Stabilized mixed three-field formulation for a generalized incompressible Oldroyd-B model

Jae Hyuk Kwack, Arif Masud, K. R. Rajagopal

Research output: Contribution to journalArticlepeer-review


This paper presents a generalization of the incompressible Oldroyd-B model based on a thermodynamic framework within which the fluid can be viewed to exist in multiple natural configurations. The response of the fluid is viewed as a combination of an elastic component and a dissipative component. The dissipative component leads to the evolution of the underlying natural configurations, while the response from the natural configuration to the current configuration is considered elastic and therefore non-dissipative. For an incompressible fluid, it is necessary that both the elastic behavior as well as the dissipative behavior is isochoric. This is achieved by ensuring that the determinant of the stretch tensor associated with the elastic response meets the constraint that its determinant is unity. A new stabilized mixed method is developed for the velocity, pressure and the kinematic tensor fields. Analytical models for fine scale fields are derived via the solution of the fine-scale equations facilitated by the Variational Multiscale framework that are then variationally embedded in the coarse-scale variational equations. The resulting method inherits the attributes of the classical SUPG and GLS methods, while a significant new contribution is that the form of the stabilization tensors is explicitly derived. A family of linear and quadratic tetrahedral and hexahedral elements is developed with equal-order interpolations for the various fields. Numerical tests are presented that validate the incompressibility of the elastic stretch tensor, show optimal L2 convergence for the conformation tensor field, and present stable response for high Weissenberg number flows.

Original languageEnglish (US)
Pages (from-to)704-734
Number of pages31
JournalInternational Journal for Numerical Methods in Fluids
Issue number9
StatePublished - Mar 30 2017


  • VMS: variational multiscale
  • biofluidics
  • biomechanics
  • incompressible flow
  • non-Newtonian
  • stabilized method

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Computer Science Applications
  • Applied Mathematics


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