A Three-Dimensional Velocity Field Related to a Generalized Third-Grade Fluid Model

Author:

Carapau Fernando12ORCID,Correia Paulo12ORCID,Rodrigues Gracino23

Affiliation:

1. Department of Mathematics, University of Évora, 7000-671 Évora, Portugal

2. Center for Research in Mathematics and Applications (CIMA), University of Évora, 7000-671 Évora, Portugal

3. Coordination of Degree in Mathematics, Campus Maceió, Federal Institute of Education, Science and Technology of Alagoas (IFAL), R. Mizael Domingues, 530-Centro, Maceió 57020-600, AL, Brazil

Abstract

In this work, we propose a new three-dimensional constitutive equation related to a third-grade fluid. This proposal is based on experimental work for which the viscosity term and the terms related to viscoelasticity may depend on the shear rate, in accordance with a power-law type model. The numerical implementation of this fluid model is rather demanding in terms of computational calculation and, in this sense, we use the Cosserat theory related to fluid dynamics, which makes the transition from the three-dimensional fluid model to a one-dimensional fluid model for a specific geometry under study which, in this case, is a straight tube with constant circular cross-section. Based on this approximation theory, the one-dimensional fluid model is solved by assuming an ordinary differential equation involving: an unsteady mean pressure gradient; an unsteady volume flow rate; the Womersley number; and the viscosity and viscoelasticity parameters. Consequently, for specific data, and using the Runge–Kutta method, we can obtain the solution for the unsteady volume flow rate and we can present simulations to the three-dimensional velocity field.

Funder

Fundação para a Ciência e Tecnologia, FCT

Publisher

MDPI AG

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