**Recent Advances in Complex Fluids Modeling**

Luís L. Ferrás, Maria L. Morgado, Magda Rebelo, Rosalía T. Leiva, António Castelo, Gareth H. McKinley and Alexandre M. Afonso

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.82689

### Abstract

In this chapter, we present a brief description of existing viscoelastic models, starting with the classical differential and integral models, and then focusing our attention on new models that take advantage of the enhanced properties of the Mittag-Leffler function (a generalization of the exponential function). The generalized models considered in this work are the fractional Kaye-Bernstein, Kearsley, Zapas (K-BKZ) integral model and the differential generalized exponential Phan-Thien and Tanner (PTT) model recently proposed by our research group. The integral model makes use of the relaxation function obtained from a step-strain applied to the fractional Maxwell model, and the differential model generalizes the familiar exponential Phan-Thien and Tanner constitutive equation by substituting the exponential function of the trace of the stress tensor by the Mittag-Leffler function. Since the differential model is based on local operators, it reduces the computational time needed to predict the flow behavior, and, it also allows a simpler description of complex fluids. Therefore, we explore the rheometric properties of this model and its ability (or limitations) in describing complex flows.

Keywords: Mittag-Leffler, viscoelastic, memory function, fractional calculus, rheology
