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Existence results for the flow of viscoelastic fluids with a differential constitutive law. (English) Zbl 0729.76006

This paper examines the flow of incompressible viscoelastic fluids which obey a constitutive law of the form \[ \tau +\lambda_ 1({\mathcal D}_ a\tau /{\mathcal D}t)+\beta (\tau,D)=2\eta (D+\lambda_ 2({\mathcal D}D/{\mathcal D}t)), \] where \(\tau\) is the extra stress tensor, D the rate of deformation tensor, \(\beta\) a nonlinear tensor valued function, \(\lambda_ 1\) a relaxation time, \(\lambda_ 2\in [0,\lambda_ 1)\) a retardation time and \({\mathcal D}_ a/{\mathcal D}t\) denotes an objective derivative. Prescribing initial data for the velocity and stress fields u and \(\tau\) together with data for any external body force present and assuming the no-slip boundary condition for u, the unknown functions are u, \(\tau\) and pressure p. To make the problem more tractable the author splits \(\tau\) into its Newtonian and purely elastic parts and introduces non-dimensional variables.
Employing a fixed point argument and an energy inequality, the author first establishes the local existence of a unique regular solution of the resulting system of equations. Global existence of this solution follows provided the prescribed data are sufficiently small. Further, the problem linearized about the zero solution is found to admit a unique global solution, which is uniformly bounded in time for any initial data, and a stability result for small solutions is obtained. Finally it is asserted that the results hold for similar models with several relaxation times.
This paper is likely to appeal more to the applied analyst than the practical rheologist.

MSC:

76A10 Viscoelastic fluids
76E99 Hydrodynamic stability
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