Mathematical Results for Some Models of Viscoelastic Fluids

We give a short survey of several results concerning mathematical analysis of models describing flows of incompressible rate type fluids. After recalling the system of governing equations, we first present the result of P.-L. Lions and N. Masmoudi [4] about global existence of solution to a special case of Oldroyd model. Then we present results of F.-H. Lin, C. Liu and P. Zhang [3] and Y. Chen and P. Zhang [1] about local existence of solutions and global existence of small solutions of another rate type fluid model. Introduction Biological fluids (blood, synovial fluid), asphalt and some geomaterials (sand, clay) are materials with complicated structure consisting of many components. None the less we can still model them as a continuum if we are interested in the behavior of this material as a whole body. Since these materials consist of many components they have some non-newtonian characteristics such as shear thinning or thickening, stress relaxation and so on. We can describe these properties using models of viscoleastic fluids of the rate type. There are several ways how to achieve suitable models of viscoelastic fluids. The classical way is described for example in the book of Renardy, Hrusa and Nohel [7]. There is also more general way of achieving models of viscoelastic (and other non-newtonian) fluids which is due to Rajagopal and Srinivasa [6]. Both methods differ in the way of achieving constitutive equations for the stress tensor. Standard approach which we will follow imposes directly a constitutive equation on the stress tensor. The procedure of Rajagopal and Srinivasa is different, the constitutive equation for the stress tensor comes out as a consequence of a general principle, namely the maximization of the so-called rate of dissipation with respect to the symmetric part of the velocity gradient. However, we will not go into details in deriving constitutive equations as it is not the main purpose of this contribution. In the whole text we will denote scalars by small letters (a, b, ν, p, ...), vectors by bold small letters (v, w, x, ...) and tensors by bold large letters (T, D, W, ...). The motion of a fluid is described by balance laws and by constitutive equations. The standard continuum mechanics yields the following equations. We will deal only with incompressible materials and we will consider only isothermal flows. Therefore the continuum equation is in the form of divv = 0. The balance of linear momentum yields ∂v ∂t + (v · ∇)v = divT+ f , where T is the Cauchy stress tensor and f is the external body force. For simplicity we will set f = 0 from now on. Finally, the balance of angular momentum yields the symmetry of the stress tensor T. The main question now is to derive some more equations for the stress tensor T which complete our system of equations. This is done by constitutive relations. As mentioned before, we will consider only viscoelastic fluids of the rate type. The Cauchy stress tensor T in this models is related to the velocity gradient ∇v by a system of differential equations. Constitutive equations have to satisfy some physically reasonable constraints such WDS'07 Proceedings of Contributed Papers, Part III, 198–203, 2007. ISBN 978-80-7378-025-8 © MATFYZPRESS