On planar flows of viscoelastic fluids of Giesekus type

Viscoelastic rate-type fluid models of higher order are used to describe the behaviour of materials with complex microstructure: geomaterials like asphalt, biomaterials such as vitreous in the eye, synthetic rubbers such as styrene butadiene rubber. A standard model that belongs to the category of viscoelastic rate-type fluid models of the second order is the model due to Burgers, which can be viewed as a mixture of two Oldroyd-B models of the first order. This viewpoint allows one to develop the whole hierarchy of generalized models of the Burgers type. We study one such generalization that can be viewed as a combination (mixture) of two Giesekus viscoelastic models having in general two different relaxation mechanisms. We prove, in two spatial dimensions, long-time and large-data existence of weak solutions to the considered generalization of the Burgers model subject to no-slip boundary condition. We also provide, as a particular case, a complete proof of global-in-time existence of weak solutions to the Giesekus model in two spatial dimensions.

Automated summary

This article discusses the use of higher-order viscoelastic rate-type fluid models to describe the behavior of materials with complex microstructures. The Burgers model, which is a second-order viscoelastic rate-type fluid model, is introduced as a standard model that can be viewed as a mixture of two first-order Oldroyd-B models. The article focuses on studying a generalization of the Burgers model that combines two Giesekus viscoelastic models with different relaxation mechanisms. The existence of weak solutions to this generalized model, subject to no-slip boundary conditions, is proven in two spatial dimensions. A complete proof of global-in-time existence of weak solutions to the Giesekus model in two spatial dimensions is also provided as a specific case.