Lotti Ekert 
Deposited On:  23 Mar 2011 09:38 
Last Modified:  29 May 2015 18:44 
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