DOI 10.1007/s10958-018-3878-x Journal of Mathematical Sciences, Vol. 232, No. 3, July, 2018 APPROXIMATION OF TWO-DIMENSIONAL VISCOELASTIC FLOWS OF GENERAL FORM N. A. Karazeeva Steklov Mathematical Institute RAS 27, nab. Fontanka, St. Petersburg 191023, Russia email@example.com UDC 517.9 We consider the initial-boundary value problem for approximations of the system of integro-diﬀerential equations generalizing the equations of motion for viscoelastic ﬂu- ids. We prove the existence and convergence theorems and give some examples of non- Newtonian ﬂuids described by the model under consideration. Bibliography:8 titles. Dedicated to the memory of Vasilii Vasil’evich Zhikov 1 Introduction We consider -approximations of the system of two-dimensional equations of motion of a con- tinuous medium v +(v ·∇)v − div σ +grad p = f, (1.1) div v =0, where σ is the deviator of the stress tensor, and the rheological equation σ =2νD + KD, (1.2) where D is the strain rate tensor and K is a nonlinear operator. Such systems include systems of equations governing the motion of non-Newtonian ﬂuids. Approximations of the form div v + p =0 (1.3) were considered for the Navier–Stokes equations in . The equations of linear viscoelastic ﬂuids of Oldroyd type and the corresponding -approximations were studied
Journal of Mathematical Sciences – Springer Journals
Published: Jun 2, 2018
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